Finite-type invariants for graphs and graph reconstructions

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Advances in Mathematics 186 (2004) 181–228

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Finite-type invariants for graphs and graph reconstructions Robin Forman
Department of Mathematics, Rice University, P.O. Box 1892, Houston, TX 77251, USA Received 15 August 2001; accepted 15 August 2003 Communicated by Laszlo Lova´sz

Abstract Many of the fundamental open problems in graph theory have the following general form: How much information does one need to know about a graph G in order to determine G uniquely. In this article we investigate a new approach to this sort of problem motivated by the notion of a finite-type invariant, recently introduced in the study of knots. We introduce the concepts of vertex-finite-type invariants of graphs, and edge-finite-type invariants of graphs, and show that these sets of functions have surprising algebraic properties. The study of these invariants is intimately related with the classical vertex- and edge-reconstruction conjectures, and we demonstrate that the algebraic properties of the finite-type invariants lead immediately to some of the fundamental results in graph reconstruction theory. r 2003 Elsevier Inc. All rights reserved. MSC: 05C60 Keywords: Graph theory; Graph reconstruction; Finite-type invariants; Nash-Williams lemma

0. Introduction Many of the fundamental open problems in graph theory have the following general form: How much information does one need to know about a graph G before one can determine G uniquely? In this paper we investigate a new approach to this problem based upon the study of graph invariants which satisfy some natural linear difference equations. Let G denote the set of all isomorphism classes of finite (simple)


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graphs. A graph invariant is a function on G (for simplicity we will consider all functions to be real-valued). Let F denote the space of all graph invariants. For any linear subspace F0 CF; we define an F0 -reconstruction of a graph G to be any graph H such that f ðGÞ ¼ f ðHÞ for every f AF0 : Say that G is F0 -reconstructible if every F0 -reconstruction of G is isomorphic to G: The basic question we ask in this paper is: Given a finite graph G; for which spaces F0 CF is G F0 -reconstructible? In other words, how many graph invariants do we need to know before we can completely determine G? In this paper we will be interested in two rather special families of subspaces of F: These spaces play the role of the polynomials on G; and we refer to these functions as being of edge- and vertex-finite type. That is, we will define two nested sequences of subspaces of F; f0g ¼ E1 CE0 CE1 CE2 C?CF; and f0g ¼ V1 CV0 CV1 CV2 C?CF; where Ek denotes the space of edge-finite-type invariants of order k; and Vn denotes the space of vertex-finite-type invariants of order n (the precise definitions will appear shortly). One theme of this paper is that properties of these spaces lead immediately to reconstruction results. For example, define an involution on F; sending f -f c ; where, for any graph G; f c ðGÞ ¼ f ðG c Þ; where G c is the complement of G: In Section 6 we prove the following theorem. Theorem 6.1. If f AEk ; then so is f c : We then show how this immediately implies Lova´sz’s Theorem [12] on edgereconstructibility (see Corollary 6.5). We can define another map on F as follows. Let X be a graph and define a map f -f X by setting, for any graph G with the same number of vertices as X ; f X ðGÞ ¼ f ðG-X Þ: This does not quite make sense if X and G are abstract graphs. To define G-X we must make some choices (e.g. a choice of an identification of the vertices of X and G). To get a graph invariant, the resulting function must then be averaged over all choices. All of this is explained in Section 7, where we prove the following result. Theorem 7.3. If f AEk ; then so is f X :

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This yields Nash-Williams’ Lemma on edge-reconstructibility [16, see Theorem 7.6]. A second theme in this paper is that the edge- and vertex-finite-type invariants are defined to be the solutions of some simple linear equations, but in fact have a surprising amount of additional structure, of which Theorems 6.1 and 7.3 are two examples. In these examples, the appropriate maps on F are constructed by first defining a map on G and then considering the dual map on F: In fact, instead of working with F as we do, one could instead work directly with L; the vector space of finite linear combinations of graphs. Note that F can be thought of as the space of linear functions on L: The idea of introducing L and then using a ‘‘linear algebra approach’’ to reconstruction problems first appeared in [18,19] (see also [11]). Before going further, we will pause to describe the motivation for this work. In [22], Vassiliev introduced an important new notion into the world of knot theory, the idea of a knot invariant of finite-type, or Vassiliev invariant (this idea seems to have been discovered independently by Gusarov [8]). A knot invariant is a function on the space of knots which depends only on the equivalence class of a knot. A knot invariant is said to be of finite-type of order k if it satisfies certain linear difference equations relating a knot to the knots that result if some crossings are switched. (This is not the form in which the definition first appeared in [22]. Instead we follow the more combinatorial approach presented in [2,3].) The fundamental question in the subject is whether knot invariants of finite-type are sufficiently numerous to distinguish knots. That is, given two inequivalent knots, is there an invariant of finite-type which assigns distinct values to these knots. It is not at all clear how one is to go about tackling this sort of question. The original idea for this paper was to try to gain some understanding of this problem by considering what appeared to be a simpler problem. Instead of knots, we consider abstract graphs. The first step is to define what it means for a graph invariant to be of finite-type. We introduce what seem to be two reasonable definitions. We distinguish these two notions by refering to vertex-finite-type invariants and edge-finite-type invariants. The fundamental question now is whether vertex- or edge-finite-type invariants are sufficient to distinguish inequivalent graphs. In fact, it is easy to see that they are, in a rather trivial way (see Theorem 4.2). A more interesting question is whether vertex- or edge-finite-type invariants are still sufficient to distinguish inequivalent graphs once one has ruled out the trivial method (all of this will be explained shortly). The answer to this question is not so obvious. In fact, as we will show, these questions are equivalent to two of the most notorious open problems in graph theory, namely the vertex- and edge-reconstruction conjectures. Vassiliev’s ideas have shown themselves to be a wonderful source of new insights and questions in the study of links and braids, as well as general three-dimensional manifolds [5,17]. We believe that these ideas may be of similar interest in the study of combinatorial problems. The ideas in this paper can be applied to the general reconstruction problems examined in, for example, [1,13,14], but for the sake of explicitness, we will work entirely in the context of graphs. However, we will concentrate in this paper on the general properties of these invariants, and our work

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will not make any use of the special properties of graphs. In a future paper we hope to show how one may use of the special properties of graphs to make a more detailed analysis of edge- and vertex-finite-type invariants. We believe that these ideas may also be of use for a variety of combinatorial problems in addition to questions of reconstructibility. See, for example, the last paragraph of this introduction. We hope to return to these issues in later work. We will now describe in more detail the contents of this paper. In this introduction we will mention the main results, and, instead of presenting the proofs—they appear in the main body of the paper—we will content ourselves with simple examples. In Section 1 we define, and begin our investigation of, graph invariants of edgefinite-type. More precisely, we define the nested sequence of subspaces of F f0g ¼ E1 CE0 CE1 CE2 C?CF:

ð0:1Þ

If a function lies in Ek ; we say that it is edge-finite-type of order k, and if f is edgefinite-type of order k; for some k; we say that f is edge-finite-type. We let E denote the vector space of all edge-finite-type invariants. Each of the spaces Ek is defined to be the solution space of a certain set of linear equations. Before stating the precise definition of the Ek ; we will present some examples. The space E1 contains only the 0 function. The space E0 consists precisely of the functions f AF such that for any graph G; f ðGÞ depends only on the number of vertices in G: Equivalently, a function f lies in E0 if and only if for every graph G with at least one edge, and every edge e of G; f ðGÞ  f ðG  eÞ ¼ 0;

ð0:2Þ

where G  e denotes the graph which results from removing from G the edge e: In particular, the vertex set of G  e is the same as the vertex set of G: The next subspace in the filtration of F is E1 : A function f lies in E1 if and only if for every graph G with at least two edges, and every pair e1 and e2 of edges of G f ðGÞ  f ðG  e1 Þ  f ðG  e2 Þ þ f ðG  e1  e2 Þ ¼ 0:

ð0:3Þ

Every function in E0 satisfies this equation. In addition, the function E which assigns to every graph G the number of edges in G also lies in E1 : We show in Section 3 that E1 consists precisely of the functions of the form f0 þ f1 E; where f0 and f1 lie in E0 (if f and g are invariants, then fg denotes the invariant which assigns to a graph G the number f ðGÞgðGÞ). Note that we can write (0.3) in the form ð f ðGÞ  f ðG  e1 ÞÞ  ð f ð½G  e2 Þ  f ð½G  e2   e1 ÞÞ ¼ 0

ð0:4Þ

making it clear that we are considering a discrete analog of a derivative of a derivative, i.e. a second derivative.

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We can now state the general definition. A function f lies in Ek if and only if, for every graph G with at least k þ 1 edges, and every set a of k þ 1 edges of G; X ð1Þjbj f ðG  bÞ ¼ 0: ð0:5Þ |DbDa

The left-hand side of these equations are discrete analogs of ðk þ 1Þth partial derivatives of f : It is easy to see from the appropriate generalization of 0.4 that Ek DEkþ1 for each k: This is all done more precisely in Sections 2 and 3. Eq. (0.5) appears implicitly in much of the work in graph reconstructions, namely in the guise of the method of inclusion/exclusion (e.g. when counting subgraphs). In this paper, these equations will form the central point from which all of the theory emerges. In Section 2 we begin our presentation of the basic properties of edge-finite-type invariants. The name ‘‘finite-type’’ refers, in other settings, to the fact that finite-type invariants are completely determined by a finite set of data. This is not quite true for edge-finite-type invariants (although it is true for the vertex-finite-type invariants we will present later), but we do have the following simple result, which follows easily from the definition. Let Gpk denote the set of isomorphism classes of graphs with pk edges. Theorem 2.4. If f is edge-finite-type of order k, then f is completely determined by its restriction to Gpk : More interesting is the following converse. Theorem 2.5. Any function f on Gpk has a unique extension to a function on all of G which is edge-finite-type of order k. In Section 3, we present some examples of edge-finite-type invariants. In particular, we have the following theorem. Theorem 3.1. Let H be a graph with k edges, and let SH denote the function which assigns to each graph G the number of spanning subgraphs of G which are isomorphic to H. Then the function SH is edge-finite-type of order k, but not order k  1: Moreover, we prove the following. Theorem 3.3. The functions SH ; as H ranges over all graphs with pk edges, form a basis of Ek as a vector space. In Section 4 we begin our investigation of reconstruction problems. Theorem 3.1 has the following immediate corollary. Theorem 4.2. Let G be a graph with k edges, then G is Ek -reconstructible.

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The question now is whether a graph G with k edges is Ek1 -reconstructible. We recall a more classical notion of reconstructibility. For any graph G with k edges, the edge deck of G is the unordered set of k unlabeled graphs G  e as e ranges over the edges of G: G is said to be edge-reconstructible if it uniquely determined by its edge deck. This notion was introduced by Harary in [9], where he states the edgereconstruction conjecture. Conjecture. Every graph with at least four edges is edge-reconstructible. More generally, for rpk one can define the notion of the r-edge deck of G to be the  unordered set of kr graphs of the form G  a as a ranges over all sets of r edges of G: We say that G is r-edge-reconstructible if it is uniquely determined by its r-edge-deck. Theorem 3.3 has the following corollary. Theorem 4.6. Let G be a graph with k edges. Then G is Ek1 -reconstructible if and only if it is edge-reconstructible. More generally, G is Ekr -reconstructible if and only if G is r-edge-reconstructible. For any n and k; let Gn denote the set of isomorphism classes of graphs with n vertices, and Gkn the isomorphism classes of graphs with precisely n vertices and k edges. In this paper, we find it useful to introduce the real vector space L of all finite linear combinations of elements of G: Just as in the case of G; we will use superscripts to indicate restrictions on the number of edges, and subscripts to indicate restrictions on the number of vertices. For example, Lkn will denote all linear combinations of elements in Gkn : Every graph invariant f AF can be extended linearly to all of L: For f AF; let Kerð f ÞDL denote the zero set of f : For each k; let KerðkÞCL denote the subspace KerðkÞ ¼

\

Kerð f Þ:

f AEk

Then H is an Ek -reconstruction of G if and only if G  HAKerðkÞ: The edgereconstruction conjecture can now be restated in the following fashion. Conjecture. Let GaH be any graphs with kX4 edges. Then G  HeKerðk  1Þ: This leads up to the following more precise problem: Given n and koK; what is k KerðkÞ-LK n ? In particular, the condition Kerðk  rÞ-Ln ¼ 0 is sufficient, but not necessary, to imply that every graph with k edges and n-vertices is r-edgereconstructible. As we see below, this is the case when kX 12 n2 þ 2r : Much of the classical approach to the subject of edge-reconstructibility is based upon the study of the functions SH : These functions provide an interesting and useful basis for the spaces Ek : One of the novelties of this paper is that we will often work without reference to a basis.

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One of the main techniques in graph reconstructions is, in the language of this paper, to find methods of constructing graph invariants which are edge-finite-type. The following theorem is central to many of the constructions. Theorem 2.7. Let f and g be edge-finite-type invariants of order k1 and k2 resp. Define a new invariant fg by setting, for every graph G, fgðGÞ ¼ f ðGÞgðGÞ: ThenSfg is edge finite-type of order k1 þ k2 : That is, the filtration (0.1) gives the space E ¼ k Ek the structure of a graded algebra. For example, the function Ek which associates to each graph G the kth power of the number of edges in G; lies in Ek : It is interesting to express this function in terms of the basis of functions provided in Theorem 3.3. The function Ek assigns to each graph G the number of ordered k-tuples ðe1 ; e2 ; y; ek Þ of (not necessarily distinct) edges of G: We can partition these k-tuples of edges according to the spanning subgraph H of G formed by the edges fe1 ; e2 ; y; ek g (ignoring multiplicity). The number of k-tuples which get associated to a graph H is equal to Tðk; EðHÞÞ; i.e. the number of ways of mapping f1; 2; y; kg to f1; 2; y; EðHÞg so that every element in f1; 2; y; EðHÞg is in the image. Therefore, X Ek ¼ Tðk; EðHÞÞSH ; H

where the sum is over all graphs H (although only those with pk edges will contribute to the sum). Another important operation is the complementation map. For any graph G; let G c denote the complement of G: That is, G c is the graph with the same vertex set as G; and two vertices are adjacent in Gc if and only if they are not adjacent in G: For any graph invariant f ; define a map f c by setting, for each graph G; f c ðGÞ ¼ f ðG c Þ: The first step in our investigation of this operation occurs in Section 5. For any graph G and any set a of edges of G; let G a denote the linear combination of graphs Ga ¼

X

ð1Þjbj ðG  bÞ:

|DbDa

The main point is that an invariant f is edge-finite-type of order k if and only if for each graph G and each set a of k þ 1 edges of G; f ðG a Þ ¼ 0:

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Generalizing this idea somewhat, for any graph G and any set a1 of edges of G; and any set a2 of edges not in G; let G a1 ;a2 denote the linear combination X X ð1Þjb1 jþjb2 j ðG  b1 þ b2 Þ Ga1 ;a2 ¼ b1 Da1 b2 Da2

so that G a;| is equivalent to G a : Theorem 5.1. Let f be a finite-type invariant of order k. Then for any graph G, any a1 DEðGÞ and any a2 DEðG c Þ; if ja1 j þ ja2 jXk þ 1 then f ðG a1 ;a2 Þ ¼ 0: We note that for any graph G and any collection a of edges of G; ðG a Þc ¼ ðG c Þ|;a : Together with Theorem 5.1 we can deduce the following important result. Theorem 6.1. If f is an edge-finite-type invariant of order k, then so is f c : c Let us present some examples. Consider first the function E: The  function E assigns to a graph G the number of edges in G c : Thus, Ec ðGÞ ¼ VðGÞ  EðGÞ; where 2

V assigns to every graph G the number of vertices in G: More succinctly,   Ec ¼ V2  E: The first term in this summand is in E0 ; and the second is in E1 ; and hence the sum is in E1 : Note that E þ Ec AE0 : Let us consider a more complicated example. Fix n42 and let H denote the graph which has n vertices, and two edges which have one endpoint in common. For any graph G with n-vertices, we can see that ( P 1 dðvÞðdðvÞ  1Þ if G has n vertices; SH ðGÞ ¼ 2 v 0 otherwise; where the above sum is over all of the vertices v of G; and dðvÞ denotes the degree of c the vertex v: We see that if G does not have n vertices then SH ðGÞ ¼ 0: If G does have n-vertices, then 1X ðn  1  dðvÞÞðn  2  dðvÞÞ 2 v X 1 1X ¼ ðn  1Þðn  2Þn  ðn  1Þ dðvÞ þ dðvÞðdðvÞ  1Þ; 2 2 v v

c ðGÞ ¼ SH

so that, restricted to Gn ; 1 c ¼ nðn  1Þðn  2Þ  2ðn  1ÞE þ SH : SH 2

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The first term in this summand is in E0 ; the second is in E1 ; and the third is in E2 ; and hence the sum is in E2 : Note that SH  ðSH Þc AE1 : Theorem 6.1 has the following corollary. Corollary 6.3. A graph G is Ek -reconstructible if and only if Gc is Ek -reconstructible. Combining this result with Corollaries 4.2 and 6.3 yields the following important result. Corollary 6.5. Let G is a graph with n vertices and k edges.  (i) If k4 12 n2 then G is edge-reconstructible.  (ii) If kX 12 n2 þ r for some positive rA 12 Z then G is 2r-edge-reconstructible. The Corollary 6.5(i) is known as Lova´sz’s Theorem [12], and part (ii) first appeared in [7]. As mentioned earlier, Corollary 6.5(ii) is equivalent to the fact that  if G and H are two inequivalent graphs with n vertices and k edges, with kX 12 n2 þ r; then G  HeKerðk  2rÞ: In fact, Corollaries 4.2 and 6.3 imply the stronger result  1 n that if kX 2 2 þ r then Kerðk  2rÞ-Lkn ¼ 0: From the complementation map f -f c we can construct the symmetrization map ps ð f Þ ¼ 12 ð f þ f c Þ; and the antisymmetrization map pa ð f Þ ¼ 12 ð f  f c Þ: These satisfy the usual relations ps ps ¼ ps ; pa pa ¼ pa ; ps pa ¼ pa ps ¼ 0: It follows from Theorem 6.1 that if f is an edge-finite-type invariant of order k; then so are ps ð f Þ and pa ð f Þ: In the examples considered after the statement of Theorem 6.1 we noticed that ps ðEÞAE0 and pa ðSH ÞAE1 : In fact, we have the following general statement. Theorem 6.9. Let f be a an edge-finite-type invariant of order k. If k is odd, then ps ð f Þ is edge-finite-type of order k  1: If k is even, then pa ð f Þ is edge-finite-type of order k  1: Theorem 6.9 leads to the following strengthening of Corollary 6.5(i) (this is actually what Lova´sz proves in [12]). Corollary 6.10. Let G and H be graphs with the same edge-deck. If SG ðG c Þ ¼ SG ðH c Þ ¼ 0; then G ¼ H: From the above comments we see that we can piece together the ingredients to form a differential complex E:

@5

@4

@3

@2

@1

@0

? ! E4 ! E3 ! E2 ! E1 ! E0 ! 0;

where @k ¼ ps if k is odd, and pa if k is even.

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It is natural to consider the homology H
ðEÞ of this complex. However, it is rather easy to see that the homology is trivial. Theorem 6.11. H
ðEÞ ¼ 0: More explicitly, if f AEk satisfies @k ð f Þ ¼ 0; then @kþ1 ðaf EÞ ¼ f for some aAE0 : (The function a is given explicitly in Section 6). (Note that from Theorem 2.7, if f AEk ; then af EAEkþ1 :) A closer look at this complex allows one to give a nice reformulation of the edge-reconstruction problem. Theorem 6.12. Let G and H be graphs with k edges. Then G and H have the same edge deck if and only if @k ðAðGÞSG  AðHÞSH Þ ¼ 0; where AðGÞ denotes the number of automorphisms of the graph G. Note that for any graph H with the same number of vertices as G; AðGÞSG ðHÞ is the number of embeddings of G into H: As this theorem indicates, for many purposes the functions AðGÞSG ; as G ranges over all graphs with k edges, form a more natural basis for Ek than the functions SG : In Section 7 we investigate other ways of constructing edge-finite-type graph invariants. We begin this section by considering, not the set of abstract graphs with n vertices, but rather the set SGðnÞ of spanning graphs of Kn ; the complete graph on n vertices. In particular, we do not identify isomorphic subgraphs, so that n jSGðnÞj ¼ 2ð2Þ : The main idea, implicit in Nash-William’s presentation in [16], is to begin with an edge-finite-type function f on the space of spanning graphs of Kn : That is, it is edge-finite-type in the sense that it satisfies the appropriate difference equations, but it is not necessarily an invariant—f may depend not only on the isomorphism class of the graph but also the embedding in Kn : We can then construct an edge-finite-type invariant f% by setting, for any graph G; % fðGÞ ¼ Averageff ðfðG 0 ÞÞg; where G 0 is any spanning subgraph of Kn which is isomorphic to G; and the average is taken over all automorphisms f of Kn : Note that since the difference equations are linear, this averaging process preserves the property of being edge-finite-type of order k; for each k: The next step is to construct interesting edge-finite-type functions on the space of SGðnÞ: Theorem 7.3. Let X be a spanning subgraph of Kn with k edges, and f an edge-finitetype function of order k on SGðnÞ: Define the function f X on SGðnÞ by setting, for any GASGðnÞ; f X ðGÞ ¼ f ðG-X Þ: Then f X is a finite-type function of order k.

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Now let X be any spanning subgraph of Kn with k edges, and Y a spanning subgraph of X with fewer than k edges. Let f denote the unique edge-finite-type function of order k  1 on SGðnÞ which satisfies f ðY Þ ¼ 1; and f ðZÞ ¼ 0 for each Y aZASGðnÞ with EðZÞok: (Such a function exists by the appropriate analogue to Theorem 2.5.) It is easy to check that f ðY Þ ¼ 1; f ðX Þ ¼ ð1Þkcþ1 ; where c is the number of edges in Y ; and f ðZÞ ¼ 0 for every other subgraph of X : Combining the operations discussed above, we see that f X is an edge-finite-type invariant of order k  1: This is the celebrated theorem of Nash-Williams [16]. Theorem 7.5 (Nash-Williams). Let X and Y be as above, and let G and H be two graphs with k edges. If G and H have the same edge-deck then for any spanning subgraphs G0 and H 0 of Kn ; such that G 0 is isomorphic to G and H 0 is isomorphic to H, jffAAutðKn Þ such that fðG 0 Þ-X ¼ Y gj þ ð1Þkcþ1 jffAAutðKn Þ such that fðG 0 Þ-X ¼ X gj ¼ jffAAutðKn Þ such that fðH 0 Þ-X ¼ Y gj þ ð1Þkcþ1 jffAAutðKn Þ such that fðH 0 Þ-X ¼ X gj: One should see the surveys [4,6,16,20], and the numerous references therein, to see how this result easily yields sufficient conditions for a graph to be edgereconstructible. In particular, the Nash-Williams Theorem immediately implies the following result of Mu¨ller [15]. Theorem (Mu¨ller). Suppose that G is a graph with n vertices and k edges, and k41 þ log2 ðn!Þ; then G is edge-reconstructible. There are other interesting methods of constructing new edge-finite-type invariants from old ones. For example, instead of taking the intersection with a fixed graph, as in Theorem 7.3, one can take the symmetric difference. Fix a spanning subgraph X of Kn : For any spanning subgraph G of Kn ; define a spanning subgraph GDX by setting EðGDX Þ ¼ ½EðGÞ  EðX Þ,½EðX Þ  EðGÞ: We now dualize this operation. For any function f on the set of n-labelled graphs, define a function f DX by setting, for any graph G f Dx ðGÞ ¼ f ðGDX Þ: Theorem 7.7. If f is an edge-finite-type function of order k on the set of SGðnÞ; then so is f DX :

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The function obtained by averaging f DX over all automorphisms of Kn is then an edge-finite-type invariant of order k: This construction yields a wide variety of interesting invariants. However, it is rather difficult to piece these invariants together to reach any clear conclusions. In practice, one often begins with a pair of graphs with the same number of vertices and edges, and one wishes to construct an edge-finite-type invariant which distinguishes them. That is, one is really only interested in the restriction of the invariant to GK n ; for some K and n: However, from the definition, one must know the values of the invariant on Gkn for all kpK in order to determine whether the invariant is edge-finite-type. It would be convenient instead, if given a function on K GK n ; to be able to check directly whether it is the restriction to Gn of a edge-finitetype invariant. We provide such a test in Section 8. The idea is that although the edge-finite-type are defined to satisfy difference equations in the ‘‘vertical’’ directions, i.e. involving graphs with different number of edges, one can show that they also satisfy difference equations in the ‘‘horizontal’ directions, i.e. involving graphs with the same number of edges, and that these horizontal equations are strong enough to characterize edge-finite-type invariants. The work in this section is reminiscent of the ‘‘balance equations’’ of Krasikov and Roditty [11]. First, some new definitions. Let G a graph. A direction at G is a pair ðe; e0 Þ of edges, where eAG and e0 eG: (The phrase direction at G is shorthand for ‘‘direction at G which is tangent to GK ’’, where K ¼ EðGÞ:) Given two directions ðe1 ; e01 Þ; ðe2 ; e02 Þ at G; we say they are orthogonal if e1 ae2 and e01 ae02 : If ðe; e0 Þ is a direction at G; we can, beginning at G; move in the direction ðe; e0 Þ; and the result is the graph G  e þ e0 : We denote this graph by G þ ðe; e0 Þ: More generally, if ðe1 ; e01 Þ; ðe2 ; e02 Þ; y; ðek ; e0k Þ is any set of pairwise orthogonal directions at G; then we can form the graph

Gþ

k X

ðei ; e0i Þ ¼ G  fe1 ; e2 ; y; ek g þ fe01 ; e02 ; y; e0k g:

i¼1

Theorem 8.1. Let G be a graph with K edges. Let D ¼ fðe1 ; e01 Þ; ðe2 ; e02 Þ; y; ðekþ1 ; e0kþ1 Þg denote a set of k þ 1 pairwise orthogonal directions at G. If f is an edge-finite-type invariant of order k, then X

X ð1Þ f G þ ðei ; e0i Þ jIj

IDf1;2;3;y;kþ1g

! ¼ 0:

ð8:6Þ

iAI

Much more interestingly, there is a converse. Theorem 8.2. Let GK n denote the set of graphs with K edges and n vertices. A function f K on Gn is the restriction to GK n of an edge-finite-type invariant of order k if and only if

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0 0 0 for every graph GAGK n ; every set D ¼ fðe1 ; e1 Þ; ðe2 ; e2 Þ; y; ðekþ1 ; ekþ1 Þg of k þ 1 pairwise orthogonal directions at G, Eq. (8.6) holds.

 In particular, if Kpk or, more importantly, n2  Kpk then the condition is K vacuous, and every function on GK n is the restriction to Gn of an edge-finite-type invariant of order k; so that Theorem 8.2 immediately implies Lova´sz’s Theorem. I believe that the study of Eqs. (8.6) and their solutions provides grounds for much further study. In Sections 9–13 we present the corresponding theory of vertex-finite-type invariants and show that the corresponding relationship holds with the vertexreconstruction problem of Kelly [10] and Ulam [21]. The reader should see these sections for definitions and examples. Some of the theory of vertex-finite-type invariants is a bit easier than for edge-finite-type invariants, but with less satisfying results. For example, we have the following analog of Theorem 6.1. Theorem 11.2. A function f is vertex-finite-type of order n if and only if f c is vertexfinite-type of order n. The proof of this theorem is more straightforward than the proof of Theorem 6.1. On the other hand, Theorem 6.1 had as an immediate corollary Lova´sz’s theorem, which is much deeper than the following well-know result, which is the corresponding corollary to Theorem 11.2. Corollary 11.5. (i) A graph G is vertex-reconstructible if and only if Gc is vertexreconstructible. (ii) A graph G is m-vertex-reconstructible if and only if G c is m-vertexreconstructible. After this result, the theories of edge- and vertex-finite-type invariants begin to diverge. There is no obvious analogue of the projection @k from Ek to Ek1 for vertex-finite-type invariants, and, although one can apply the method of NashWilliams to construct vertex-finite-type invariants, the resulting functions are much more difficult to work with. The precise reason is that if G and H are spanning subgraphs of Kn with k edges, then G-H will generally have ok edges, and hence can be well studied via edge-finite-type invariants of order k  1: However, G-H will have n vertices, and hence it is more difficult to investigate these graphs using vertex-finite-type invariants of order n  1: The restrictions to GN of vertex-finitetype invariants satisfy a set of difference equations similar to those in (8.6). I think that there is room for much interesting analysis, and I hope to return to these topics in later work. I believe that the study of finite-type graph invariants can find applications to a variety of questions. For example, perhaps one can use these ideas to gain some insight in Ramsey theory. One of the fundamental problems in Ramsey theory is the study of the function Rðn; nÞ; which is defined to be the smallest integer N such that

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every graph with N vertices either has an induced subgraph isomorphic to Kn ; the complete graph on n vertices, or ½n; the disjoint union of n vertices. Define a function Rn;n : G-Z by setting for any graph G; Rn;n ðGÞ ¼ #finduced subgraphs of G isomorphic to Kn g þ #finduced subgraphs of G isomorphic to ½ng: Then Rðn; nÞ is the smallest integer N such that the restriction of Rn;n to GN is strictly positive. It follows from the work in Sections 9–13 that Rn;n is vertex-finite-type of order n; and hence satisfies a large collection of difference equations on GN : This may be of some interest.

1. Notation and some basic observations In this section we present the basic definitions and notation we will follow for the remainder of this paper. Much of this has already been explained in the introduction. We let G denote the set of isomorphism classes of graphs. We will often restrict attention to graphs with a limited number of vertices or edges. We will use subscripts to indicate restrictions on the number of vertices, and superscripts to indicate restrictions on the number of edges. For example, Gn will denote the set of isomorphism classes of graphs with precisely n vertices, and Gok will denote the set of isomorphism classes of graphs with fewer than k edges. We will sometimes combine the notation, so that, for example, G4k pn will denote the set of all isomorphism classes of graphs which have pn vertices and 4k edges. We hope the meaning of all such notation will be clear to the reader. In what follows, it will often be useful to consider linear combinations of graphs. With that in mind, we let L denote the set of finite linear combinations of isomorphism classes of graphs. In this paper, we will always consider the coefficients to be the real numbers R; although one can just as well work over Q: Working over Z would introduce some minor complications which are interesting, but would lead us astray from our main points. As in the case of G; will use subscripts to indicate restrictions on the number of vertices, and superscripts to indicate restrictions on the number of edges. For example, Ln will denote linear combinations of elements of 4k Gn ; L4k pn will denote linear combinations of elements of Gpn ; etc. A graph invariant is a (real-valued) function on G; and we will let F denote the set of all such functions. We note that any map F AF can be extended linearly to a linear map on L: Conversely, any linear map from L to R can be restricted to G: For this reason, we will not distinguish between maps from G to R and linear maps from L to R: For any graph GAG (we will usually not distinguish between isomorphic graphs), we let V ðGÞ denote the set of vertices of G; and EðGÞ the set of edges. If a is a set of edges of G; we let G  a denote the graph which results from the removing from G

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the edges in a; so that V ðG  aÞ ¼ V ðGÞ: A subgraph H of G is said to be a spanning subgraph if H ¼ G  a for some aDEðGÞ; or, equivalently, if V ðHÞ ¼ V ðGÞ: If x is a set of vertices of G; we let G  x denote the graph which results from removing from G the vertices in x and all edges incident to these vertices. Some linear combinations of graphs will play an important role in what follows, and we introduce them here. For any graph G and any set xDV ðGÞ; we let Gx denote the linear combination X ð1Þjyj ðG  yÞ; Gx ¼ yDx

where, for any set y; jyj denotes the number of elements in y: Similarly, for any graph G set aDEðGÞ; we let G a denote the linear combination X ð1Þjbj ðG  aÞ: ð1:1Þ Ga ¼ bDa

Many of the algebraic properties of the objects we will introduce follow immediately from a simple identity involving these linear combinations. Theorem 1.1. For any graph G and any disjoint sets a1 ; a2 DEðGÞ we have X ð1Þjb2 j ðG  b2 Þa1 : G a1 þa2 ¼ b2 Da2

Proof. The proof follows from the simple calculation X ð1Þjbj ðG  bÞ G a1 þa2 ¼ bDa1 þa2

¼

X

X

ð1Þjb1 jþjb2 j ðG  b1  b2 Þ

b2 Da2 b1 Da1

¼

X b2 Da2

¼

X

ð1Þ

jb2 j

X

! ð1Þ

jb1 j

ð½G  b2   b1 Þ

b1 Da1

ð1Þjb2 j ðG  b2 Þa1 :

&

b2 Da2

The special case in which a2 consists of a single edge is of particular significance, so for future reference we highlight it here. Corollary 1.2. For any graph G; and aCEðGÞ and any eAEðGÞ such that eea we have Gaþe ¼ G a  ðG  eÞa :

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2. Edge-finite-type functions In this section we define, and begin our investigation of, invariants of edge-finitetype. The main new ingredient of this section is the nested sequence f0g ¼ E1 CE0 CE1 CE2 C?CF; of spaces of graph invariants. The functions in Ek are called edge-finite-type of order k and are defined as follows. Definition 2.1. A function f AF is in Ek if and only if, for every graph G with at least k þ 1 edges, and every set aAEðGÞ satisfying jajXk þ 1; we have f ðG a Þ ¼ 0; where G a is defined as in (1.1). If aS function f lies in Ek for some k; then we say that f is edge-finite-type. Let E ¼ k Ek denote the space of all edgefinite-type functions. We first note that the definition of an edge-finite-function can be slightly simplified. Theorem 2.2. Let f AF: Then the following two conditions are equivalent: (i) For every graph G with at least k þ 1 edges, and every subset aDEðGÞ with jajXk þ 1 we have f ðG a Þ ¼ 0: (ii) For every graph G with at least k þ 1 edges, and every subset aDEðGÞ with jaj ¼ k þ 1 we have f ðG a Þ ¼ 0: Proof. It is clear that condition (i) implies condition (ii). Assume that f satisfies condition (ii). We will see that it must also satisfy condition (i). Let G be any graph with at least k þ 1 edges, and a any set of edges with jajXk þ 1: We must see that f ðG a Þ ¼ 0: Let a1 be any subset of a with ja1 j ¼ k þ 1 and let a2 ¼ a  a1 : From Theorem 1.1 we see that ! X jb j a f ðG a Þ ¼ f ðG a1 þa2 Þ ¼ f ð1Þ 2 ðG  b2 Þ 1 ¼

X

b2 Da2

ð1Þ

jb2 j

f ððG  b2 Þa1 Þ ¼ 0:

&

b2 Da2

We think of Gk as being the horizontal slices of G; with Gk1 lieing below Gk2 if k1 ok2 : Loosely speaking, a function f is in Ek if all ðk þ 1Þth partial derivatives in

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the downward directions vanish. This is equivalent to requiring that all kth partial derivatives in the downward directions be constant in the downward directions, and hence constant on all of Gn : That is, for any f AEk ; any graph G; any set aAEðGÞ with jaj ¼ k; and any eAEðGÞ  a we have that f ðG a Þ  f ððG  eÞa Þ ¼ f ðG a  ðG  eÞa Þ ¼ f ðG a,feg Þ: Thus, f is in Ek if and only if for every G; a and e as above f ðG a Þ ¼ f ððG  eÞa Þ: We highlight this fact for future reference. Theorem 2.3. Let f AF: Then f AEk if and only if one of the following equivalent conditions holds: (i) For every graph G; and every spanning subgraph H of G with k edges, f ðG EðHÞ Þ ¼ f ðH EðHÞ Þ: (ii) For every pair of graphs G1 and G2 ; with spanning k-edge subgraphs H1 and H2 ; respectively, if H1 is isomorphic to H2 ; then EðH1 Þ

f ðG1

EðH2 Þ

Þ ¼ f ðG2

Þ:

In this paper, we will state most of our results in terms of F: However, many of these results have a dual formulation in terms of L: For example, one can restate Definition 2.1 and Theorem 2.2 as follows. We consider the nested family of subspaces L ¼ Lð1Þ+Lð0Þ+Lð1Þ+?; where LðkÞ denotes the span of the set of elements of the form G a where G is any graph with at least k þ 1 edges, and aDEðGÞ satisfies jajXk þ 1: Let L
ðkÞ denote the span of the Ga where G has at least k þ 1 edges and jaj ¼ k þ 1: Definition 2.1 states that f AEk 3 f ðcÞ ¼ 0

for all cALðkÞ:

Theorem 2.2 states that LðkÞ ¼ L
ðkÞ: The term finite-type, when applied in the original context of knot invariants, referred to the fact that a finite-type invariant is completely determined by a finite

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amount of information. For edge-finite-type functions we do not quite have such a finiteness statement. However, we can make the following statement. Theorem 2.4. If f is an edge-finite-function of order k; then f is completely determined by its restriction to Gpk : Proof. If f is a function which is edge-finite-type   of order k and G is a graph which EðGÞ has at least k þ 1 edges, then for any aA kþ1 we have f ðGÞ ¼ f ðGÞ  f ðG a Þ ¼ f ðG  G a Þ: Note that G  Ga is a linear combination of graphs with fewer than jEðGÞj vertices. Repeating this process, we can eventually write f ðGÞ completely in terms of f ðG 0 Þ for subgraphs G 0 with fewer than k þ 1 edges. & One of the main results of this section is the converse to Theorem 2.4. Theorem 2.4 states that any f AEk is completely determined by its restriction to Gpk : In fact, any function on Gpk is the restriction of an edge-function-type function of order k: Theorem 2.5. For any k and any function f : Gpk -R there is a unique function F : G-R which is edge-finite-type of order k and which satisfies F ðGÞ ¼ f ðGÞ for each GAGpk : Proof. The uniqueness of F follows from Theorem 2.3, so we concern ourselves here only with the existence of F : We will establish existence by explicitly constructing a function which satisfies the desired properties. This construction will be done inductively on the GK ’s. We will show that for any KXk and any FK : GpK -R; there is an extension of FK to a function FKþ1 : GpðKþ1Þ -R; which is also edge-finite-type of order k:

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This is sufficient to complete the theorem since, assuming the above inductive step, we can simply let F be the limit of the FK ’s. That is, given any GAG; the values FK ðGÞ are constant for K large enough. Set F ðGÞ to be the value of FK ðGÞ for large K: Then F satisfies the conclusion of the theorem. To extend FK : GpK -R; to a function FKþ1 on GpðKþ1Þ it is sufficient to define a value for FKþ1 ðGÞ for each graph G with K þ 1 edges. Let a be a set of k þ 1 edges of G: Then G  G a ALpK so FK ðG  G a Þ is defined (where we have extended FK linearly to LpK ). We set FKþ1 ðGÞ ¼ FK ðG  G a Þ:

ð2:1Þ

It is the content of the following lemma that this value is independent of the choice of a: Assuming this for the moment, we finish the proof of the theorem. Formula (2.1) defines FKþ1 on GKþ1 : For GAGpK set FKþ1 ðGÞ ¼ FK ðGÞ; and extend FKþ1 linearly to all of LpKþ1 : Then FKþ1 is clearly an extension of FK ; so all that remains is to show thatFKþ1is edge-finite-type of order k: Let G be a graph with pðK þ 1Þ vertices, and aA

EðGÞ k

: We need to see that FKþ1 ðGa Þ ¼ 0:

If G has fewer than K þ 1 edges, then G a ALpK so FKþ1 ðG a Þ ¼ FK ðG a Þ ¼ 0 by the inductive hypothesis. If G has precisely K þ 1 edges, then FKþ1 ðG a Þ ¼ FKþ1 ðG a  GÞ þ FKþ1 ðGÞ ¼ FKþ1 ðG  Ga Þ þ FKþ1 ðGÞ ¼  FK ðG  G a Þ þ FKþ1 ðGÞ ¼ 0 from (2.1).

&

We now prove the lemma referred to in the above proof. Lemma 2.6. Given KXk; suppose FK : GpK -R   is edge-finite-type of order k: For any GAGKþ1 and a1 ; a2 A EðGÞ ; kþ1 FK ðG  G a1 Þ ¼ FK ðG  G a2 Þ: Proof. It is sufficient to consider the case a2 ¼ a1  e1 þ e2 for some edges e1 Aa1 and e2 ea1 : Let a ¼ a1  e1 ¼ a2  e2 : Then by repeated application of Corollary 1.2 and

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the linearity of FK we see that FK ðG  G a1 Þ  FK ðG  Ga2 Þ ¼ FK ðG a2  G a1 Þ ¼ FK ð½G a2 þe1 þ ðG  e1 Þa2   ½Ga1 þe2 þ ðG  e2 Þa1 Þ ¼ FK ððG  e1 Þa2  ðG  e2 Þa1 Þ ¼ FK ððG  e1 Þa2 Þ  FK ððG  e2 Þa1 Þ ¼ 0  0 ¼ 0:

&

It is a fundamental question in knot theory whether all knot invariants can be approximated by knot invariants of finite-type. In our setting we have such an approximation property. Say a sequence of fi AF has a limit f AF if for every graph G there is an N such that for iXN; fi ðGÞ ¼ f ðGÞ: Corollary 2.7. Every f AF is a limit of edge-finite-type functions. Proof. Given f AF; for k ¼ 1; 2; 3; y; let fk AF denote the unique edge-finite-type function of order k which satisfies fk ðGÞ ¼ f ðGÞ for each GAGpk : It is then clear that f is the limit of the fk ’s. & Until now, we have been presenting the vector space structure of the spaces Ek : We end this section by showing that filtration (2.1) gives E the structure of a graded algebra. Given two functions f1 ; f2 AF; we define their product, by setting for every graph G; ð f1 f2 ÞðGÞ ¼ f1 ðGÞf2 ðGÞ: We can then extend f1 f2 linearly to a function on all of L: Note that it is not true that ð f1 f2 ÞðcÞ ¼ f1 ðcÞ f2 ðcÞ for every cAL: For any k1 and k2 ; let Ek1  Ek2 denote the set of all functions f ¼ f1 f2 where f1 AEk1 and f2 AEk2 : Theorem 2.8. Ek1  Ek2 DEk1 þk2 : This theorem belongs in this section, since it can be proved in a straightforward manner from the definition. However, the proof is a complicated, and uninsightful, calculation. Instead, we will give an easier proof in Section 3 once we have presented a useful basis for the Ek ’s. Theorem 2.8 can be seen as a corollary of the following result which we will also prove in Section 3. Recall from Theorem 2.3 that an invariant f is edge-finite-type of order k if and only if all kth partial derivatives in the downward directions are constant. Therefore, to prove that fgAEk1 þk2 whenever f AEk1 and gAEk2 ; it is sufficient to calculate the ðk1 þ k2 Þth partial derivatives of fg in the downward directions. Theorem 2.9. Let f and g be edge-finite-type invariants of order k1 and k2 resp. Let G be a graph with at least k1 þ k2 edges, and aDEðGÞ be a set of k1 þ k2 edges of G:

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Then fgðG a Þ ¼

X

f ðG b1 ÞgðG ab1 Þ:

ð2:2Þ

b1 Ca jb1 j¼k1

We note, before going further, that it is often convenient, when discussing edgefinite-type invariants, to fix a number n; and then to consider only graphs with n vertices. The point is that if G has n vertices, then for any aCEðGÞ; Ga is a linear combination of graphs with n vertices. Thus, one can define a function f : Gn -R to be edge-finite-type   of order k if for every graph G with n vertices and Xk edges, and every aA

EðGÞ k

f ðGa Þ ¼ 0: Equivalently, a function f : Gn -R is edge-finite-type of

order k; if F : G-R is edge-finite-type of order k in the sense of Definition 2.1, where F is defined by setting F ðGÞ ¼ f ðGÞ for any graph with n vertices, and 0 otherwise.

3. Examples of edge-finite-type functions In this section we present the basic examples of edge-finite-type functions. Other examples will be constructed in Section 7. Let H be a graph with n vertices and k edges, and let SH : Gn -Z denote the function which assigns to every graph G with n vertices, the number of subgraphs of G which are isomorphic to H: Set SH ðGÞ ¼ 0 if VðGÞan: Equivalently, we define SH ðGÞ to be the number of spanning subgraphs of G which are isomorphic to H: Theorem 3.1. If H has k edges, then SH : G-Z is edge-finite-type of order k; but not order k  1: Proof. It is easy to see that SH is not edge-finite-type of order k  1: Namely, jEðHÞj ¼ k; and SH ðH EðHÞ Þ ¼ 1a0: Now we will see that SH is edge-finite-type of order k: Let G be a graph with at  EðGÞ least k þ 1 edges, and let aA kþ1 : We must see that SH ðG a Þ ¼ 0:

ð3:1Þ

If G has n0 an vertices, then SH ðG  bÞ ¼ 0 for every set b of edges of G; so (3.1) is clearly satisfied. Assume that G has n vertices. Suppose that SH ðGÞ ¼ m; and let H1 ; H2 ; y; Hm denote the m distinct subgraphs of G which are isomorphic to H: Let

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Ei CEðGÞ denote the edges of Hi : For each i ¼ 1; 2; y; m; define a function Si on the spanning subgraphs of G as follows. For any spanning subgraph G 0 of G; let Si ðG 0 Þ ¼ 1 if and only if Ei DEðG0 Þ and 0 otherwise. Then, for any subgraph of G; SH ðG 0 Þ ¼

m X

Si ðG 0 Þ:

i¼1

Therefore, in order to prove (3.1), it is sufficient to prove that for each i; Si ðG a Þ ¼ 0: Since we will need to apply this result again later in the paper, we state it separately as a lemma. Lemma 3.2. Let G be a graph with Xk þ 1 edges. Let E 0 CEðGÞ be a set of pk edges. Define a function S on the spanning subgraphs of G by setting, for any spanning subgraph G 0 of G; SðG 0 Þ ¼ 1 if E 0 DEðG 0 Þ; and 0 otherwise. If aCEðGÞ satisfies jajXk þ 1; then SðGa Þ ¼ 0: Proof. Let a1 ¼ a  E 0 : Note that a1 a|: Then X SðG a Þ ¼ ð1Þjbj sðG  bÞ: bDa

Since SðG  bÞ ¼ 1 if bDa1 ; and 0 otherwise, we have that X SðG a Þ ¼ ð1Þjbj ¼ 0: & bDa1

Let H be a graph with n vertices and pk edges. By Theorem 2.4, there is a unique function fHk which is edge-finite-type of order k; and which satisfies fHk ðH 0 Þ ¼ 1 if H ¼ H 0 and 0 otherwise. These functions fkH provide a natural basis for Ek : If H has exactly k edges, then SH satisfies these properties, so we must have fHk ¼ SH : If H has P fewer than k edges, then SH ¼ fHk þ H 0 SH ðH 0 ÞfHk 0 ; where the sum is over all graphs H 0 with n vertices and jEðHÞjojEðH 0 Þjpk: In this manner, one can deduce the following theorem. Theorem 3.3. For each k; the set of functions SH ; jEðHÞjpk; forms a basis of Ek (as a vector space). We can now use this basis to prove Theorems 2.8 and 2.9. Proof of Theorem 2.8. To prove that Ek1  Ek2 CEk1 þk2 it is sufficient, by Theorem 3.3, to show that if H1 is a graph with pk1 edges, and H2 is graph with pk2 edges, then SH1 SH2 AEk1 þk2 : If H1 and H2 do not have the same number of vertices, then SH1 SH2 ¼ 0; so we now suppose that H1 and H2 have n vertices. For any graph H with n vertices, let cðH; H1 ; H2 Þ denote the number of coverings of H by H1 and H2 ; i.e. the number of pairs ðG1 ; G2 Þ of spanning subgraphs of H such that G1 is

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isomorphic to H1 ; G2 is isomorphic to H2 ; and EðHÞ ¼ EðG1 Þ,EðG2 Þ: The function SH1 SH2 assigns to any graph G the number of pairs ðG1 ; G2 Þ of spanning subgraphs with G1 isomorphic to H1 and G2 isomorphic to H2 : Partitioning these pairs according to the spanning subgraph H with EðHÞ ¼ EðG1 Þ,EðG2 Þ we see that X SH1 SH2 ðGÞ ¼ cðH; H1 ; H2 ÞSH ðGÞ; H

where the sum is over all graphs H with n vertices, or simply X SH1 SH2 ¼ cðH; H1 ; H2 ÞSH :

ð3:2Þ

H

Since cðH; H1 ; H2 Þ ¼ 0 if H has more than k1 þ k2 edges, we see from Theorem 3.1 that SH1 SH2 AEk1 þk2 : & Proof of Theorem 2.9. Since both sides of the formula (2.2) are linear in both f and g; it is sufficient to check the formula for f and g elements of the basis provided by Theorem 3.3. Let H1 and H2 be graphs with n vertices and pk1 and pk2 edges, resp. We must see that for any graph G with n vertices and at least k1 þ k2 edges, and any subset aCEðGÞ consisting of k1 þ k2 edges, X SH1 SH2 ðGa Þ ¼ SH1 ðG b1 ÞSH2 ðG ab1 Þ: ð3:3Þ b1 Ca jb1 j¼k1

If H1 has ok1 edges, or H2 has ok2 edges, then both sides of this formula are 0, so it is sufficient to check this formula in the case that H1 and H2 have precisely k1 and k2 edges, respectively. We now assume this. We will prove (3.3) by showing that both sides are equal to the number of pairs of spanning subgraphs ðG1 ; G2 Þ of G such that G1 is isomorphic to H1 ; G2 is isomorphic to H2 ; and EðG1 Þ,EðG2 Þ ¼ a: Denote this number by cða; H1 ; H2 Þ: Note that since we are assuming that jaj ¼ k1 þ k2 ¼ EðH1 Þ þ EðH2 Þ; these conditions imply that EðG1 Þ-EðG2 Þ ¼ |: Let us begin first with the left-hand side. From Theorems 2.3 and 2.8 we can see that the left-hand side of (3.3) is unchanged if an edge not in a is removed from G: Thus, it is sufficient to calculate the left-hand side in the case that EðGÞ ¼ a: From (3.2) we see that X SH1 SH2 ðG a Þ ¼ cðH; H1 ; H2 ÞSH ðG a Þ: H

If H has fewer than k1 þ k2 edges, then SH AEk1 þk2 1 so that SH ðG a Þ ¼ 0: Therefore X SH1 SH2 ðG a Þ ¼ cðH; H1 ; H2 ÞSH ðGa Þ: EðHÞ¼k1 þk2

We have assumed that EðGÞ ¼ jaj ¼ k1 þ k2 : This implies that for any H with EðHÞ ¼ k1 þ k2 ; and any nonempty set b of edges of G; SH ðG  bÞ ¼ 0:

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Thus SH1 SH2 ðGa Þ ¼

X

cðH; H1 ; H2 ÞSH ðGÞ:

EðHÞ¼k1 þk2

For the H’s that appear in the summand, we have EðHÞ ¼ k1 þ k2 ¼ jaj ¼ EðGÞ: Therefore, for such H’s, SH ðGÞ ¼ 1 if G ¼ H and 0 otherwise. This yields the desired formula that in the case EðGÞ ¼ a we have SH1 SH2 ðG a Þ ¼ cðG; H1 ; H2 Þ: However, if EðGÞ ¼ a; then cðG; H1 ; H2 Þ ¼ cða; H1 ; H2 Þ as defined in the previous paragraph. Thus, for any graph G with EðGÞ+a SH1 SH2 ðG a Þ ¼ cða; H1 ; H2 Þ: Now consider the right-hand side of (3.3). From Theorem 2.3(i), if EðH1 Þ ¼ jb1 j; then SH1 ðG b1 Þ is unchanged if edges not in b1 are removed from G: Similarly, SH2 ðG ab1 Þ is unchanged if edges in b1 are removed from G: Thus, we can write the right-hand side as X SH1 ððGðb1 ÞÞb1 Þ SH2 ðGðb2 Þb2 Þ; b1 ;b2 Ca jb1 j¼k1 ;jb2 j¼k2 b1 -b2 ¼|

where Gðb1 Þ is the spanning graph of G with edge set b1 ; and Gðb2 Þ is defined similarly. For each set b1 appearing in this sum, we have that jb1 j ¼ k1 ¼ EðH1 Þ; so that SH1 ððGðb1 ÞÞb1 Þ ¼ SH1 ðGðb1 ÞÞ ¼



1

if H1 ¼ Gðb1 Þ;

0

otherwise

and similarly SH2 ððGðb2 ÞÞb2 Þ ¼



1 if H2 ¼ Gðb2 Þ; 0 otherwise:

Putting these observations together we see that the right-hand side of (3.3) is the number of ways of partitioning a into sets b1 and b2 such that Gðb1 Þ ¼ H1 and Gðb2 Þ ¼ H2 : This is precisely cða; H1 ; H2 Þ: & We end this section with the presentation of an important identity that will have an important use later (see Theorem 6.13). Our goal is to better understand the relationship between finite-type invariants and the operation of replacing a graph by its complement. For any function f defined on G; define a new function f c by setting,

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for any graph G; f c ðGÞ ¼ f ðG c Þ: For any graph G; let AðGÞ denote the number of automorphisms of G; i.e. the number of permutations of the vertices of G which preserve all adjacency relations. Note that for any graph G; AðGÞ ¼ AðG c Þ: Theorem 3.4. For any two graphs G and H with the same number of vertices, SGc ðHÞ ¼

AðHÞ c S ðGÞ: AðGÞ H

Proof. Let n denote the number of vertices of G and H: Let NðG; HÞ denote the number of pairs ðG 0 ; H 0 Þ of spanning subgraphs of Kn such that G 0 is isomorphic to G; H 0 is isomorphic to H; and EðG0 Þ-EðH 0 Þ ¼ |: It is easy to see that NðG; HÞ ¼ SG ðKn ÞSH ðG c Þ; and that SG ðKn Þ ¼ n!=AðGÞ: Therefore NðG; HÞ ¼ n!SH ðGc Þ=AðGÞ: By symmetry, we also have NðG; HÞ ¼ n!SG ðH c Þ=AðHÞ; and the result follows.

&

4. Edge reconstructions Definition 4.1. Let F : G-S be any function, where S is any set. Let G be a graph. We say that F ðGÞ is Ek -reconstructible if, for any graph H; f ðHÞ ¼ f ðGÞ for every f AEk implies that F ðHÞ ¼ F ðGÞ: For any graph G; let wG : G-f0; 1g denote the function which equals 1 on all graphs isomorphic to G and 0 otherwise. We say that G is Ek -reconstructible if wG ðGÞ is Ek -reconstructible. Equivalently, we say that G is Ek -reconstructible if, for any graph H; f ðGÞ ¼ f ðHÞ for every f AEk implies that G ¼ H:

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Note that VAE0 and EAE1 ; so for kX0 VðGÞ is Ek -reconstructible, and for kX1 EðGÞ is Ek -reconstructible. If G and H are two graphs with the same number of vertices and edges, then G is isomorphic to H if and only if SG ðHÞ40: The next theorem now follows immediately from Theorem 3.1. Theorem 4.2. If G is a graph with kX1 edges, then G is Ek -reconstructible. We also mention a more classical notion of reconstructibility. Given a graph G with k edges, the edge deck of G is the unordered list of the k unlabeled graphs of the form G  e as e ranges over the edges of G: Definition 4.3. Let F : G-S be any function, where S is any set. Let G be a graph. We say that F ðGÞ is edge-reconstructible, if for any graph H with the same edge deck as G; F ðHÞ ¼ F ðGÞ: We say that G is edge-reconstructible if wG ðGÞ is edgereconstructible. Equivalently, we say that G is edge-reconstructible if every graph H with the same edge deck as G is isomorphic to G: The next theorem relates these two types of reconstructions. Theorem 4.4. Let F : G-S be any function, and G a graph with kX1 edges. Then F ðGÞ is edge-reconstructible if and only if F ðGÞ is Eðk1Þ -reconstructible. In particular, G is edge-reconstructible if and only if G is Eðk1Þ -reconstructible. Proof. The edge deck of G is completely determined by the values SH ðGÞ as H ranges over all graphs with n vertices and k  1 edges, and hence, by Theorem 3.1, the edge deck is Eðk1Þ -reconstructible. Conversely, the edge deck determines the values SH ðGÞ as H ranges over all graphs with n vertices and k  1 edges, which in turn determine the values SH ðGÞ as H ranges over all graphs with fewer than k edges. Thus, by Theorem 3.3, the edge deck determines the value of f ðGÞ for every f AEk1 : & One can consider a more general notion than that defined in Definition 4.3. If G is a graph with k edges, then given any r; 1prpk; the r-edge deck of G is the unordered  list of the kr unlabeled graphs of the form G  a; as a ranges over all subsets of EðGÞ of size r: Definition 4.5. Let F : G-S be any function, where S is any set. Let G be a graph. We say that F ðGÞ is r-edge-reconstructible if for any graph H with the same r edge deck as G; F ðHÞ ¼ F ðGÞ: We say that G is r-edge-reconstructible if for every graph H with the same r-edge deck as G is isomorphic to G: Theorems 3.1 and 3.3 imply the following generalization of Theorem 4.3.

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Theorem 4.6. Let F -S be any function, and G a graph with k edges. Then F ðGÞ is r-edge-reconstructible if and only if F ðGÞ is EðkrÞ -reconstructible. In particular, only if G is r-edge-reconstructible if and only if G is EðkrÞ -reconstructible.

5. Additional equations satisfied by edge-finite-type invariants In Section 2, we defined a function of edge-finite-type to be one which satisfies linear equations involving a graph G and the graphs that result from removing edges from G: In this section we will see that such functions also satisfy linear equalities involving a graph G and the graphs that result from removing some edges while adding others. More explicitly, if a1 is a set of edges of G; and a2 is a set of edges of G c ; we let G a1 ;a2 denote the linear combination X X G a1 ;a2 ¼ ð1Þjb1 jþjb2 j ðG  b1 þ b2 Þ: b1 Da1 b2 Da2

Note that G a;| is equivalent to what we have been calling G a : The main result of this section is the observation that edge-finite-type invariants satisfy linear equations involving these new linear combinations. The key observation is that these new linear combinations are not really new at all. This is the content of the following observation, which we state as a lemma for future reference. Lemma 5.1. For any graph GAG and any a1 DEðGÞ and any a2 DEðG c Þ; G a1 ;a2 ¼ ð1Þja2 j ðG þ a2 Þa1 þa2 :

This immediately implies the following theorem. Theorem 5.2. Let f : G-R be an edge-finite-type function of order k. Then for any graph GAG and any a1 DEðGÞ and any a2 DEðG c Þ; if ja1 j þ ja2 jXk þ 1 then f ðG a1 ;a2 Þ ¼ 0:

6. Complements and Lova´sz’s theorem In this section we focus our attention on the complementation map G-Gc : The identity ðG  eÞc ¼ G c þ e; for any graph G and any eAEðGÞ; extends to give the following identity for any graph G and any disjoint a1 ; a2 DEðGÞ ðG a1 ;a2 Þc ¼ ðG c Þa2 ;a1 : From Theorem 5.2 we learn the following result.

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Theorem 6.1. Let f : G-R be an edge-finite-type function of order k. Then f c is edgefinite-type of order k. Corollary 6.2. If, for any F : G-R; and any graph G, F ðGÞ is Ek -reconstructible, then F c ðG c Þ is also Ek -reconstructible. Corollary 6.3. If a graph G is Ek -reconstructible, then so is G c : Now recall that every graph G is EEðGÞ -reconstructible. Corollary 6.4. If a graph G has at least 12 is EEðGÞ2r -reconstructible.

n 2

þ r edges, for some positive rA12Z; then G

 Corollary 6.5. (i) If a graph G has more than 12 n2 edges, then G is edge-reconstructible  (ii) If a graph G has at least 12 n2 þ r edges, for some positive rA12Z; then G is 2r-edgereconstructible. In fact, one can make a stronger statement (see the comments at the end of Section 4). Corollary 6.6. Let Ker ðcÞ ¼ then

T

f AEc

Kerð f Þ: If kX12

n 2

þ r; for some positive rA12Z;

Ker ðcÞ-Lkn ¼ 0 for each cX12

n 2

 r:

Corollary 6.5(i) appeared in [12], and is often referred to as Lova´sz’s Theorem. Corollary 6.5 (ii) appeared in [11]. In [12], Lova´sz proved more than just Corollary 6.5(i), he proved the following statement. Theorem 6.7. Suppose G has the property that for every edge-reconstruction H of G, SG ðH c Þ ¼ 0 then G is edge-reconstructible. We will soon present a proof of this theorem from our point of view. First, we note that this theorem implies 6.5(i), but is stronger. We give just one example of the sort of result which follows immediately from Theorem 6.7. Corollary 6.8 follows from the easily proved fact that the degree sequence of G is edge-reconstructible. Corollary 6.8. Let G be a graph with n vertices, and d (resp. D) the minimal (resp. maximal) degree of the vertices in G. If d þ DXn then G is edge-reconstructible. Theorem 6.7 will follow from a study of the relationship between an edge-finitetype invariant f ; and the edge-finite-type invariant f c : We have seem in Theorem 6.1

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that if f is edge-finite-type of order k; then so is f c : There is much more that can be said, as can be seen in the following striking result. Theorem 6.9. Let f be an edge-finite-type function of order k. Then f þ ð1Þkþ1 f c is edge-finite-type of order k  1: Proof. Let G be a graph with at least k edges, and aDEðGÞ be a set of edges with jaj ¼ k: We must show that ð f þ ð1Þkþ1 f c ÞðG a Þ ¼ 0: First note that f c ðGa Þ ¼ f c ðGa;| Þ ¼ f ððG a;| Þc Þ ¼ f ððG c Þ|;a Þ ¼ f ðð1Þjaj ðG c þ aÞa Þ ¼ ð1Þjaj f ððG c þ aÞa Þ ¼ ð1Þk f ððGc þ aÞa Þ; where the fourth equality follows from Lemma 5.1. Therefore ð f þ ð1Þkþ1 f c ÞðG a Þ ¼ f ðG a Þ  ð1Þk f c ðGa Þ ¼ f ðG a Þ  f ððGc þ aÞa Þ ¼ 0; with the last equality following from Theorem 2.3(ii). & We can now give a proof of Theorem 6.7. Proof of Theorem 6.7. Let G be a graph with k edges. By Theorem 6.9, the function f ¼ SG þ ð1Þkþ1 ðSG Þc is edge-finite-type of order k  1: Let H be an edge reconstruction of G: Then we must have f ðGÞ ¼ f ðHÞ: On the other hand, by hypothesis SG ðG c Þ ¼ SG ðH c Þ ¼ 0; so that f ðGÞ ¼ SG ðGÞ þ ð1Þkþ1 ðSG Þc ðGÞ ¼ SG ðGÞ þ ð1Þkþ1 SG ðG c Þ ¼ 1 þ 0 ¼ 1; and f ðHÞ ¼ SG ðHÞ þ ð1Þkþ1 ðSG Þc ðHÞ ¼ SG ðHÞ þ ð1Þkþ1 SG ðH c Þ ¼ SG ðHÞ: Therefore, SG ðHÞ ¼ 1 which implies that G ¼ H:

&

For each k; let @k : Ek -Ek1 denote the map which sends f AEk to @k ð f Þ ¼ 12ð f þ ð1Þkþ1 f c Þ: The following fact is simple to check. Lemma 6.10. @k1 3@k ¼ 0:

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From Lemma 6.10 the complex @kþ1

@kþ1

@k

@k1

E : ? ! Ekþ1 ! Ek ! Ek1 ! y forms a differential complex, so that we may consider the homology Hk ðEÞ ¼

Ker ð@k Þ : Imð@kþ1 Þ

Of course, it is hard to imagine that we would find any interesting homology studying the space of all graphs. Theorem 6.11. The sequence E is exact. That is, for each k, Hk ðEÞ ¼ 0: More explicitly, if f AEk satisfies @k ð f Þ ¼ 0; then @kþ1 ðgÞ ¼ f ; where g ¼ V2  Ef : 2

Proof. Let f be an edge-finite-type function of order k; and suppose that @k ð f Þ ¼ þ ð1Þkþ1 f c Þ ¼ 0: Let g ¼ V2 Ef : The function V2  is edge-finite-type of order 0

1 2ð f

2

2

and E is edge-finite-type of order 1. Hence by Theorem 2.7, V2 Ef is edge-finite-type 2

of order k þ 1: Moreover, 2

@kþ1 VEf 2

!

1 2 ¼ V½Ef þ ð1Þk Ec f c  2 2 i  1 h V k c c c ¼ V E f  E f þ ð1Þ 2 2

1 ¼ f  V½Ec ð f þ ð1Þkþ1 f Þ ¼ f :

&

2

Using these ingredients, we can present an interesting dual formulation of the notion of an edge-reconstruction of a graph. We begin by recalling the following simple fact (see Theorem 2.5). For any k and graph H with pk edges, let fHk denote the unique edge-finite-type invariant which satisfies fHk ðHÞ ¼ 1 and fHk ðH 0 Þ ¼ 0 for all graphs H 0 aH which have pk edges. Then the fHk ; as H ranges over all graphs with pk edges, form a basis for Ek :

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Lemma 6.12. For any graph G with k edges, @k ðSG Þ ¼

ð1Þkþ1 2AðGÞ

X

c AðHÞSH ðGÞfHk1

EðHÞok

Proof. Any f AEk1 can be written as X f ðHÞfHk1 : f ¼ EðHÞok

We apply this formula to @k ðSG Þ; using the fact that SG ðHÞ ¼ 0 for any graph H with ok edges, to find X 1 SGc ðHÞfHk1 : @k ðSG Þ ¼ ð1Þkþ1 2 EðHÞok The lemma now follows from Theorem 3.4.

&

Theorem 6.13. Let G1 and G2 be two graphs with n vertices and k edges. Then G1 and G2 have the same edge deck if and only if @k ðAðG1 ÞSG1  AðG2 ÞSG2 Þ ¼ 0: Proof. As H ranges over all graphs with ok edges, the invariants SH form a basis for Ek1 : The map f -f c restricts to a linear isomorphism from Ek1 to itself. c Therefore, the invariants SH ; as H ranges over all graphs with ok edges, also form a basis for Ek1 : From the previous lemma, we see that @k ðAðG1 ÞSG1  AðG2 ÞSG2 Þ ¼ 0 if and only if c c SH ðG1 Þ ¼ SH ðG2 Þ for every graph H with ok edges. By the previous paragraph, this is true if and only if f ðG1 Þ ¼ f ðG2 Þ for every f AEk1 ; and this, by Theorem 4.4, is true if and only if G1 and G2 have the same edge deck. &

7. The Nash-Williams lemma In this section we present some new methods of constructing edge-finite-type functions. One application will be a proof of the Lemma of Nash-Williams [16]. We begin with Kn ; the complete graph on n vertices. Let SGðnÞ denote the set of spanning subgraphs of Kn : If G is any spanning subgraph of Kn ; then for any collection a of edges of G; G  a is also a spanning subgraph. Given f : SGðnÞ-R; say f is Kn -edge-finite-type of order k if for every GASGðnÞ with at least k þ 1 vertices, and every set aDEðGÞ with jajXk þ 1; we have f ðGa Þ ¼ 0: Just as in Theorem 2.2, it is sufficient to consider aDEðGÞ with jaj ¼ k þ 1: We denote the set of kn -edge-finite-type of order k by Ek ðnÞ: The following theorem can be proved by the same means as Theorems 2.3 and 2.4.

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Theorem 7.1. A function f AEk ðnÞ is completely determined by its restriction to the graphs GASGðnÞ which have pk edges. Conversely, given any function on SGpk ðnÞ; there is a unique Kn -edge-finite-type of order k which has these initial values. We note that f AEk ðnÞ may not be a edge-finite-type invariant in the sense of our previous sections, because it may not depend only on the isomorphism type of G: For any graph H; let AutðHÞ denote the set of automorphisms of H: Note that jAutðKn Þj ¼ n!: Every element fAAutðKn Þ acts on SGðnÞ; and hence on the functions on SGðnÞ; sending f to f f ; where, for any graph G; f f ðGÞ ¼ f ðfðGÞÞ: If f is Kn -edge-finite-type of order k; then so is f f : If we average over all of AutðKn Þ; then the resulting function f% is a graph invariant. That is, we have the following theorem. Theorem 7.2. Let f be a Kn -edge-finite-type of order k. Define a function f% on Gn as follows. For any graph G with n vertices, choose a graph G 0 ASGðnÞ such that G 0 is isomorphic to G. Set X % f ðfðG 0 ÞÞ: fðGÞ ¼ fAAutðKn Þ

% % k: Extend f% to all of G by setting fðGÞ ¼ 0 if VðGÞan: Then fAE In what follows we will be constructing functions on Gn : It is to be understood that they are to be extended to G by setting their values to be 0 on Gm ; for man: We now introduce an important construction of edge-finite-type functions. Let X be a spanning subgraph of Kn : If G is any element of SGðnÞ; then X -GASGðnÞ: We have the following simple facts (note that part (ii) follows from part (i)). Lemma 7.3. (i) If G and X are spanning subgraphs of Kn ; and e is an edge of G, then G-X eeEðX Þ; ðG  eÞ-X ¼ ðG-X Þ  e eAEðX Þ: (ii) With G and X as in part (i), if a is any set of edges of G 0 aD / EðX Þ; a G -X ¼ a ðG-X Þ aDEðX Þ; where Ga -X denotes the linear combination resulting from intersecting each graph in the linear combination G a with X : For any function f : SGðnÞ-R; define a function fX : SGðnÞ-R by setting, for any GASGðnÞ fX ðGÞ ¼ f ðX -GÞ: Lemma 7.3 implies the following theorem.

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Theorem 7.4. If f is a Kn -edge-finite-type function of order k, and X is a spanning subgraph of Kn ; then fX is also Kn -edge-finite-type of order k. Let us piece together what we have just proved. Combining the previous two theorems, we have the following. Theorem 7.5. Let f be a Kn -edge-finite-type function of order k, and X a spanning subgraph of Kn : We define a function F : SGðnÞ-Z as follows. For any graph G with n vertices, let G 0 be a subgraph of Kn which is isomorphic to G, and set X f ðfðG 0 Þ-X Þ: F ðGÞ ¼ fAAutðKn Þ

Then F is an edge-finite-type invariant of order k. Now let us get more specific. Let HDX be spanning subgraphs of Kn ; and suppose that H has pk  1 edges and X has precisely k edges. Let f be the unique function in Ek1 ðnÞ such that if H 0 ASGðnÞ has pk  1 edges then f ðH 0 Þ ¼ 1 if H 0 ¼ H and 0 otherwise. Then X f ðX Þ ¼ f ðX  X EðX Þ Þ ¼  ð1Þjaj f ðX  aÞ ¼ ð1ÞkjEðHÞjþ1 : |aaDEðX Þ

Now let G be any graph with n vertices, and let G 0 be any spanning subgraph of Kn which is isomorphic to G: Applying Theorem 7.5 to the function f leads to the conclusion that the function which sends G to jffAAutðKn Þ such that EðfðG 0 ÞÞ-EðX Þ ¼ EðHÞgj þ ð1ÞkjEjþ1 jffAAutðKn Þ such that EðfðG 0 ÞÞ-EðX Þ ¼ EðX Þgj is edge-finite-type of order k  1: Letting X ¼ G0 leads us to the following theorem, known as the Nash-Williams Lemma. Theorem 7.6. Let G and H be graphs with n vertices and k edges, let G 0 and H 0 ; resp., be spanning subgraphs of Kn which are isomorphic to G and H, resp. Let ECEðG0 Þ be any subset with jEjpk  1: If G and H have the same edge deck, then jffAAutðKn Þ such that EðfðG 0 ÞÞ-EðG0 Þ ¼ Egj þ ð1ÞkjEjþ1 jffAAutðKn Þ such that fðG0 Þ ¼ G 0 gj ¼ jffAAutðKn Þ such that EðfðH 0 ÞÞ-EðG 0 Þ ¼ Egj þ ð1ÞkjEjþ1 jffAAutðKn Þ such that fðH 0 Þ ¼ G 0 gj:

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This result is central to the subject of edge-reconstructibility. Many of the graphs which are known to be edge-reconstructible were shown to be so by a counting argument that has the Nash-Williams lemma at its core. See the surveys [4,6,16,20] for examples. We close this section with another construction of edge-finite-type invariants. Let G and H be spanning subgraphs of Kn : We define the symmetric difference of G and H; which we denote by GDH to be the spanning subgraph of Kn whose edge set is ðEðGÞ  EðHÞÞ-ðEðHÞ  EðGÞÞ: Then we have the following simple lemma. Lemma 7.7. (i) If G and H are spanning subgraphs of Kn ; and e is an edge of G, then ðGDHÞ þ e eAEðHÞ; ðG  eÞDH ¼ ðGDHÞ  e eeEðHÞ: (ii) For any spanning subgraphs G and H of Kn ; and any set of edges a of G, we have G a DH ¼ ðGDHÞaEðHÞ;a-EðHÞ : We now define a dual operation on functions. Given a spanning subgraph X of Kn ; and a function f on SGðnÞ; let f DX denote the function defined by f DX ðGÞ ¼ f ðGDX Þ: Lemmas 5.1 and 7.7 imply the following theorem. Theorem 7.8. With all notation as above, if f is edge-finite-type of order k, then so is f DX : Therefore, given f AEk ðnÞ and X ASGðnÞ; the invariant resulting from averaging f DX over AutðKn Þ is edge-finite-type of order k: This construction yields a wide variety of edge-finite-type invariants, but it is not clear how to best use these functions to extract information about a given graph G:

8. Analysis on GK If we wish to know whether a graph G with K edges is Ek -reconstructible, that is, distinguishable from all other graphs with K edges by functions in Ek ; the relevant information is the restriction of functions in Ek to GK : An important question then arises— Given k and K; which functions on GK are restrictions of functions in Ek ? Working directly with the definition is rather inconvenient, since to check that a function is edge-finite-type, one must compare its value for a graph with the values of the subgraphs resulting from removing edges. In this section we present a necessary and sufficient condition, which does not require the consideration of graphs with fewer edges, for a function on GK to be the restriction of a function in Ek :

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First, some definitions. Let G a graph. A direction at G is a pair ðe; e0 Þ of edges, where eAG and e0 eG: We say two directions ðe1 ; e01 Þ; ðe2 ; e02 Þ at G are orthogonal if e1 ae2 and e01 ae02 : If ðe1 ; e01 Þ; ðe2 ; e02 Þ; y; ðek ; e0k Þ is any set of pairwise orthogonal directions at G; then we can form the graph Gþ

k X

ðei ; e0i Þ ¼ G  fe1 ; e2 ; y; ek g þ fe01 ; e02 ; y; e0k g:

i¼1

Let G be a graph with n vertices. Let D ¼ fðe1 ; e01 Þ; ðe2 ; e02 Þ; y; ðek ; e0k Þg denote a set  of k pairwise orthogonal directions at G: (This requires that kpEðGÞp n2  k:Þ We then define ! X X jIj ½D 0 G ¼ ð1Þ Gþ ðei ; ei Þ : iAI

IDf1;2;y;kg

If D ¼ fðe1 ; e01 Þ; ðe2 ; e02 Þ; y; ðek ; e0k Þg is a set of k pairwise orthogonal directions at G; and a is a set of edges of G; we say that a is disjoint from D if a-fe1 ; e2 ; y; ek ; e01 ; e02 ; y; e0k g ¼ |: If a consists of a single edge e; we will sometimes write eeD to indicate that feg is disjoint from D: If a set of edges a is disjoint from D then we can form the linear combination ! X X X jbjþjIj a;½D 0 G ¼ ð1Þ Gbþ ðei ; ei Þ : iAI

bDa IDf1;2;y;kg

Our first result is that if f is an edge-finite-type function, then it satisfies a collection of linear difference equation involving directions tangent to the GK ’s. Theorem 8.1. Let f be an edge-finite-type invariant of order k. Then for any graph G with at least k þ 1 edges, any set D ¼ fðe1 ; e01 Þ; ðe2 ; e02 Þ; y; ðed ; e0d Þg of dpk þ 1 pairwise orthogonal directions at G, and any set a of k þ 1  d edges of G which is disjoint from D, f ðGa;½D Þ ¼ 0:

ð8:1Þ

Proof. The proof is by induction on d: If d ¼ 0 then (8.1) is just the definition of an edge-finite-type invariant of order k: Now assume that d40 and that (8.1) is true for all sets of directions D0 with jD0 jod: With all notation as in the theorem, let D0 ¼ fðe1 ; e01 Þ; ðe2 ; e02 Þ; y; ðed1 ; e0d1 Þg; By induction, we have 0

0

0

f ðGa;½D  Þ  f ððG  ed Þa;½D  Þ ¼ f ðG aþed ;½D  Þ ¼ 0:

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Similarly, 0

0

0

0

f ððG  ed þ e0d Þa;½D  Þ  f ððG  ed Þa;½D  Þ ¼ f ððG  ed þ e0d Þaþed ;½D  Þ ¼ 0: Therefore 0

0

f ðG a;½D Þ ¼ f ðG a;½D  Þ  f ððG  ed þ e0d Þa;½D  Þ ¼ 0:

&

We think of the GK ’s as being horizontal slices of G; and hence refer to the values f ðG ½D Þ; as G ranges over GK and ½D ranges over the sets of ðk þ 1Þ-pairwise orthogonal directions at G; as the kth partial derivatives of f in the horizontal directions on GK : The main theorem of this section is the converse to Theorem 8.1. Theorem 8.2. Let f be a function on GK ; suppose that for every GAGK ; and every set D ¼ fðe1 ; e01 Þ; ðe2 ; e02 Þ; y; ðekþ1 ; e0kþ1 Þg of pairwise orthogonal directions at G we have f ðG ½D Þ ¼ 0:

ð8:2Þ

Then f is the restriction to GK of an edge-finite-type function of order k. Proof. The proof is by induction on k: If k ¼ 0; then Eqs. (8.2) imply that f must be K constant on GK n for each n; and hence restricted to G depends only on the number of vertices, which implies that f is edge-finite-type of order 0. Now suppose that the theorem is true for all k0 ok; with kX1: To prove that the theorem is true for k we use induction on K: If Kpk; then the theorem is true, since the hypotheses are vacuous, and, by Theorem 2.5 any function on Gpk can be extended to an edgefinite-type function of order k: Now suppose it is true for all K 0 oK; with KXk þ 1: The main idea is to construct the extension of f to GK1 ; and then to apply the inductive hypothesis to this extension. We claim that there is a function g : GK1 -R with the property that for any GAGK ; any set D0 of k pairwise orthogonal directions at G; and any edge eAG such that eeD0 0

0

f ðG ½D  Þ ¼ gððG  eÞ½D  Þ:

ð8:3Þ

We will establish the existence of such a g shortly. For now we will assume the existence of g and complete the proof. Note that any such function g must have vanishing ðk þ 1Þth partial derivatives in the horizontal directions on GK1 : That is, for any GAGK1 and any set D of k þ 1 pairwise orthogonal directions, write D ¼ D0 ,ðe; e0 Þ for some set D0 of k pairwise

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orthogonal directions at G: Then 0

0

gðG ½D Þ ¼ gðG½D  Þ  gððG  e þ e0 Þ½D  Þ 0

0

¼ f ððG þ e0 Þ½D  Þ  f ðððG  e þ e0 Þ þ eÞ½D  Þ ¼ 0: Therefore, by the inductive hypothesis on K; g is the restriction to GK1 of an invariant g˜ on all of G which is edge-finite-type of order k: By Theorem 8.1, for any GAGK any set D0 of k pairwise orthogonal directions at G; and any edge eAG such that eeD0 ; 0

0

0

gðG ˜ ½D  Þ  gððG ˜  eÞ½D  Þ ¼ gðG ˜ feg;½D  Þ ¼ 0; so that 0

0

0

0

gðG ˜ ½D  Þ ¼ gððG ˜  eÞ½D  Þ ¼ gððG  eÞ½D  Þ ¼ f ðG ½D  Þ: This shows that the function f  g˜ : GK -R has vanishing kth partial derivatives in the horizontal directions on GK : By the inductive hypothesis on k; there is an edgefinite-type invariant h : G-R of order k  1 (and hence of order k) such that, restricted to GK ; h ¼ f  g: ˜ Now we see that the function F ¼ h þ g˜ is an edgefinite- type invariant of order k whose restriction to GK is precisely f : All that remains is to prove the existence of g: One can do this rather abstractly. Namely, (8.3) shows what the kth derivatives of g have to be, and one can show that they are the kth derivatives of an invariant. Instead we will take a rather pedestrian approach, and construct g directly. We will construct the restriction of g to each  GK1 separately, so fix a value of n: The theorem is trivially true if K ¼ n2 ; so we n n n may assume that Ko 2 : Let a ¼ 2  ðK  1 þ kÞ and set for any graph GAGK1 ; n gðGÞ ¼

1X f ðG þ eÞ: a eeG

ð8:4Þ

An important point is that this defines an invariant. That is, if G1 and G2 are isomorphic graphs with K  1 edges, then gðG1 Þ ¼ gðG2 Þ: This is trivially true, since if f : G1 -G2 is an isomorphism, and e is an edge not in G1 ; then G1 þ e is isomorphic to G2 þ fðeÞ; and hence f ðG1 þ eÞ ¼ f ðG2 þ fðeÞÞ: This is a rather simple observation, but it is the crucial point in some attempts to generalize this theorem. (See for example, the final paragraph of this paper.) Now we have to see that for any graph GAGK1 ; any set D0 of k pairwise orthogonal directions at G; and any edge eeG such that eeD0 ; 0

0

gðG½D  Þ ¼ f ððG þ eÞ½D  Þ:

ð8:5Þ

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Note that for any e0 eG with e0 eD0 and e0 ae; 0

0

0

0

f ððG þ eÞ½D  Þ  f ððG þ e0 Þ½D  Þ ¼ f ððG þ eÞfðe;e Þg,½D  Þ ¼ 0; which shows that the right-hand side of (8.5) is independent of e (among all eeG 0 such that eeD0 ), so it is sufficient to prove that gðG ½D  Þ is equal to the average of the right-hand side over all such e: There are a such e’s, so that we must show that 0

gðG ½D  Þ ¼

0 1 X f ððG þ eÞ½D  Þ: a eeG;eeD0

ð8:6Þ

We will prove this by a direct calculation. 0

gðG ½D  Þ ¼

X

ð1ÞjIj g G 

[

ei þ

iAI

IDf1;2;y;kg

X 1 ð1ÞjIj ¼ a IDf1;2;y;kg

[

! e0i

iAI

X eeðG

S iAI

ei þ

S

f G e0 iAI i

[

ei þ

iAI

Þ

For a fixed IDf1; 2; y; kg we can partition feeðG  three sets

S

iAI

[

! e0i þ e :

iAI

ei þ

S

iAI

e0i Þg into the

fejeeG and eefe01 ; e02 ; y; e0k gg,fej j jAIg,fe0j j jAf1; 2; y; kg  Ig: Therefore, we can write 2 !3 X X [ [ 1 0 gðG ½D  Þ ¼ 4 ð1ÞjIj f G ei þ e0i þ e 5 a IDf1;2;y;kg 0 iAI iAI eeG;eeD 2 !3 X X [ [ 1 þ4 ð1ÞjIj f G ei þ e0i þ ej 5 a IDf1;2;y;kg jAI iAI iAI 2 !3 X X [ [ 1 jIj 0 0 ð1Þ f G ei þ ei þ ej 5 þ4 a IDf1;2;y;kg iAI iAI jAf1;2;y;kgI ¼ S1 þ S2 þ S3 :

ð8:7Þ

For any IDf1; 2; y; kg and any jAI we have the identity [ [ [ [ G ei þ e0i þ ej ¼ G  ei þ iAI

iAI

iAIf jg

e0i þ e0j :

ð8:8Þ

iAIf jg

Every term in the sum S2 in (8.7) involves a graph of the form of the left-hand side of (8.8), and every term in the sum S3 in (8.7) involves a graph of the form of the righthand side of (8.8). In this way, the terms in S2 can be paired up with the terms in S3 :

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Because of the factor ð1ÞjIj each term is paired with a term of equal magnitude P and opposite sign. Hence S2 þ S3 ¼ 0: The sum S1 is precisely 1a eeG;eeD0 f ððG þ 0

eÞ½D  Þ; which established (8.6).

&

We observe, before leaving this section that Theorem 8.2 immediately implies   the 1 n generalized Lova´sz’s Theorem (Theorem 6.5) that every graph with at least 2 2 þ r edges (for some positive rA12Z) is 2r-edge-reconstructible, since, if G has at least   1 n exist 12 n2  r þ 1 pairwise orthogonal directions at 2 2 þ r edges, then there do not   G: Hence, for such K and k ¼ 12 n2  r; the hypotheses of Theorem 8.2 are vacuously  true. Thus, every function on GK ; where KX12 n2 þ r is the restriction of an edge finite-type invariant of order k ¼ 12 n2  r: In particular, for any GAGK ; the characteristic function of G is the restriction of an edge-finite-type function of order  1 n  r; so G is E1 n -reconstructible, and hence (by Theorem 4.6) is 2r-edge2 2 2ð2Þr reconstructible.

9. Vertex-finite-type functions In the previous sections of this paper, we considered subspaces En of the space of graph invariants which were those functions which satisfied certain linear equations involving graphs and their spanning subgraphs, or, equivalently, the subgraphs resulting from removing edges. In the remainder of this paper we present the analogous theory for induced subgraphs. We will not include proofs, since the proofs are quite analogous to the proofs for edge-finite-type invariants. First some notation. If v is a vertex of a graph G; G  v denotes the result of removing from G the vertex v and all edges incident to v: We will also use the notation G  v when v is a set of vertices of G: An induced subgraph of G is any subgraph of G of the form G  v for some set v of vertices of G: If x is any set of vertices of G; we let Gx denote the linear combination Gx ¼

X

ð1Þjyj ðG  yÞ:

yDx

Definition 9.1. We define a nested family of subspaces of F f0g ¼ V1 CV0 CV1 CV2 C?CF:

ð9:1Þ

An invariant f AF is in Vn if and only if, for every graph G with at least n þ 1 vertices, and every set xAV ðGÞ satisfying jxjXn þ 1; we have f ðGx Þ ¼ 0: Functions in Vn are said to be vertex-finite-type of order n; and if f AVn for some n; we say that

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f is vertex-finite-type. Let V ¼ functions.

S

n

Vn denote the space of all vertex-finite-type

We first note that the definition of an vertex-finite-type invariant can be slightly simplified. Theorem 9.2. Let f AF: Then the following two conditions are equivalent (i) For every graph G with at least n þ 1 vertices, and every subset xDV ðGÞ with jxjXn þ 1 we have f ðGx Þ ¼ 0: (ii) For every graph G with at least n þ 1 vertices, and every subset xDV ðGÞ with jxj ¼ n þ 1 we have f ðGx Þ ¼ 0:

For a proof see the proof of Theorem 2.1. The analogues of Theorems 2.2, 2.3 and 2.7 also hold in this context. The reader can easily adapt the proofs in Section 2 to the new setting. Theorem 9.3. If f is an vertex-finite-type function of order n, then f is completely determined by its restriction to Gpn : Note that f AVn is completely determined by a finite amount of data. Moreover, there are no restrictions on this ‘‘initial data’’. Theorem 9.4. For any n and any function f : Gpn -R there is a unique function F : G-R which is vertex-finite-type of order n and which satisfies F ðGÞ ¼ f ðGÞ for each GAGpn : Just as one might expect from our analysis of edge-finite-type invariants, filtration 9.1 gives V the structure of a graded algebra. Theorem 9.5. Vn1  Vn2 DVn1 þn2 :

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10. Examples of vertex-finite-type invariants In this section we present some examples of functions which are vertex-finite-type of order n: In what follows, we let ½n denote the graph consisting of n isolated vertices. n ¼ 1: A function f is vertex-finite-type of order 1 if and only if for every graph G we have f ðG| Þ ¼ 0: Since G| ¼ G; we see that f must be the 0 function, so that V1 ¼ f0g: n ¼ 0: A function f is vertex-finite-type of order 0 if and only if for every graph G with at least one vertex, and every vertex v of G we have f ðGv Þ ¼ 0: That is, for every such G and v; f ðGÞ  f ðG  vÞ ¼ 0; so that, beginning with any graph G; f is unchanged if a vertex is removed. Since any graph G can be transformed into the empty graph by a finite number of vertex removals, we see that for any graph G and any function f which is vertex-finite-type of order 1, f ðGÞ ¼ f ð½0Þ; so that f must be constant on all of G: On the other hand, it is quite clear that any constant function lies in V0 : Therefore V0 consists of precisely the constant functions. n ¼ 1: Let f be a vertex-finite-type invariant of order 1. By Theorem 9.3, f is completely determined by the value of f ð½0Þ and f ð½1Þ: It is then easy to see that the function V; which assigns to each graph G the number of vertices in G; is vertexfinite-type of order 1. Multiplying V by a constant results in another vertex-finitetype invariant of order 1, as does adding a constant function (since V0 CV1 ). Therefore the function f˜ ¼ f ð½0Þ þ ð f ð½1Þ  f ð½0ÞÞV is a vertex-finite-type function of order 1, and has the same ‘‘initial values’’ as f (i.e. the same values on the graphs [0] and [1]), and therefore we must have that f˜ ¼ f : That is, for any f AV1 ; and any graph G; f ðGÞ ¼ f ð½0Þ þ ð f ð½1Þ  f ð½0ÞÞVðGÞ: In particular, V1 is generated (as a vector space) by the constant functions and the function V: n ¼ 2: From Theorem 9.3, we know that any invariant f AV2 is completely determined by its values on all graphs with p2 vertices. There are four such graphs, namely [0], [1], [2], and K2 : In particular, V2 is four dimensional. The subspace V1 CV2 is two dimensional, so we must find two additional, linearly independent elements in V2 : Theorem 9.4 provides one such invariant, namely V2 ; that is, the

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invariant which assigns to every graph G the number ðVðGÞÞ2 : It is easy to check that E is also vertex-finite-type of order 2, and these invariants are linearly independent. For any graph H; let sV H : G-Z denote the function which assigns to every graph G the number of induced subgraphs of G which are isomorphic to H: The following theorem can be proved in the same manner as Theorem 3.1 Theorem 10.1. If H has n vertices, then sV H is vertex-finite-type of order n, but not order n  1: One can prove that the functions sV H ; as H ranges over all graphs with pn vertices, are linearly independent. This implies the following theorem. Theorem 10.2. For each n, the functions fsV H jVðHÞpng form a basis of Vn (as a vector space). We end this section with another look at vertex-finite-type invariants of order 0, 1 and 2. n ¼ 0: From Theorem 10.2 we know that V0 is spanned by the function sV ½0 : Since, for any graph G; sV ½0 ðGÞ ¼ 1; we see that V0 consists precisely of the constant functions. n ¼ 1: From Theorem 10.2 we know that V1 is spanned by the functions sV ½0 and V sV ½1 : As we saw in the previous case, s½0 is the constant function which assigns the

number 1 to every graph. It is easy to see that sV ½1 assigns to every graph G the number of vertices in G; i.e. sV ½0 ¼ V: V V V n ¼ 2: From Theorem 10.2 we know that the functions sV ½0 ; s½1 ; sK2 and s½2 c V provide a basis for V2 : We can easily see that sV K2 ðGÞ ¼ EðGÞ; and s½2 ðGÞ ¼ E ðGÞ ¼   VðGÞ  EðGÞ: Previously, we saw that the functions 1, V; V2 and E forms a basis 2

for V2 : These two bases are related by the simple observation that for any graph G; EðGÞ þ Ec ðGÞ ¼



VðGÞ 2



1 ¼ ðV2 ðGÞ  VðGÞÞ: 2

There is another natural basis for Vn : For any graph H; let sH be the function which assigns to every graph G the number of subgraphs isomorphic to H: Theorem 10.3. For every graph H with n vertices, sH is vertex-finite-type of order n but not order n  1: Once again, it can be checked that the invariants sH ; as H ranges over all graphs with pn vertices, are linearly independent. This implies the following theorem.

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Theorem 10.4. For each n, the functions fsH jVðHÞpng form a basis of Vn (as a vector space). For example, the functions s½0 ; s½1 ; s½2 and sK2 form a basis for V2 : Comparing with the basis provided by Theorem 10.2 we note that s½0 ¼ sV ½0 ;

s½1 ¼ sV ½1 ¼ V;

and

 s½2 ¼

V 2



sK 2 ¼ sV K2 ¼ E;

V ¼ sV ½2 þ sK2 :

11. Vertex reconstructions In this section we present the vertex versions of the reconstruction problems we considered in Section 5. We begin with the analogous definitions. Definition 11.1. Let F : G-S be any function, where S is any set. Let G be a graph. We say that F ðGÞ is Vn -reconstructible, if for any graph H; f ðHÞ ¼ f ðGÞ for every f AVn implies that F ðHÞ ¼ F ðGÞ: For any graph G; let wG : G-f0; 1g denote the function which equals 1 on all graphs isomorphic to G and 0 otherwise. We say that G is Vn -reconstructible if wG ðGÞ is Vn -reconstructible. Equivalently, we say that G is Vn -reconstructible if for any graph H; f ðGÞ ¼ f ðHÞ for every f AVn implies that G ¼ H: The next theorem follows immediately from Theorem 9.4. Theorem 11.2. If G is a graph with n vertices, then G is Vn -reconstructible. There is a corresponding classical notion of reconstructibility. Given a graph G with n vertices, the vertex deck of G is the unordered list of the n unlabeled graphs of the form G  v as v ranges over the vertices of G: Definition 11.3. Let F : G-S be any function, where S is any set. Let G be a graph. We say that F ðGÞ is vertex-reconstructible, if for any graph H; if H and G have the same vertex deck, then F ðHÞ ¼ F ðGÞ: We say that G is vertex-reconstructible if wG ðGÞ is vertex-reconstructible. Equivalently, we say that G is vertex-reconstructible if for any graph H; if G and H have the same edge deck, then G ¼ H: Theorem 5.4 has the following direct analogues. Theorem 11.4. Let F -S be any function, and G a graph with n vertices. Then F ðGÞ is vertex-reconstructible if and only if F ðGÞ is Vðn1Þ -reconstructible. In particular, G is vertex-reconstructible if and only if G is Vðn1Þ -reconstructible.

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Just as in the case of edge-reconstructions defined in Section 4, the notion of vertex-reconstructions can be generalized. If G is a graph with n vertices, then given  any r; 1prpn; the r-vertex deck of G is the unordered list of the nr unlabeled graphs of the form G  x; as x ranges over all subsets of V ðGÞ of size r: Definition 11.5. Let F : G-S be any function, where S is any set. Let G be a graph. We say that F ðGÞ is r-vertex-reconstructible if for any graph H with the same rvertex deck as G; F ðHÞ ¼ F ðGÞ: We say that G is r-vertex-reconstructible if for every graph H with the same r-vertex deck as G is isomorphic to G: Theorem 10.1 implies the following analog of Theorem 4.4. Theorem 11.6. Let F -S be any function, and G a graph with n vertices. Then F ðGÞ is r-vertex-reconstructible if and only if F ðGÞ is VðnrÞ -reconstructible. In particular, only if G is r-vertex-reconstructible if and only if G is VðkrÞ -reconstructible. The fundamental question here is the vertex-reconstruction conjecture. Conjecture 11.7. Every graph with at least four vertices is vertex-reconstructible. 12. Complements of vertex-finite-type functions In Section 6 we proved that the complementation map preserves edge-finite-type functions. An immediate corollary was the strong form of Lova´sz’s Theorem. In this section we prove observe that the corresponding statement is also true for vertexfinite-type functions. The proof is somewhat easier in this case, and the conclusion is less striking. For any graph G and any vertex v of G; ðG  vÞc ¼ ðG c Þ  v: More generally, this identity holds if v denotes a set of vertices of G: This observation immediately yields the following lemma. Lemma 12.1. For any graph G and any xCV ðGÞ; ðGx Þc ¼ ðG c Þx : Recall that the complementation map induces a map from F to F; sending a function f to f c ; where, for any graph G; f c ðGÞ ¼ f ðG c Þ: The following theorem is an immediate corollary of the previous lemma. Theorem 12.2. If f AVn then f c AVn : Theorem 12.3. Let F -S be any function, and G a graph. Then for any n, F ðGÞ is Vn reconstructible if and only if F c ðG c Þ is Vn -reconstructible.

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Theorem 12.4. Let G be any graph. Then for any n, G is Vn -reconstructible if and only if G c is Vn -reconstructible. Combining this with Theorems 10.4 and 10.6 yields the following corollary, which seems to be well-known. Corollary 12.5. Let G be any graph. Then for any n, G is vertex-reconstructible if and only if Gc is vertex-reconstructible. More generally, for any r40; G is r-vertex reconstructible if and only if G c is r-vertex reconstructible.

13. More constructions of vertex-finite-type invariants Just as in Section 7, we begin with KN ; the complete graph on N vertices. Let GðNÞ denote the set of subgraphs of KN : We note that now we do not identify isomorphic P   n subgraphs, so that jGðNÞj ¼ N N 2ð2Þ : n¼0 n

If G is a subgraph of KN ; then for any collection x of vertices of G; G  x is also canonically an element of GðNÞ: Given f : GðNÞ-R; say f is KN -vertex-finite-type  

of order n if for every GAGðNÞ with at least n þ 1 vertices, and every set xA

V ðGÞ nþ1

;

we have f ðGx Þ ¼ 0: We denote the set of KN -vertex-finite-type of order n by Vn ðNÞ: The following analogue of Theorems 9.2 and 9.3 holds in this context. Theorem 13.1. A function f AVn ðNÞ is completely determined by its restriction to the graphs GAGðNÞ which have pn vertices. Conversely, given any function on Gpn ðNÞ; there is a unique KN -vertex-finite-type of order n which has these initial values. For any f : GðNÞ-R; define the symmetrization of f over AutðKN Þ by f:% That is, X f f: f% ¼ fAAutðKN Þ

The symmetrized function f% can be viewed as a function on GpN : Namely, given any graph G with pN vertices, choose any subgraph G 0 of KN which is isomorphic to G: Since any two such G 0 ’s are isomorphic via some element of AutðKN Þ; we see that % % 0 Þ: If f % 0 Þ is independent of our choice of G 0 : Therefore, we may set fðGÞ to be fðG fðG is vertex-finite-type of order n; then so is each f f ; and hence so is f:% Thus we have the following theorem. Theorem 13.2. Let f : GðNÞ-R be a KðNÞ-vertex-finite-type function of order n, then f% : GpN -R is vertex-finite-type of order n. We now investigate some interesting KN -vertex-finite-type functions. Fix npN; and HAGðNÞ be a subgraph with on vertices. Let snH denote the unique

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KN -vertex-finite-type function of order n  1 satisfying, for all graphs H 0 with on vertices, snH ðH 0 Þ ¼ 1 if H 0 ¼ H and 0 otherwise. (Note that in this setting, H 0 ¼ H does not mean graph isomorphism, but equality as subgraphs of KN :) The following lemma follows easily from the definitions. Lemma 13.3. Let H and G be subgraphs of KN satisfying VðHÞon and VðGÞ ¼ n: Then snH ðGÞ ¼ ð1Þnþ1VðHÞ if H is an induced subgraph of G, and 0 otherwise. Now, let H be a subgraph of KN with on vertices, and consider s%nH : More explicitly, if G has on vertices, then s%nH ðGÞ ¼ 0 if G is not isomorphic to H; and jffAAutðKðNÞÞ such that fðHÞ ¼ Hgj if G is isomorphic to H: If G has exactly n vertices, and G0 is a subgraph of KN which is isomorphic to G; then s%nH ðGÞ ¼ ð1Þnþ1VðHÞ jffAAutðKN Þ such that H is an induced subgraph of fðG 0 Þgj: Theorem 13.2 implies the following result. Theorem 13.4. The function s%nH : Gpn -R is vertex-finite-type of order n  1: We can generalize this idea by considering symmetric difference. That is, let F be a subset of EðKN Þ: For any subgraph G of Kn ; let FG denote the subset of F consisting of those edges with both endpoints in V ðGÞ: We define the symmetric difference of G and F ; denoted by GDF ; to be the subgraph of KN with the same vertex set as G; and satisfying EðGDF Þ ¼ ðEðGÞ  F Þ,ðFG  EðGÞÞ: Note that if F ¼ |; then GDF ¼ G; while if F ¼ EðKN Þ; then GDF ¼ G c : We observe that for any set xCV ðGÞ ðG  xÞDF ¼ ðGDF Þ  x:

ð13:1Þ

This implies the following lemma. Lemma 13.5. If G is a subgraph of KN ; and F DEðKN Þ; then for any xDEðGÞ; Gx DF ¼ ðGDF Þx : This symmetric difference operator induces a map on Vn ðNÞ: That is, given any F CEðKN Þ and any f AVn ðNÞ; define a function f DF by setting, for any subgraph G of KN ; f DF ðGÞ ¼ f ðGDF Þ: Theorem 13.6. Let f : GðNÞ-R be a KN -vertex-finite-type function of order n. Then for any F DEðKN Þ; f DF is KN -vertex-finite-type of order n, and hence averaging over AutðKN Þ results in a vertex-finite-type invariant of order n. The theorems in this section allow the construction of a wide variety of vertex-finite-type invariants. These invariants do not seem to lead to reconstruction results in the same direct manner as Nash-Williams’ Lemma does in the case of

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edge-finite-type invariants. Still, I believe that much progress can be made by further study of these invariants, and I hope to return to this topic in later work.

14. Analysis on GN In this final section we show that vertex-finite-type invariants satisfy linear equations involving derivatives tangent to GN : This section should be compared to Section 8, where the corresponding results are proved for edge-finite-type invariants. The theory will be slightly more complicated here, since we must work with induced subgraphs, rather than general subgraphs. Let X be any graph, and VGðX Þ the set of induced subgraphs of X : Any GAVGðX Þ is completely identified by its vertices, so, for any V DEðX Þ; let GðV ÞAVGðX Þ denote the induced subgraph of X with vertex set V : For any N; let VGðX ÞN denote the elements of VGðX Þ which have precisely N vertices. For any GðV Þ; a v-direction at GðV Þ is a pair ðv; v0 Þ of vertices, where vAV and v0 eV : (The phrase v-direction at GðV Þ is shorthand for ‘‘vertexdirection at GðV Þ which is tangent to VGðX ÞN ’’, where N ¼ jV j:) Given two vdirections ðv1 ; v01 Þ; ðv2 ; v02 Þ at GðV Þ; we say they are orthogonal if v1 av2 and v01 av02 : If ðv; v0 Þ is a v-direction at GðV Þ; we can, beginning at GðV Þ; move in the v-direction ðv; v0 Þ; and the result is the graph GðV  v þ v0 Þ: We denote this graph by GðV Þ þ ðv; v0 Þ: More generally, if ðv1 ; v01 Þ; ðv2 ; v02 Þ; y; ðvn ; v0n Þ is any set of pairwise orthogonal v-directions at GðV Þ; then we can form the graph GðV Þ þ

n X ðvi ; v0i Þ ¼ GðV  fv1 ; v2 ; y; vn g þ fv01 ; v02 ; y; v0n gÞ: i¼1

Theorem 14.1. Let GAVGðX Þ; and let D ¼ fðv1 ; v01 Þ; ðv2 ; v02 Þ; y; ðvnþ1 ; v0nþ1 Þg denote a set of n þ 1 pairwise orthogonal directions at G. If f is an vertex-finite-type invariant of order n, then ! X X jIj 0 ð1Þ f G þ ðvi ; vi Þ ¼ 0: ð14:1Þ IDf1;2;3;y;nþ1g

iAI

This result can be proved in the same manner as Theorem 8.1. In Section 8, we prove a converse to Theorem 8.1. However, it is not so easy to phrase a converse to Theorem 14.1. For example, suppose we attempt to prove the following converse to Theorem 14.1: If f : VGðX ÞN -R is an invariant (i.e. f assigns the same number to isomorphic graphs) and satisfies (14.1) for every G and every set D of n þ 1 vdirections, then f is the restriction to VGðX ÞN of a vertex-finite-type invariant of order n: The first step might be to construct an extension to VGðX ÞN1 (see our construction of g in the proof of Theorem 8.2). However, if we follow the construction in that case (see (8.4)) and try to define gðGðV ÞÞ; for jV j ¼ N  1; to be some average of the f ðGðV þ vÞÞ’s as v ranges over the vertices not in V ; then there seems to be no reason to expect the resulting g to be an invariant.

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R. Forman / Advances in Mathematics 186 (2004) 181–228

Acknowledgments This work was partially supported by the National Science Foundation and the National Security Agency. The author is extremely grateful to the anonymous referee for his thoughtful comments, and especially for suggesting the new proof of Theorem 6.9, that greatly improved the presentation of this work.

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