The neighborhood inclusion structure of a graph

M&l. Comput. Modding Vol. 17,No. 11,pp. 75-28,1993 Printed in Great Britain. All rights reserved

THE

0895-7177/93 $6.00+0.00 Copyright@ 1993 Pergamon Press Ltd

NEIGHBORHOOD INCLUSION OF A GRAPH

STRUCTURE

FRANK BoEscH+, CHARLES SUFFEL~ AND RALPH

TINDELL

Stevens Institute of Technology Hoboken, NJ 07030, U.S.A. FRANK

HARARY*

New Mexico State University Las Cruces, NM 88003, U.S.A. Abstract-Motivated by connections with problems in network reliability, we explore the relationship between vertex neighborhoods N, in a graph, focusing on neighborhood equality and inclusion (suitably modified for adjacent vertices). We survey recent work which shows that solutions to certain reliability extremal problems must be graphs in which the neighborhood inclusion relation is total; such graphs are known in the literature as threshold graphs. We then define a mixed graph M(G) determined by the neighborhood inclusion relation on the vertices of a graph G and also a digraph D(G) obtained from h/l(G) by identifying points with the “same” neighborhood in G. We characterize M(G) and D(G) for threshold graphs and for trees.

1. THRESHOLD

GRAPHS

AND

RELIABILITY

EXTREMAL

PROBLEMS

The neighborhood of a point u in graph G, denoted by N,(G) (or just NU), is the set of points adjacent to u in G. Notice that N,(G) is a deleted neighborhood in that u $ N,G. The neighborhood equivalence relation =_N on G is the equivalence relation on the vertex set V of G in which u 3~ v iff N,\(v) = N,\(u). The strict subneighborhood relation
graphs were introduced

by Chv&tal and Hammer

in 1973 [2]. Recently,

graphs

these graphs

were found to play an important role in the study of two extremal problems arising in the theory of network reliability. The first problem is to minimize the number of spanning trees in a connected graph with a fixed number

n of nodes and e of edges; a solution

to this problem

is called a min-t

then G has a unique nontrivial

block, and that

graph. 1.1. [See [31.) If G is a min-t graph, block is a threshold graph.

THEOREM

Theorem 1.1 is only a partial statement of the result of Bogdanowicz and Boesch, in that they show that the nontrivial block of a min-t graph is unique and give its explicit structure. The second extremal problem considered here is that of finding a graph with the maximum number of length-2 paths among graphs with the same number of nodes and edges. These graphs are studied in the paper [4] by Ahlswede and Katona, where it is shown that one of two graph constructions always yields an optimal graph. We next note that the optimal graphs for this problem are threshold graphs. Research supported N00014-90-J-1860*.

in part by NSF grant ECS-8600782+,

ONR contract N00014-87-K-0376$,

and ONR contract

Typeset 25

by &+S-‘&X

F. BOESCH, et al.

26

THEOREM 1.2. If G has the maximum number of length-2 paths among all graphs with the same number of nodes and edges, then G is a threshold graph. PROOF. First we note that we may count the number Pz(G) of length-2 paths in G via their midpoints. To obtain a length-2 path with midpoint u, we choose as endpoints of the path any pair of vertices adjacent to u. Since there are exactly d,(d, - 1)/2 such pairs, the total number of length two paths in G is the sum over all vertices u of d,(d, - 1)/2. Now suppose that G is not a threshold graph, which is to say that there are vertices u, v, and w such that d, 5 d, and w is adjacent to u but not to u. Construct a new graph G’ from G by removing the edge from u to w and adding an edge from w to v. Note that the only vertices whose degrees have changed are u and v, namely dU(G’) = d,(G) - 1 and d, (G’) = d,(G) + 1. It is then easy to see that &(G’) = P,(G) + d, - (d, - 1) = Pz(G) + (d, - d,) + 1. Since d, - d, > 0, G’ has more length-2 paths than G, a contradiction. 2. THE NEIGHBORHOOD

INCLUSION

MIXED

GRAPH

AND DIGRAPH

OF G

The basic properties of the relations =_N,
The relation =_N on the vertices of a graph G is an equivalence relation. The relation
or

edgeless.

If G = (V, E) is a graph, the neighborhood quotient graph G/ EN of G has as vertices all the EN-classes [u] of vertices u of G. Distinct equivalence classes [u], [v] are adjacent in G/ =_N iff u and u are adjacent in G. While it is tempting to claim that no two points of G/ 3~ have the same neighborhood, this is not so. If G = Kr,s + z (x a new edge), then G/ =_N is Ki,z and the two degree-one points of K 1,~ have the same neighborhood. A graph G is said to be N-irreducible if G and G/ EN are isomorphic. It is clear that for every graph G there is a unique N-irreducible graph pN(G) obtainable from G by a sequence of quotients by the relation =_N. th e neighborhood reduction, or N-reduction, of G. The graph G = Kr,a + x has We call IN successive quotients KI,~, Ka, and K1, with K1 = plv(G). Threshold graphs are characterized in terms of their neighborhood reductions in Corollary 2.2 below. Corresponding to any graph G we have an associated neighborhood inclusion mixed graph M(G) : M(G) has the same point set as G, has a directed edge from u to v if an only if u
is the transitive

is

inclusion

closure of a directed

path.

The characterization of M(G) for threshold graphs G now requires only that we characterize the equivalence class size function !&. It is clear from definitions that in a threshold graph,

Neighborhood inclusionstructure

27

u EN u if and only if d, = d,. Thus we define the degree partition of the vertex set of a graph G as follows: if the distinct positive degrees of G are ordered by 0 < dl < . . < d,, we define DO to be the set of isolated points of G and Di to be the set of vertices of degree di, 1 < i 5 m. Note that DO may be empty. We will refer to the sets Di as the degree partition sets of G. We may now state the following structure theorem for threshold graphs. (See (21.) If Do, DI, . . , D, is the degree partition of the vertex set of graph G, then G being a threshold graph is equivalent to the condition that u E Di and u E Dj are adjacent inGifandonJyifi+j>m.

THEOREM 2.1.

One of the fascinating consequences of the structure theorem for threshold graphs is that a connected threshold graph is uniquely determined by either its set of degrees or by the cardinalities ni = lDi[ of the sets in its degree partition. While every set of positive integers is (uniquely) realizable as the degree set of a connected threshold graph, there is a minor restriction on the realizable sequences no, nl, . . . , nm of degree partition set cardinalities. The following result is implicit in the discussion of threshold graphs in Golumbic’s book [I]. Let no, nl, . . . , n, be an integer sequence such that no > 0 and ni > 1 if 1 5 i 5 m. Then there exists a threshold graph whose degree partition sets satisfy 1Dil = ni, 0 < i 5 m, iff m = 0 or nrm,2, > 1. Moreover, if such a threshold graph exists, it is unique. THEOREM 2.2.

PROOF. Necessity: Let no, nl, . . . , n, be the cardinalities of the degree partition sets of a threshold graph G. We wish to show that m # 0 implies that nrrn/21 > 1. Let r = [m/2] and consider vertices u E D, and u E Dr+l. By the definition of degree partition, d, # d,. We will show that nrrni21 > 1 by showing that otherwise d, = d,. Thus, assume that nrrn/21 = 1. If m is odd, then [m/2] = r + 1 and N, = N, U (D,+l - w). But Dr+l = 21,N, = N,, and thus d, = d,, a contradiction. If m is even, then u and 2, are adjacent, [m/21 = r and N, = (N, - u) U D,. Since D, = u, N,\u = N,\v and thus d, = d,, which again is a contradiction. Suficiency: Suppose that a sequence no, nl, . . . , n, is given with no > 0 and ni 2 1 if 1 < i 5 m. If m = 0, then the edgeless graph with no points is a threshold graph with degree partition DO. Next assume m > 0 and nr,i21 > 1. Choose pairwise disjoint sets DO, D1,. . . , D, such that 1Dil = ni, 0 5 i 5 m. Let G be the graph with vertex set V = U Di and adjacencies given by condition 2 of Theorem 2.1. It then follows from Theorem 2.1 that G is the only possible graph that could be a threshold graph with degree partition set cardinalities no, nl,. . . , n,. Moreover, G is in fact a threshold graph if and only if DO, D1, . . . , D, is the degree partition of G. It is clear that NO is the set of isolated points of G. We next consider the neighborhoods of vertices from V\Do. To that end, let Vi = Dm-i+l U Dm-i+a U.. . U D, for each i, 1 < i < m. IfuEDiandlIiIlm/21,thenu~ViandN,=Viandhenced,=IV,I. IfuEDiand lm/2] < i 5 m, then u E Vi and N, = Vi\(U); thus d, = /Vii- 1. It then follows that all points in Di have the same degree di in G, 1 < i 5 m. Moreover, di+l - di = n,_l > 0 if i < lm/2] or i > [m/2], and d lrnp~+l - dLrnpJ = n,-Lrnpl - 1= ny,p - 1 > 0. Thus, we have shown that di < d2 < ’ ’ ’ < d,, and hence, DO, D1, . . . , D, is the degree partition of G. COROLLARY 2.1.

M(G)

If H is a connected graph and G is a connected threshold graph, then M(H) implies H = G.

COROLLARY 2.2.

If G is a threshold graph then IN

=

= KI.

PROOF. It is simple to see from definitions that if G is a threshold

graph, then G/ EN is a threshold graph. Therefore, we need only show that K1 is the only N-irreducible threshold graph. If DO, D1, . . . , D, is the degree partition of an N-irreducible threshold graph G, then \ni\= 1 for 0 I i 5 m. By Theorem 2.2, this is possible if and only if m = 0, which is to say that G= KI. The converse of Corollary 2.2 is not valid, as may be seen by considering the “bowtie” graph &\C4.

The mixed graphs realizable as M(G) for some connected terized by combining Proposition 2.2 and Theorem 2.2. We terization of the digraphs which are neighborhood inclusion characterization has been found (see [5]), the statement is

threshold graph G are easily characconclude by considering t,he &aracdigraphs of trees. While a complete quite complicated. For this reason,

F. BOESCH, et al.

28

we restrict ourselves here to the special case of weakly connected digraphs. Detailed proofs are omitted. We will denote by D,.,, the unidirectional complete bipartite graph, which is to say the orientation of K,,, in which every edge is directed from the part with cardinality I- to the part with cardinality s. In our usage, a cycle in a digraph need not be a directed cycle. 2.3. The neighborhood inclusion digraph D(T) of a tree T is weakly connected iff every edge of T lies on a path whose endpoints are leaves and whose length is at most 3. PROPOSITION

THEOREM 2.3. Let D be a weakly connected digraph with at least one arc. Then D = D(T) some tree if and only if all the following conditions are satisfied.

1. 2. 3. 4.

for

D is a spanning subdigraph of Da,a+b for some a 2 1 and some b 2 0.

induced cycle of D has length 4. The intersection of distinct 4-cycles in D is either empty or a single arc. For every vertex u of positive outdegree, u lies on a 4cycJe iff u is adjacent to no vertex of indegree 1. 5. If vertex v lies on a 4-cycle C, then for every in-edge x of u there is a 4-cycle C, containing x. Every

To prove the sufficiency of conditions 4, a tree T is constructed with the same vertex set as D. The leaves of this tree are the vertices of positive outdegree in D. The first problem is to identify, for each leaf u of T, the unique point n(u) adjacent in T to U. If u is adjacent to some indegree-l point v in D, then r(u) = V. Otherwise, u lies on at least one 4-cycle in D. The vertex chosen as r(u) is then a vertex of outdegree 0 which lies on all 4-cycles containing u. The details of the proof of Theorem 2.2, as well as the characterization of D(T) for arbitrary trees may be found in the technical report [5]. The next result follows easily from Theorem 2.3. 2.4. Let H be a weakly connected digraph having at least one arc and satisfying conditions (l)-(5) of Theorem 2.3. Let v be a function from the vertices of H into the positive integers. Then there is a tree with H = D(T) and u = VT if and only if (1) for every vertex of u of outdegree 0 in H, V(U) = 1; and (2) if H = D I,1 and u is the vertex of H with outdegree 1, then V(U) 2 2. THEOREM

Addendum Subsequent to writing this paper, we discovered that Peled and Srivinasan [6] have also studied the subneighborhood digraph of a tree as a partial order (the “vicinal order”). They give necessary and sufficient conditions for a given partial order on the vertex set of a given tree T to be the vicinal order obtained from T. This is not quite the same problem considered in Theorem 2.3 of this paper, since the digraph D is given (but not a candidate tree) and we must decide if there exists a tree T with D = D(T). REFERENCES Graph Theory and Perfect Graphs, Academic Press, (1980). 1. M. Golumbic,Algorithmic 2. V. Chvatal and P.L. Hammer, Set-packing and threshold graphs, Univ. Waterloo Res. Report 73-21, Univ. of Waterloo,

Waterloo,

Canada, (1973). The minimum number of spanning trees in a connected

3. F.T. Boesch and Z. Bogdanowicz,

communication). 4. R. Ahlswede and G.O.H. Katona, Graphs Acad. Sci. Hangar. 32, 97-120 (1978). 5. F.T. Boesch, C.L. Suffel and R. Tindell, Report, Stevens Institute, Hoboken, NJ, 6. U.N. Peled and M.K. Srinivasan, Vicinal Rep. 23, Chicago, (1987).

graph, (verbal

with the maximum number of adjacent pairs of edges, Acta Math. The neighborhood inclusion digraphs of trees, Stevens Technical (August 1987). order of trees, University of Illinois at Chicago Comp. Sci. Res.