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Edge-Reconstruction of Graphs With Topological Properties
Annals of Discrete Mathematics 17 (1983) 285-288 @ North-Holland Publishing Company
EDGE-RECONSTRUCTION OF GRAPHS WITH TOPOLOGICAL PROPERTIES S. FIORINI and J. LAURI The University of Malta, Malta
1. Introduction
A graph G is said to be edge-reconstructible if G is uniquely determined (up to isomorphism) from the collection (called the d e c k ) 9 ( G ) = {G - e : e E E ( G ) } of edge-deleted subgraphs of G. In this paper we deal with two kinds of graphs having certain topological properties. In Section 2 we outline how all 4-connected planar graphs are edge-reconstructible. In the last section we deal with the edge-reconstruction of certain graphs which triangulate surfaces.
2. 4-Connected planar graphs
In this section, we let G be a 4-connected planar graph. When the minimum valency 6 ( G )is 5 , reconstruction follows by [3], so that we can assume 6 ( G )to be 4. Since G - e is 3-connected for every edge e, then by Whitney’s well-known theorem, G - e has a unique embedding in the sphere. It follows that the sequence of face-valencies in non-increasing order in each G - e is uniquely determined. In fact, more can be said. Proposition 1. The face-valency sequence of G is reconstructible from 9 ( G ) . Corollary. Let G - e E 9( G ) .If e is incident in G with faces F, F’, then the pair of face -valencies p ( F ) , p ( F ‘ ) is reconstructible from 9( G ) . The main result in this section is to prove the following. Theorem A. If G is a 4-connected planar graph, then G is edge-reconstructible. 285
286
S. Fiorini, J . Lauri
To this end, we let u be a 4-vertex incident to consecutive faces F, with face-valencies p ( E ) = a, + 2 ( i = 0,1,2,3). Then the wheel-sequence W ( u )of u is (ao,a l ,a 2 ,a3).The conclusion of Theorem A hinges on certain auxiliary results on wheel-sequences, the central one being the following. Proposition 2. Let G have a 4-vertex which has a repeated term in its wheelsequence. Then G is edge -reconstructible.
3. Two reconstruction techniques In establishing the results in Section 2, use is made of the following two techniques. (i) It is easily shown (using wheel-sequences) that if a 4-connected planar graph G is not edge-reconstructible, then there exists exactly one edgereconstruction H of G not isomorphic to G. The graph H can be represented as G - vx + uy, where u is a 4-vertex. Given a fixed 4-vertex u and using successive representations of this type, we generate a sequence of graphs (Go,G I ,Gz,. . .) such that Gz, = G and G21+I =H. Under certain conditions we obtain the contradiction that G is isomorphic to G21+Ifor some j. (ii) In trying to establish that a graph G (no longer necessarily planar and 4-connected) is edge-reconstructible, it is possible sometimes to consider certain ‘configurations’ such that if G contains any one of them then G is edgereconstructible. Often, such a set of configurations is ordered in a sequence (C,, CI,C,,. . . ), called a reconstructor sequence, in such a way that the proof that the configuration Ci gives reconstruction of G depends on the fact that previous configurations Cj (j< i ) also give reconstruction of G. The configurations which we use to prove the results in Section 2 are wheel-sequences. This technique, introduced by Hoffman [7], was used by Caunter [2] and by Swart [9] in the different context of the edge-reconstruction of bidegree graphs. These same two techniques, mutatis mutandis, are also the principal tools we use in establishing the results in the following section.
4. Graphs which triangulate surfaces In this context, we deal with the following two theorems. Theorem B. If a graph has connectivity 3 and triangulates some surface, then it is edge -reconstructible.
Edge-reconstruction of graphs with topological properties
287
Theorem C . Any graph which triangulates the real projective plane is edgereconstructible. The conditions of 4-connectedness and planarity in Section 2 allow us to make heavy use of embedding properties and of Kuratowski’s characterization of planar graphs. In general, these are no longer available for surfaces of higher genera. Hence, to prove Theorem B we have to use methods which do not involve considerations of embeddings. We do this by actually edgereconstructing a class % of graphs wider than the class of graphs which triangulate surfaces and have connectivity 3. The class %’ is the class of graphs with connectivity 3 and such that if G E % and a, b, c are three vertices whose deletion disconnects G, then a7 b, c induce a 3-circuit in G. The only connecting link between the graph theoretic properties of the graphs in % and the topological properties of graphs which triangulate a surface is that for any vertex v of a graph G which triangulates some surface, the subgraph of G induced by the neighbours of v is Hamiltonian. Whereas to prove Theorem B we are able to bypass all considerations of embeddings by working with the class %, to prove Theorem C an important part is played by the topological properties of the graphs under consideration. For example, to solve the problem of edge-recognition (i.e.7 to show that we can determine from 9(G) whether or not G triangulates the real projective plane), we make use of the Kuratowski-type theorem of Archdeacon [l]and Glover et al. [6] which characterizes projective graphs in terms of a set of 103 ‘forbidden subgraphs’. A simple example of a reconstructor sequence for graphs G with minimum valency 6 used to prove Theorems B and C is shown in Fig. 1, where labels correspond to valencies of the vertices in G. 6
6+1
s
6+1
Fig. 1
The above results appear in full detail in [4] and [5]. During the course of this conference, Lovisz [8] pointed out that our main theorem of Section 2 also follows from the following result he had just proved.
Theorem (Lovisz). If a graph with average valency greater than 4 and with a sufficiently large number of vertices has a Hamiltonian path, then it is edgereconstructible.
288
S. Fiorini, J. Lauri
References [l] D.S. Archdeacon, A Kuratowski theorem for the projective plane, Doctoral Thesis, Ohio State University (1980). [2] J. Caunter, Private communication. [3] S. Fiorini, On the edge-reconstruction of planar graphs, Math. Proc. Cambridge Phil. SOC.83 (1977) 31-35. [4] S. Fiorini and J. Lauri, Edge-reconstruction of 4-connected planar graphs, J. Graph Theory 6 (1982) 33-42. (51 S. Fiorini and J. Lauri, On the edge-reconstruction of graphs which triangulate surfaces, Quart. J. Math. Oxford 33 (2) (1982) 191-214. [6] H.Glover, J. Huneke and C. Wang, 103 graphs that are irreducible for the projective plane, J. Comb. Theory, Ser. B 27 (1979) 332-370. [7] D.G. Hoffman, Notes on edge-reconstruction of bidegree graphs, unpublished. (81 L. Lovisz, Private communication. [U] E. Swart, The edge-reconstructibility of planar bidegree graphs, University of Waterloo Res. Rept. Corr. 78-44 (1978).
EDGE-RECONSTRUCTION OF GRAPHS WITH TOPOLOGICAL PROPERTIES S. FIORINI and J. LAURI The University of Malta, Malta
1. Introduction
A graph G is said to be edge-reconstructible if G is uniquely determined (up to isomorphism) from the collection (called the d e c k ) 9 ( G ) = {G - e : e E E ( G ) } of edge-deleted subgraphs of G. In this paper we deal with two kinds of graphs having certain topological properties. In Section 2 we outline how all 4-connected planar graphs are edge-reconstructible. In the last section we deal with the edge-reconstruction of certain graphs which triangulate surfaces.
2. 4-Connected planar graphs
In this section, we let G be a 4-connected planar graph. When the minimum valency 6 ( G )is 5 , reconstruction follows by [3], so that we can assume 6 ( G )to be 4. Since G - e is 3-connected for every edge e, then by Whitney’s well-known theorem, G - e has a unique embedding in the sphere. It follows that the sequence of face-valencies in non-increasing order in each G - e is uniquely determined. In fact, more can be said. Proposition 1. The face-valency sequence of G is reconstructible from 9 ( G ) . Corollary. Let G - e E 9( G ) .If e is incident in G with faces F, F’, then the pair of face -valencies p ( F ) , p ( F ‘ ) is reconstructible from 9( G ) . The main result in this section is to prove the following. Theorem A. If G is a 4-connected planar graph, then G is edge-reconstructible. 285
286
S. Fiorini, J . Lauri
To this end, we let u be a 4-vertex incident to consecutive faces F, with face-valencies p ( E ) = a, + 2 ( i = 0,1,2,3). Then the wheel-sequence W ( u )of u is (ao,a l ,a 2 ,a3).The conclusion of Theorem A hinges on certain auxiliary results on wheel-sequences, the central one being the following. Proposition 2. Let G have a 4-vertex which has a repeated term in its wheelsequence. Then G is edge -reconstructible.
3. Two reconstruction techniques In establishing the results in Section 2, use is made of the following two techniques. (i) It is easily shown (using wheel-sequences) that if a 4-connected planar graph G is not edge-reconstructible, then there exists exactly one edgereconstruction H of G not isomorphic to G. The graph H can be represented as G - vx + uy, where u is a 4-vertex. Given a fixed 4-vertex u and using successive representations of this type, we generate a sequence of graphs (Go,G I ,Gz,. . .) such that Gz, = G and G21+I =H. Under certain conditions we obtain the contradiction that G is isomorphic to G21+Ifor some j. (ii) In trying to establish that a graph G (no longer necessarily planar and 4-connected) is edge-reconstructible, it is possible sometimes to consider certain ‘configurations’ such that if G contains any one of them then G is edgereconstructible. Often, such a set of configurations is ordered in a sequence (C,, CI,C,,. . . ), called a reconstructor sequence, in such a way that the proof that the configuration Ci gives reconstruction of G depends on the fact that previous configurations Cj (j< i ) also give reconstruction of G. The configurations which we use to prove the results in Section 2 are wheel-sequences. This technique, introduced by Hoffman [7], was used by Caunter [2] and by Swart [9] in the different context of the edge-reconstruction of bidegree graphs. These same two techniques, mutatis mutandis, are also the principal tools we use in establishing the results in the following section.
4. Graphs which triangulate surfaces In this context, we deal with the following two theorems. Theorem B. If a graph has connectivity 3 and triangulates some surface, then it is edge -reconstructible.
Edge-reconstruction of graphs with topological properties
287
Theorem C . Any graph which triangulates the real projective plane is edgereconstructible. The conditions of 4-connectedness and planarity in Section 2 allow us to make heavy use of embedding properties and of Kuratowski’s characterization of planar graphs. In general, these are no longer available for surfaces of higher genera. Hence, to prove Theorem B we have to use methods which do not involve considerations of embeddings. We do this by actually edgereconstructing a class % of graphs wider than the class of graphs which triangulate surfaces and have connectivity 3. The class %’ is the class of graphs with connectivity 3 and such that if G E % and a, b, c are three vertices whose deletion disconnects G, then a7 b, c induce a 3-circuit in G. The only connecting link between the graph theoretic properties of the graphs in % and the topological properties of graphs which triangulate a surface is that for any vertex v of a graph G which triangulates some surface, the subgraph of G induced by the neighbours of v is Hamiltonian. Whereas to prove Theorem B we are able to bypass all considerations of embeddings by working with the class %, to prove Theorem C an important part is played by the topological properties of the graphs under consideration. For example, to solve the problem of edge-recognition (i.e.7 to show that we can determine from 9(G) whether or not G triangulates the real projective plane), we make use of the Kuratowski-type theorem of Archdeacon [l]and Glover et al. [6] which characterizes projective graphs in terms of a set of 103 ‘forbidden subgraphs’. A simple example of a reconstructor sequence for graphs G with minimum valency 6 used to prove Theorems B and C is shown in Fig. 1, where labels correspond to valencies of the vertices in G. 6
6+1
s
6+1
Fig. 1
The above results appear in full detail in [4] and [5]. During the course of this conference, Lovisz [8] pointed out that our main theorem of Section 2 also follows from the following result he had just proved.
Theorem (Lovisz). If a graph with average valency greater than 4 and with a sufficiently large number of vertices has a Hamiltonian path, then it is edgereconstructible.
288
S. Fiorini, J. Lauri
References [l] D.S. Archdeacon, A Kuratowski theorem for the projective plane, Doctoral Thesis, Ohio State University (1980). [2] J. Caunter, Private communication. [3] S. Fiorini, On the edge-reconstruction of planar graphs, Math. Proc. Cambridge Phil. SOC.83 (1977) 31-35. [4] S. Fiorini and J. Lauri, Edge-reconstruction of 4-connected planar graphs, J. Graph Theory 6 (1982) 33-42. (51 S. Fiorini and J. Lauri, On the edge-reconstruction of graphs which triangulate surfaces, Quart. J. Math. Oxford 33 (2) (1982) 191-214. [6] H.Glover, J. Huneke and C. Wang, 103 graphs that are irreducible for the projective plane, J. Comb. Theory, Ser. B 27 (1979) 332-370. [7] D.G. Hoffman, Notes on edge-reconstruction of bidegree graphs, unpublished. (81 L. Lovisz, Private communication. [U] E. Swart, The edge-reconstructibility of planar bidegree graphs, University of Waterloo Res. Rept. Corr. 78-44 (1978).