Reconstruction of a Pair of Graphs from their Concatenations

A concatenation of two vertex disjoint graphs is defined to be the graph obtained by identifying a vertex of one graph with a vertex of the other graph. We show that an arbitrary pair of connected graph can be uniquely reconstructed from the set of all their distinct concatenations, with only trivial exception. 1. The reconstruction problem. Let G1 and G2 be two vertex disjoint graphs, and let xi(i = 1 ,2) be an arbitrary vertex in Gi. The graph G(x1x2) obtained by merging the vertices x1 and x2 into a single vertex is called the concatenation of G1 and G2 at the points x1 and x2. The reconstruction problem after concatenation is to determine the original pair of graphs g1 and G2 given the family of all concatenations G(x1x2) obtained by varying the vertices x1 and x2 in the respective graphs. As in other graph reconstruction problems, it is understood that the vertices in G(x1x2) are unlabeled, and that the concatenated vertex in G(x1x2) is not distinguishable as such from other nodes. We show that with only trivial exceptions any pair of connected graphs is uniquely determined from its concatenations. The reconstruction theorem given here requires only that we know the distinct nonisomorphic G(x1x2)’s, and not how many times each concatenation appears among all concatenations. The original graph reconstruction problem [12] states that, for all graphs with three or more vertices, G can be uniquely reconstructed from the family of graphs {G\x}, G\x being the subgraph of G obtained by deleting the vertex x and all edges incident with x. This particular reconstruction property has been established for a large set of graphs including trees, outer planar graphs, unicyclic graphs and disconnected graphs [1]-[7]. Graph reconstructions from elementary contractions and elementary partitions were considered in [8]-[10]. In all cases, the proofs depend heavily on the particular structural properties assumed for the graphs. No unified technique is yet available for general graph reconstruction problems. 2. Preliminaries. Let G = (V, E) be a connected graph (unless otherwise specified. All graphs discussed in this paper are assumed to be connected). For V v∈ , define the cutdegree of v, denoted cd(v), to be the number of connected components in the induced subgraph G\v on V\{v}. v is a cutvertex if 1 ) ( > v cd . (Note that if G is a tree then the cutdegree of a vertex is the same as its ordinary degree.) A block of G is a maximal subgraph without cutvertices . A limb (at a vertex v) is a component of G-v together with v and all the edges in E joining v to that component. A rooted graph is a pair (G, v), where G = (V, E) is a graph and V v∈ . Two rooted graphs (G1, v1) , (G2, v2) are isomorphism if there is a graph isomorphim 2 1 : G G → φ such that 2 1) ( v v = φ . The concatenation of two rooted graphs (G1,v1) and (G2, v2), denote by (G1,v1) ○ (G2, v2), is the rooted graph (G(v1v2),v1v2), the concatenated vertex being denoted by v1v2. For a graph G, define the sequence ....) , ( ) ( 2 1 ~ α α α = G by α1 = αi(G) = number of vertices of cutdegree i (if G is trivial, α(G)=(0, 0, ...)). Define ∆(G) = max{I:αi≠ 0}; ∆(G) = 1 if G is a block. Thus, if G = (G1,v1) ○ (G2, v2), we have )) ( ) ( ( ) ( ) ( ) ( 2 1 ~ 2 ~ 1 ~ ~ v cd v cd e G G G + + + = α α α * Received by the editors July 28, 1978, and in final revised form November 15, 1979. † Logicon, Inc., Lexington, Massachusetts 02173. Now at Bell Laboratories, Murray Hill, New Jersey 07974. ‡ Department of Mathematics, Karnataka University, Dharwar Karnataka 580 003, India § Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. The work of this author was supported in part by the office of Naval Research under Contract N00014-76-C-0366. ** Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. The work of this author was supported in part by a National Science Foundation Graduate Fellowship.