A new approach to graph reconstruction using supercards

Abstract

The vertex-deleted subgraph G − v, obtained from the graph G by deleting the vertex v and all edges incident to v, is called a card of G. The deck of G is the multiset of its unlabelled vertexdeleted subgraphs. The number of common cards of G and H is the cardinality of a maximum multiset of common cards, i.e., the multiset intersection of the decks of G and H. We introduce a new approach to the study of common cards using supercards, where we define a supercard G + of G and H to be a graph that has at least one vertex-deleted subgraph isomorphic to G, and at least one isomorphic to H. We show how maximum sets of common cards of G and H correspond to certain sets of permutations of the vertices of a supercard, which we call maximum saturating sets. We then show how to construct supercards of various pairs of graphs for which there exists some maximum saturating set X contained in Aut(G + ). For certain other pairs of graphs, we show that it is possible to construct G + and a maximum saturating set X such that the elements of X that are not in Aut(G + ) are in oneto-one correspondence with a set of automorphisms of a different supercard G + λ of G and H. Our constructions cover nearly all of the published families of pairs of graphs that have a large number of common cards.

Supercards, sunshines and caterpillar graphs

“…In [9], Brown gave an intricate proof that, for large n, the number of common cards between a sunshine graph and a caterpillar of order n is at most 2 n+1 5 and, moreover, that this bound is only attained by a unique family of pairs of graphs for which n ≡ 4 (mod 5). In this paper, we prove this result using supercards, a new approach to the study of the maximum number of common cards that we introduced in [8].…”

“…So it follows from Corollary 3.4 that the possible skeletons of T − λ(w) are determined by τ U + (x ν ). In addition, τ U (x µ ) = τ U + (x ν ) by (8). It then follows from Lemma 3.1 that the possible skeletons of U − x µ are also determined by τ U + (x ν ).…”

“…We now recall the main definitions and results in the theory of supercards. Detailed explanations and proofs of the results can be found in [8].…”

“…A supercard of G and H is a graph G + , the deck of which contains at least one card isomorphic to G and at least one card isomorphic to H. In [8], we showed the existence of certain subsets of S V (G + ) of cardinality b(G, H), the elements of which are in one-to-one correspondence with the elements of D(G) ∩ D(H). We called these subsets maximum saturating sets.…”