Families of pairs of graphs with a large number of common cards

Abstract

AbstractThe 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 vertex‐deleted subgraphs. The number of common cards of G and H (or between G and H) is the cardinality of the multiset intersection of the decks of G and H. In this article, we present infinite families of pairs of graphs of order n ≥ 4 that have at least \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}\begin{eqnarray*}2\lfloor\frac{1}{3}(n-1)\rfloor\end{eqnarray*}\end{document} common cards; we conjecture that these, along with a small number of other families constructed from them, are the only pairs of graphs having this many common cards, for sufficiently large n. This leads us to propose a new stronger version of the Reconstruction Conjecture. In addition, we present an infinite family of pairs of graphs with the same degree sequence that have \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}\begin{eqnarray*}\frac{2}{3}(n+5-2\sqrt{3n+6})\end{eqnarray*}\end{document} common cards, for appropriate values of n, from which we can construct pairs having slightly fewer common cards for all other values of n≥10. We also present infinite families of pairs of forests and pairs of trees with \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}\begin{eqnarray*}2\lfloor\frac{1}{3}(n-4)\rfloor\end{eqnarray*}\end{document} and \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}\begin{eqnarray*}2\lfloor\frac{1}{3}(n-5)\rfloor\end{eqnarray*}\end{document} common cards, respectively. We then present new families that have the maximum number of common cards when one graph is connected and the other disconnected. Finally, we present a family with a large number of common cards, where one graph is a tree and the other unicyclic, and discuss how many cards are required to determine whether a graph is a tree. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 146–163, 2010