Supercards, sunshines and caterpillar graphs

Brown, Paul and Fenner, Trevor (2022) Supercards, sunshines and caterpillar graphs. Journal of Combinatorics 13 (1), pp. 41-78. ISSN 2156-3527.

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 cards. The number of common cards b(G, H) of G and H is the cardinality of the multiset intersection of the decks of G and H. A supercard G+ of G and H is a graph whose deck contains at least one card isomorphic to G and at least one card 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 apply the theory of supercards and maximum saturating sets to the case when G is a sunshine graph and H is a caterpillar graph. We show that, for large enough n, there exists some maximum saturating set that contains at least b(G, H) − 2 automorphisms of G+, and that this subset is always isomorphic to either a cyclic or dihedral group. We prove that b(G, H) ≤ 2(n+1) 5 for large enough n, and that there exists a unique family of pairs of graphs that attain this bound. We further show that, in this case, the corresponding maximum saturating set is isomorphic to the dihedral group.

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