Supercards, Sunshines and Caterpillar Graphs

TL;DR

This paper explores the relationship between certain graph structures called supercards and maximum common subgraphs, providing bounds and characterizations for specific graph classes like sunshine and caterpillar graphs.

Contribution

It introduces the concept of maximum saturating sets in supercards and characterizes their automorphism groups for sunshine and caterpillar graphs, establishing bounds on common cards.

Findings

Maximum saturating sets relate to permutations of supercard vertices.

For large graphs, these sets contain automorphisms isomorphic to cyclic or dihedral groups.

Bound b(G,H) <= 2(n+1)/5 is established for large n.

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.