Supercards, sunshines and caterpillar graphs

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 $b(G,H) \leq \frac{2(n+1)}{5}$ 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.

Keywords

graph reconstruction, reconstruction numbers, vertex-deleted subgraphs, supercards, graph automorphisms, sunshine graph, caterpillar graph

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Received 24 January 2019

Accepted 8 January 2021

Published 31 January 2022