Recognizing Trees From Incomplete Decks

Abstract

Given a graph $G$, the unlabeled subgraphs $G-v$ are called the cards of $G$. The deck of $G$ is the multiset $\left\{G-v:v\in V\left(G\right)\right\}$. Wendy Myrvold showed that a disconnected graph and a connected graph both on $n$ vertices have at most $\lfloor \frac{n}{2}\rfloor +1$ cards in common and found (infinite) families of trees and disconnected forests for which this upper bound is tight. Bowler, Brown, and Fenner conjectured that this bound is tight for $n\ge 44$. In this article, we prove this conjecture for sufficiently large $n$. The main result is that a tree $T$ and a unicyclic graph $G$ on $n$ vertices have at most $\lfloor \frac{n}{2}\rfloor +1$ common cards. Combined with Myrvold's work, this shows that it can be determined whether a graph on $n$ vertices is a tree from any $\lfloor \frac{n}{2}\rfloor +2$ of its cards. On the basis of this theorem, it follows that any forest and nonforest also have at most $\lfloor \frac{n}{2}\rfloor +1$ common cards. Furthermore, the main ideas of the proof for trees are used to show that the girth of a graph on $n$ vertices can be determined based on any $\frac{2n}{3}+1$ of its cards. Lastly, we show that any $\frac{5n}{6}+2$ cards determine whether a graph is bipartite.