Recognizing trees from incomplete decks

For a given graph, the unlabeled subgraphs $G-v$ are called the cards of $G$ and the deck of $G$ is the multiset $\{G-v: v \in V(G)\}$. Wendy Myrvold [Ars Combinatoria, 1989] showed that a non-connected graph and a connected graph both on $n$ vertices have at most $\lfloor \frac{n}{2} \rfloor +1$ cards in common and she found (infinite) families of trees and non-connected forests for which this upper bound is tight. Bowler, Brown, and Fenner [Journal of Graph Theory, 2010] conjectured that this bound is tight for $n \geq 44$. 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. Based on this theorem, it follows that any forest and non-forest also have at most $\lfloor \frac{n}{2} \rfloor +1$ common cards. Moreover, we have classified all except finitely many pairs for which this bound is strict. 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.