Recognizing Trees From Incomplete Decks

Given a graph G, the unlabeled subgraphs G - ν are called the cards of G. The deck of G is the multiset {G - ν: ν ∈ V (G)}. Wendy Myrvold showed that a disconnected graph and a connected graph both on n vertices have at most⌊5-2⌋ +  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 ≥  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 ⌊n-2⌋ + 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  of its cards. On the basis of this theorem, it follows that any forest and nonforest also have at most  common cards. Furthermore, the main ideas of the proof for trees are used to show that the girth of a graph on  vertices can be determined based on any 2n/3 + 1 of its cards. Lastly, we show that any 5n/6 + 2 cards determine whether a graph is bipartite.