Reconstruction from $k$-decks for graphs with maximum degree 2

Abstract

The $k$-deck of a graph is its multiset of induced subgraphs on $k$ vertices. We prove that $n$-vertex graphs with maximum degree $2$ have the same $k$-decks if each cycle has at least $k+1$ vertices, each path component has at least $k-1$ vertices, and the number of edges is the same. Using this for lower bounds, we obtain for each graph with maximum degree at most $2$ the least $k$ such that it is determined by its $k$-deck. For the $n$-vertex cycle this value is $\lfloor n/2 \rfloor$, and for the $n$-vertex path it is $\lfloor n/2 \rfloor+1$. Also, the least $k$ such that the $k$-deck of an $n$-vertex graph always determines whether it is connected is at least $\lfloor n/2 \rfloor +1$.