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

TL;DR

This paper investigates the graph reconstruction problem from $k$-decks for graphs with maximum degree 2, establishing bounds on $k$ for unique determination of such graphs and their connectivity.

Contribution

It provides new bounds on the minimum $k$-deck size needed to uniquely identify graphs with maximum degree 2 and determine their connectivity.

Findings

Graphs with maximum degree 2 are determined by their $k$-decks if cycles and paths meet certain size conditions.

For cycles, the minimal $k$ is approximately half the number of vertices.

For paths, the minimal $k$ is about half plus one the number of vertices.

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 $\lfloor n/2 \rfloor +1$.