Token sliding independent set reconfiguration on block graphs

TL;DR

This paper presents a polynomial time algorithm for reconfiguring independent sets via token sliding on block graphs, extending previous results from trees to a broader class of graphs.

Contribution

It introduces the first polynomial time algorithm for independent set reconfiguration on block graphs, generalizing the known algorithm for trees.

Findings

Polynomial time algorithm for block graphs

Extension of tree algorithms to block graphs

First such algorithm for this class of graphs

Abstract

Let $S$ be an independent set of a simple undirected graph $G$. Suppose that each vertex of $S$ has a token placed on it. The tokens are allowed to be moved, one at a time, by sliding along the edges of $G$, so that after each move, the vertices having tokens always form an independent set of $G$. We would like to determine whether the tokens can be eventually brought to stay on the vertices of another independent set $S'$ of $G$ in this manner. In other words, we would like to decide if we can transform $S$ into $S'$ through a sequence of steps, each of which involves substituting a vertex in the current independent set with one of its neighbours to obtain another independent set. This problem of determining if one independent set of a graph ``is reachable'' from another independent set of it is known to be PSPACE-hard even for split graphs, planar graphs, and graphs of bounded treewidth. Polynomial time algorithms have been obtained for certain graph classes like trees, interval graphs, claw-free graphs, and bipartite permutation graphs. We present a polynomial time algorithm for the problem on block graphs, which are the graphs in which every maximal 2-connected subgraph is a clique. Our algorithm is the first generalization of the known polynomial time algorithm for trees to a larger class of graphs (note that trees form a proper subclass of block graphs).