The Solitaire Clobber game and the correducibility of k-connected graphs

Abstract

Clobber is a two-player combinatorial game played on a graph with black and white stones. The Solitaire Clobber game is a single-player variant of Clobber that was introduced by Demaine et al. (2004). In the initial position, a black or white stone is placed at each vertex of an undirected graph. A player picks up a stone and clobbers a stone of different color located at an adjacent vertex; the clobbered stone is replaced by the stone that has been moved. Dantas et al. (2020) defined the $k$-correducibility of a graph as follows: a graph $G$ is $k$-correducible if for every non-monochromatic initial configuration of stones on $G$ and every subset $S$ of $V\left(G\right)$ with $\left|S\right|\le k$, there exists a procedure for moving stones such that $S$ is empty of stones. Dantas et al. (2020) showed that a graph is $\ell$-correducible if and only if it is $\ell$-connected for $\ell =1,2$ and every $\left(k+\lceil \frac{k}{2}\rceil \right)$-connected graph is $k$-correducible for $k\ge 1$.

We improve their result and show that every $k$-connected graph is $k$-correducible for $k\ge 1$. Moreover, our connectivity condition is sharp, that is, there are infinitely many $(k-1)$-connected graphs that are not $k$-correducible.