The Solitaire Clobber game and correducibility of graphs

Abstract

The Solitaire Clobber game is a one-player combinatorial game played on a graph. A stone, black or white, is placed at each vertex of the graph. A move in the game consists in picking up a stone and clobbering another one of different color located at an adjacent vertex; the clobbered stone is replaced by the stone that has been moved. The game has been extensively investigated in relation to the problem of minimizing the number of stones remaining on the graph when no further move in the game is possible. We study a different problem: we define a graph G to be k-correducible if for every non-monochromatic initial configuration of stones on G and every subset S of V(G) of size at most k, there is a Solitaire Clobber game that empties S. For i=1,2, we show that a graph is i-correducible if and only if it is i-connected. Furthermore, for each k, we prove that each (k+⌊k2⌋)-connected graph is k-correducible.