Impartial Solitaire Clobber played on Powers of Paths

Impartial Solitaire Clobber (ISC) is a variant of the two-player game Clobber introduced by Albert et al. in 2002. It is a one-player game with the following rules: given a graph G = ( V , E ) , for each vertex v ∈ V , we assign a black or a white stone. A move consists in picking up a stone and clobbering another one of the opposite color located on an adjacent vertex. The clobbered stone is removed from the graph and it is replaced by the picked one. The goal is to find a succession of moves that minimizes the number of the remaining stones. In this paper, we study the ISC when G is a power of a path, and we prove that any non-monochromatic configuration of stones can be reduced to a single stone for a r-power of a path, r ⩾ 3 and some circulant graphs.