Strong reducibility of powers of paths and powers of cycles on Impartial Solitaire Clobber.

Electronic Notes in Discrete Mathematics 01/2011; 37:177-182. DOI: 10.1016/j.endm.2011.05.031
Source: DBLP

ABSTRACT We consider the Impartial Solitaire Clobber which is a one-player combinatorial game on graphs. The problem of determining the minimum number of remaining stones after a sequence of moves was proved to be NP-hard for graphs in general and, in particular, for grid graphs. This problem was studied for paths, cycles and trees, and it was proved that, for any arrangement of stones, this number can be computed in polinomial time. We study a more complex question related to determining the color and the location of the remaining stones. A graph G is strongly 1-reducible if: for any vertex v of G, for any arrangement of stones on G such that G\v is non-monochromatic, and for any color c, there exists a succession of moves that yields a single stone of color c on v. We investigate this problem for powers of paths Pnr and for powers of cycles Cnr and we prove that if r⩾3, then Pnr (resp. Cnr) is strongly 1-reducible; if r=2, then Pnr, is not strongly 1-reducible.

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    ABSTRACT: The Clobber game was introduced by Albert et al. in 2002, then its solitaire version that we are interested in was presented by Demaine et al. in 2004. Solitaire Clobber is played on a graph , by placing a stone, black or white, on each vertex of the graph. A move consists in picking up a stone and clobbering another one of opposite color located on an adjacent vertex; the clobbered stone is removed from the graph and is replaced by the picked one. The goal is to minimize the number of stones remaining when no further move is possible. We investigate a more restrictive question related to the color and the location of the remaining stones. A graph is strongly -reducible if, for any vertex , any initial configuration that is not monochromatic outside , can be reduced to one stone on of either color. This question was studied by Dorbec et al. (2008) for multiple Cartesian product of cliques (Hamming graphs). In this paper, we generalize this result by proving that if we have two strongly -reducible connected graphs and (both graphs with at least seven vertices) then is strongly -reducible.
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