New results about impartial solitaire clobber
RAIRO - Operations Research - Recherche Opérationnelle, Volume 43 (2009) no. 4, p. 463-482

Impartial Solitaire Clobber is a one-player version of the combinatorial game Clobber, introduced by Albert et al. in 2002. The initial configuration of Impartial Solitaire Clobber is a graph, such that there is a stone placed on each of its vertex, each stone being black or white. A move of the game consists in picking a stone, and clobbering an adjacent stone of the opposite color. By clobbering we mean that the clobbered stone is removed from the graph, and replaced by the clobbering one. The aim is to make a sequence of moves leaving the minimum number of stones on the graph; this number is called the reducibility value of the configuration. As any one-player game, Solitaire Clobber is essentially an optimization problem, whose resolution may give bounds on the two-player version of the game. As an optimization problem, Solitaire Clobber can be considered as a constrained version of the underlying optimization problem related to hamiltonian path. This enables to show that Solitaire Clobber is NP-hard. Solitaire Clobber was already studied in various graph structures, including paths, cycles, trees, and Hamming graphs. In this paper we investigate the problem in complete multipartite graphs. In particular, we give a linear-time algorithm computing the reducibility value of any configuration in complete multipartite graphs. We also address some extremal questions related to Solitaire Clobber in general graphs.

DOI : https://doi.org/10.1051/ro/2009029
Classification:  91A46,  91A46,  0SC35
Keywords: combinatorial games, games invoking graphs, extremal problem, solitaire clobber
@article{RO_2009__43_4_463_0,
     author = {Duch\^ene, Eric and Gravier, Sylvain and Moncel, Julien},
     title = {New results about impartial solitaire clobber},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {4},
     year = {2009},
     pages = {463-482},
     doi = {10.1051/ro/2009029},
     zbl = {pre05632103},
     mrnumber = {2573994},
     language = {en},
     url = {http://www.numdam.org/item/RO_2009__43_4_463_0}
}
Duchêne, Eric; Gravier, Sylvain; Moncel, Julien. New results about impartial solitaire clobber. RAIRO - Operations Research - Recherche Opérationnelle, Volume 43 (2009) no. 4, pp. 463-482. doi : 10.1051/ro/2009029. http://www.numdam.org/item/RO_2009__43_4_463_0/

[1] M.H. Albert, J.P. Grossman, R.J. Nowakowski and D. Wolfe, An introduction to Clobber. INTEGERS 5 (2005). | MR 2192079 | Zbl 1139.91013

[2] L. Beaudou, E. Duchêne and S. Gravier, A survey about Solitaire Clobber (submitted).

[3] V.D. Blondel, C. De Kerchove, J.M. Hendrickx and R. Jungers, Solitaire Clobber as an optimization problem on words. INTEGERS 7 (2008). | MR 2425633 | Zbl pre05557919

[4] B. Bollobás, Random graphs. Cambridge University Press (2001). | MR 1864966 | Zbl 0979.05003

[5] E.D. Demaine, M.L. Demaine and R. Fleischer, Solitaire Clobber. Theor. Comput. Sci. 313 (2004) 325-338. | MR 2056930 | Zbl 1066.91016

[6] P. Dorbec, E. Duchêne and S. Gravier, Solitaire Clobber played on Hamming graphs. Integers, J. Comb. No. Theory 7 (2008). | MR 2393376 | Zbl pre05557918

[7] A. Itai, C.H. Papadimitriou and J.L. Szwarcfiter, Hamilton paths in grid graphs. SIAM J. Comput. 11 (1982) 676-686. | MR 677661 | Zbl 0506.05043

[8] I. Peterson, Getting clobbered, Science News 161 (2002), http://www.sciencenews.org/articles/20020427/mathtrek.asp.

[9] M. Ruszinkó, private communication.