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A361517 The value of n for which the two-player impartial {0,1}-Toggle game on a generalized Petersen graph GP(n,2) with a (1,0)-weight assignment is a next-player winning game.

%I A361517 #21 Nov 18 2024 16:38:22
%S A361517 3,4,5,11,17,27,35,37,49,59,69,81,91,103,115,123,135,137,167,175,189,
%T A361517 199,207,287,295,307,361,1051,2507,2757,2917,3057,3081,7255,7361,7871,
%U A361517 16173
%N A361517 The value of n for which the two-player impartial {0,1}-Toggle game on a generalized Petersen graph GP(n,2) with a (1,0)-weight assignment is a next-player winning game.
%C A361517 The two-player impartial {0,1}-Toggle game is played on a simple connected graph G where each vertex is assigned an initial weight of 0 or 1.
%C A361517 A Toggle move consists of selecting a vertex v and switching its weight as well as the weights of each of its neighbors. This move is only legal provided the weight of vertex v is 1 and the total sum of the vertex weights decreases.
%C A361517 In the special case G=GP(n,2), a (1,0)-weight assignment is one in which each vertex of the outer polygon is assigned weight 1 and each vertex of the inner polygon(s) is assigned weight 0.
%D A361517 E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.
%H A361517 Eugene Fiorini, Maxwell Fogler, Katherine Levandosky, Bryan Lu, Jacob Porter, and Andrew Woldar, <a href="https://arxiv.org/abs/2411.08247">On the Nature and Complexity of an Impartial Two-Player Variant of the Game Lights-Out</a>, arXiv:2411.08247 [math.CO], 2024. See p. 17.
%H A361517 E. Fiorini, M. Lind, A. Woldar, and T. W. H. Wong, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Wong/wong31.html">Characterizing Winning Positions in the Impartial Two-Player Pebbling Game on Complete Graphs</a>, Journal of Integer Sequences, 24(6), 2021.
%H A361517 Katherine Levandosky, <a href="/A361517/a361517.txt">CGSuite Program</a>.
%e A361517 For n = 3, the {0,1}-Toggle game on GP(3,2) with a (1,0)-weight assignment is a next-player winning game.
%e A361517 For n = 5, the {0,1}-Toggle game on GP(5,2) with a (1,0)-weight assignment is a next-player winning game.
%o A361517 (CGSuite) # See Levandosky link
%Y A361517 Cf. A071426, A340631, A346197, A346401, A346637.
%K A361517 nonn,more
%O A361517 3,1
%A A361517 _Eugene Fiorini_, Maxwell Fogler, _Katherine Levandosky_, _Bryan Lu_, _Jacob K. Porter_ and _Andrew Woldar_, Mar 14 2023