Graphs and Combinatorics (2011) 27:109–119 DOI 10.1007/s00373-010-0958-1 ORIGINAL PAPER A Note on Lights-Out-Puzzle: Parity-State Graphs Severino V. Gervacio · Hiroshi Maehara Received: 9 November 2007 / Revised: 18 June 2010 / Published online: 1 July 2010 © Springer 2010 Abstract A state of a graph G is an assignment of 0 or 1 to each vertex of G. A move of a state consists of choosing a vertex and then switching the value of the vertex as well as those of its neighbors. Two states are said to be equivalent if one state can be changed to the other by a series of moves. A parity-state graph is defined to be a graph in which two states are equivalent if and only if the numbers of 1’s in the two states have the same parity. We characterize parity-state graphs and present some constructions of parity-state graphs together with applications. Among other things, it is proved that the one-skeleton of the 3-polytope obtained from a simple 3-polytope by cutting off all vertices is a parity-state graph. Keywords Lights-out-puzzle · Parity-state graph · Solvable graph · Dependency 1 Introduction Let G be a finite graph with vertex set V (G).A state on G is a map s : V (G) → Z 2 = {0, 1}. The constant map s(v) = 1 (resp. s(v) = 0) for all v ∈ V is called all-one-state (resp. all-zero-state). For v ∈ V (G), the move f v is a transformation of states s on G S. V. Gervacio Department of Mathematics, De La Salle University, 2401 Taft Avenue, 1004 Manila, Philippines H. Maehara (B ) Research Institute of Educational Development, Tokai University, 2-28-4 Tomigaya, Shibuya-ku, Tokyo 151-8677, Japan e-mail: maehara@tokai-u.jp 123