Graphs and Combinatorics

, Volume 27, Issue 1, pp 109–119

# A Note on Lights-Out-Puzzle: Parity-State Graphs

• Severino V. Gervacio
• Hiroshi Maehara
Original Paper

## 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

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