Lit-only σ-game on pseudo-trees

Abstract

a b s t r a c tA configuration of a graph is an assignment of one of two states, ON or OFF, to each vertex of it. A regular move at a vertex changes the states of the neighbors of that vertex. A valid move is a regular move at an ON vertex. A pseudo-tree is a graph obtained from a tree by attaching zero or more loops. The following result is proved in this note: given any starting configuration x of a pseudo-tree, if there is a sequence of regular moves which brings x to another configuration in which there are ℓ ON vertices then there must exist a sequence of valid moves which takes x to a configuration with at most ℓ+2 ON vertices. We provide an example to show that the upper bound ℓ + 2 is sharp. Some related problems and conjectures are also reported.