Lit-only sigma-game on pseudo-trees

A 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 validmoveswhich takes x to a configurationwith atmost l+2ONvertices.We provide an example to show that the upper bound l + 2 is sharp. Some related problems and conjectures are also reported. © 2010 Elsevier B.V. All rights reserved. 1. σ-game and lit-only σ-game We consider graphs without multiple edges but which may have loops. That is, for any graph G with vertex set V (G), its edge set E(G) is a subset of  V (G) 2  ∪ V (G). We say that there is a loop at a vertex v of G provided {v} ∈ E(G) and we often record an edge {u, v} as uv when u ≠ v. Two vertices u and v, possibly equal, are adjacent in G if {u, v} ∈ E(G). The neighbors of v in G, denoted NG(v), is the set of vertices adjacent to v in G. For any v ∈ V (G), χv ∈ F V (G) 2 stands for the binary function for which χv(u) = 1 if u = v and χv(u) = 0 otherwise. For any U ⊆ V (G), we set χU to be ∑ v∈U χv ∈ F V (G) 2 . The elements of F (G) 2 will often be regarded as binary row vectors in an obvious way. Let us call each element of F V (G) 2 a configuration of G. We can think of a configuration x as an assignment of one of two states, ON or OFF, to the vertices of G such that v is ON if x(v) = 1 and v is OFF if x(v) = 0. The light number (Hamming weight) of a configuration x, written as |x|, refers to the number of vertices which are assigned the ON state by x. A regular move at a vertex v on a graph G transforms a configuration x to x + χNG(v) and we write x→G y to designate that we can make successive regular moves to go from x to y. If we make regular moves at all the vertices of G, then we go from x to σ(x) that satisfies σ(x)(v) = x(v) + −