Two-lit trees for lit-only σ-game

Abstract

A configuration of the lit-only σ -game on a finite graph is an assignment of one of two states, on or off, to all vertices of . Given a configuration, a move of the lit-only σ -game on allows the player to choose an on vertex s of and change the states of all neighbors of s. Given any integer k, we say that is k-lit if, for any configuration, the number of on vertices can be reduced to at most k by a finite sequence of moves. Assume that is a tree with a perfect matching. We show that is 1-lit and any tree obtained from by adding a new vertex on an edge of is 2-lit.

Lit-only sigma-game on nondegenerate graphs

“…Apply Q to either side of (9). Using (6), (13) and Q(α s ) = 1 to evaluate the resulting equation, we obtain that st∈R Q(α ∨ t ) = 1.…”

Solutions to All‐Colors Problem on Graph Cellular Automata

“…A graph-theoretic proof was given by Erikisson et al [19]. More results and related references can be referred to [20][21][22][23][24][25].…”