Difference between minimum light numbers of sigma-game and lit-only sigma-game

TL;DR

This paper investigates the difference in minimum light numbers between sigma-game and lit-only sigma-game on trees, establishing a sharp upper bound and exploring related results and conjectures.

Contribution

It proves that for any configuration on a tree, the minimal number of valid moves needed is at most two more than the minimal number of regular moves, and shows this bound is sharp.

Findings

The upper bound of  + 2 is sharp.

A configuration on a tree can be transformed with at most two more valid moves than regular moves.

The paper discusses related results and conjectures in the area.

Abstract

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. The following result is proved in this note: given any starting configuration $x$ of a tree, if there is a sequence of regular moves which brings $x$ to another configuration in which there are $\ell$ on vertices then there must exist a sequence of valid moves which takes $x$ to a configuration with at most $\ell +2$ on vertices. We provide example to show that the upper bound $\ell +2$ is sharp. Some relevant results and conjectures are also reported.