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

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

2009-04-20

arxiv.org/abs/0904.3050v1

Mathematics

Combinatorics

14 pages, 1 figure

Scientific paper

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.

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