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{\bf John Goldwasser, Xinmao Wang and Yaokun Wu}
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{\bf Minimum Light Numbers in the $\sigma$-Game and Lit-Only $\sigma$-Game on Unicyclic and Grid Graphs}
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Consider a graph each of whose vertices is either in the ON state or
in the OFF state and call the resulting ordered bipartition into ON
vertices and OFF vertices a configuration of the graph. A regular move at a vertex
changes the states of the neighbors of that vertex and hence sends
the current configuration to another one. A valid move is a regular
move at an ON vertex. For any graph $G,$ let $\mathcal{D}(G)$ be
the minimum integer such that
given any starting configuration $\bf x$ of $G$ there must exist a sequence of valid
moves which takes $\bf x$ to a configuration with at most $\ell
+\mathcal{D}(G)$ ON vertices provided there is a sequence of
regular moves which brings $\bf x$ to a configuration in which
there are $\ell$ ON vertices. The shadow graph $\mathcal{S}(G)$ of
a graph $G$ is obtained from $G$ by deleting all loops. We prove
that $\mathcal{D}(G)\leq 3$ if $\mathcal{S}(G)$ is unicyclic and
give an example to show that the bound $3$ is tight. We also prove
that $\mathcal{D}(G)\leq 2$ if $ G $ is a two-dimensional grid graph
and $\mathcal{D}(G)=0$ if $\mathcal{S}(G)$ is a two-dimensional
grid graph but not a path and $G\not= \mathcal{S}(G)$.
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