非退化图上的Lt唯一σ-对策

摘要：

图$Γ$上的仅点亮$σ$ -游戏的配置是将两个状态之一（{ \it开}或{ \it关}）分配给$Γ.$的每个顶点。给定配置，$Γ$上的仅点亮$σ$ -游戏的移动允许玩家选择$Γ$的{ \it开}顶点$s$并改变$s.$的所有邻居的状态。给定整数$k$，如果对于任何配置，通过移动的有限序列，{ \it on}顶点的数目可以减少到至多$k$，则基础图$Γ$被称为$k$ -lit。本文给出了非退化图$Γ$（非线图）上的仅限点亮的$σ$ -对策的轨道的描述。我们证明了这些图$Γ$是2-lit的，并给出了$Γ$是1-lit的线性代数判据。

英文摘要：

A configuration of the lit-only $σ$-game on a graph $Γ$ is an assignment of one of two states, {\it on} or {\it off}, to each vertex of $Γ.$ Given a configuration, a move of the lit-only $σ$-game on $Γ$ allows the player to choose an {\it on} vertex $s$ of $Γ$ and change the states of all neighbors of $s.$ Given an integer $k$, the underlying graph $Γ$ is said to be $k$-lit if for any configuration, the number of {\it on} vertices can be reduced to at most $k$ by a finite sequence of moves. We give a description of the orbits of the lit-only $σ$-game on nondegenerate graphs $Γ$ which are not line graphs. We show that these graphs $Γ$ are 2-lit and provide a linear algebraic criterion for $Γ$ to be 1-lit.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。