仅亮sigma博弈的双亮树
Two-lit trees for lit-only sigma-game
摘要：
有限图$Γ$上的仅点亮$σ$ -游戏的配置是将两个状态之一（开或关）分配给$Γ.$的所有顶点。给定配置，$Γ$上的仅点亮$σ$ -游戏的移动允许玩家选择$Γ$的顶点上的$s$并改变$s.$的所有邻居的状态给定任何整数$k$，我们说$Γ$是$k$ -lit，如果对于任何配置，通过移动的有限序列可以将上顶点的数目减少到至多$k$。假设$Γ$是一个完美匹配的树。我们证明了$Γ$是1-lit的，并且通过在$Γ$的一条边上添加一个新的顶点而从$Γ$得到的任何树都是2-lit的.
英文摘要：
A configuration of the lit-only $σ$-game on a finite graph $Γ$ is an assignment of one of two states, on or off, to all vertices of $Γ.$ Given a configuration, a move of the lit-only $σ$-game on $Γ$ allows the player to choose an on vertex $s$ of $Γ$ and change the states of all neighbors of $s.$ Given any integer $k$, we say that $Γ$ 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 $Γ$ is a tree with a perfect matching. We show that $Γ$ is 1-lit and any tree obtained from $Γ$ by adding a new vertex on an edge of $Γ$ is 2-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。
