sigma-对策与仅亮sigma-对策最小光数的区别
Difference between minimum light numbers of sigma-game and lit-only sigma-game
摘要：
一个图的构形是指它的每个顶点被赋予开或关两种状态中的一种，在一个顶点上的规则移动会改变该顶点的邻居的状态。有效的移动是在顶点上的常规移动。本文证明了如下结果：给定树的任意起始构形$x$，如果存在一个正规移动序列使$x$到达另一个构形，在该构形中顶点上有$\ell$，则一定存在一个有效移动序列使$x$到达顶点上至多有$\ell +2$的构形.通过实例证明了上界$\ell +2$是尖锐的.文中还报道了一些有关的结果和图表。
英文摘要：
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.
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。
