关于线性图的压局猜想的证明
Proving the Pressing Game Conjecture on Linear Graphs
摘要：
黑白图上的按压游戏如下：给定一个图$G(V,E)$，其顶点被着色为黑色和白色，任何黑色顶点$v$都可以被按压，其效果如下：（a）$v$的所有邻居改变颜色，即白色邻居变成黑色，而\emph{vice versa}，（b）$v$的所有邻居对改变连通性，即连接对变成不连接，未连接的顶点变成连接的，（c）最后，$v$变成一个分离的白色顶点。游戏的目的是将$G$转换为一个全白的空图。已知的一个结果是：在压制对策中，如果$G$的每个分支至少包含一个黑顶点，并且对于一个固定图，任何成功的变换都具有相同的压制顶点数，则全白空图是可达的.挤压博弈的猜想是，任何成功的挤压路径都可以通过微小的改变转化为任何其他成功的挤压路径。在这里，我们证明猜想的线性图。连接到基因组重排和排序有符号的排列与逆转也进行了讨论。
英文摘要：
The pressing game on black-and-white graphs is the following: Given a graph $G(V,E)$ with its vertices colored with black and white, any black vertex $v$ can be pressed, which has the following effect: (a) all neighbors of $v$ change color, i.e. white neighbors become black and \emph{vice versa}, (b) all pairs of neighbors of $v$ change connectivity, i.e. connected pairs become unconnected, unconnected ones become connected, (c) and finally, $v$ becomes a separated white vertex. The aim of the game is to transform $G$ into an all white, empty graph. It is a known result that the all white empty graph is reachable in the pressing game if each component of $G$ contains at least one black vertex, and for a fixed graph, any successful transformation has the same number of pressed vertices. The pressing game conjecture is that any successful pressing path can be transformed into any other successful pressing path with small alterations. Here we prove the conjecture for linear graphs. The connection to genome rearrangement and sorting signed permutations with reversals is also discussed.
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。
