论文标题
零强迫和其他$ x $ -SET参数的焦油重新配置图的同构和属性
Isomorphisms and properties of TAR reconfiguration graphs for zero forcing and other $X$-set parameters
论文作者
论文摘要
$ x $ -TAR（令牌添加/删除）重新配置图作为其顶点集，可满足某些属性$ x $，如果通过添加或删除一个元素从另一个设置获得两个集合之间的边缘。本文考虑了$ x $ -tar图，用于$ x- $的基本图$ g $的顶点，其中$ g $的$ x $ - set必须满足某些条件。主导集，动力主导套装，零强迫套装和积极的半决赛零强迫套件都是$ x $ - 集的示例。对于图形$ g $和$ g'$，没有孤立的顶点，这表明$ g $和$ g'$具有同构$ x $ -tar重新配置图，并且仅当且仅当$ g'$的顶点重新标记为$ g $和$ g'$具有完全相同的$ x $ -set。引入了$ x $ -irrevant顶点的概念，以促进分析$ x $ -tar图同构。此外，给出了与零强制焦油图的连接性有关的结果。我们介绍了超过已知的连接性参数的已知下限的图。
An $X$-TAR (token addition/removal) reconfiguration graph has as its vertices sets that satisfy some property $X$, with an edge between two sets if one is obtained from the other by adding or removing one element. This paper considers the $X$-TAR graph for $X-$ sets of vertices of a base graph $G$ where the $X$-sets of $G$ must satisfy certain conditions. Dominating sets, power dominating sets, zero forcing sets, and positive semidefinite zero forcing sets are all examples of $X$-sets. For graphs $G$ and $G'$ with no isolated vertices, it is shown that $G$ and $G'$ have isomorphic $X$-TAR reconfiguration graphs if and only if there is a relabeling of the vertices of $G'$ such that $G$ and $G'$ have exactly the same $X$-sets. The concept of an $X$-irrelevant vertex is introduced to facilitate analysis of $X$-TAR graph isomorphisms. Furthermore, results related to the connectedness of the zero forcing TAR graph are given. We present families of graphs that exceed known lower bounds for connectedness parameters.
