图的棱柱的永恒控制
Eternal domination on prisms of graphs
摘要：
图$G$的永恒支配集是支配$G$的顶点（或“守卫”）的集合，并且可以防御任何无穷系列的顶点攻击，其中通过将一个守卫沿着边从其当前位置移动到被攻击的顶点来防御攻击。最小永恒支配集的大小记为$γ^\infty(G)$，称为$G$的永恒控制数。本文回答了Klostermeyer和Mynhardt的一个猜想[Discussiones Mathematicae Graph Theory，vol. 35，pp. 283-300]，证明了存在无穷多个图$G$使得$γ^\infty(G)=θ(G)$和$γ^\infty(G \Box K_2)<θ(G \Box K_2)$，其中$θ(G)$表示$G$的团覆盖数.
英文摘要：
An eternal dominating set of a graph $G$ is a set of vertices (or "guards") which dominates $G$ and which can defend any infinite series of vertex attacks, where an attack is defended by moving one guard along an edge from its current position to the attacked vertex. The size of the smallest eternal dominating set is denoted $γ^\infty(G)$ and is called the eternal domination number of $G$. In this paper, we answer a conjecture of Klostermeyer and Mynhardt [Discussiones Mathematicae Graph Theory, vol. 35, pp. 283-300], showing that there exist there are infinitely many graphs $G$ such that $γ^\infty(G)=θ(G)$ and $γ^\infty(G \Box K_2)<θ(G \Box K_2)$, where $θ(G)$ denotes the clique cover number of $G$.
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。
