图中的超k-安全集
On Ultra $k$-Secure Sets in Graphs
AI总结：
本文引入超k-安全集和图的超k-安全数概念，给出其刻画并计算完全多部图与类网格图的超k-安全数，是超安全性概念的推广研究。
AI中文摘要：
在图G=(V,E)中，设S是V的非空子集。对任意x∈S，顶点x及其在S内部的邻居是x的防御者，而在S外部的邻居是x的攻击者。对S的一次攻击是由S中各顶点的两两不相交的攻击者集合构成的|S|元组，而防御则是防御者的对应集合。若S存在一种防御，使得对其每个顶点而言，防御者数量至少等于攻击者数量，则称对S的攻击是可防御的。若对S的每一次攻击都是可防御的，则称集合S是安全集。进一步，若S存在一种防御可成功抵御对S的每一次攻击，则称S是超安全集。对于整数k≥0，k-安全集S是这样的安全集：对S的任意一次攻击，存在S的一种防御，使得对每个在S外有邻居的S顶点，都有k个额外防御者。作为超安全性的推广，本文引入了超k-安全集和图的超k-安全数的概念，得到了超k-安全集的一个刻画，并利用该刻画计算了完全多部图和类网格图的超k-安全数。
英文摘要：
In a graph $G=(V,E)$, let $S$ be a non empty subset of $V$. For any $x\in S$, the vertex $x$ and its neighbors inside $S$ are defenders of $x$, whereas those lying outside $S$ are attackers on $x$. An attack on $S$ is an $|S|$-tuple of pairwise disjoint sets of attackers on vertices of $S$, whereas a defense is that of defenders. An attack on $S$ is defendable if $S$ has a defense, which provides at least as many defenders as attackers for each of its vertices. The set $S$ is a secure set if every attack on $S$ is defendable. Further, $S$ is an ultra secure set if $S$ has a defense which successfully defends $S$ against every attack. For an integer $k\geq 0$, a $k$-secure set $S$ is a secure set in which for any attack on $S$, there is a defense of $S$ with $k$ additional defenders for every vertex of $S$ having neighbors outside $S$. As a generalization of ultra security, in this paper ultra $k$-secure sets and ultra $k$-security number of a graph are introduced. A characterization of ultra $k$-secure sets is obtained and using which ultra $k$-security numbers of complete multipartite graphs and grid-like graphs are computed.
