arXiv 1909.027182019-09math.CO

点权图安全集问题的稳定结构

Stable structure on safe set problems in vertex-weighted graphs

Shinya Fujita, Tadashi Sakuma, Boram Park

摘要：

设$G$是一个图，$w$是$V(G)$上的一个正实值权函数.对于$V(G)$的每个子集$S$，设$w(S)=\sum_{v \in S} w(v).$一个非空子集$S \subset V(G)$是$(G,w)$的加权安全集，如果对于由$S$诱导的子图的每个分量$C$和$G-S$的每个分量$D$，只要$C$和$D$之间存在边，我们就有$w(C) \geq w(D)$。如果由一个加权安全集$S$导出的$G$的子图是连通的，则称这个集$S$为$(G,w)$的一个连通加权安全集. $(G,w)$的加权安全数$\mathrm{s}(G,w)$和连通加权安全数$\mathrm{cs}(G,w)$分别是$(G,w)$的所有加权安全集和所有连通加权安全集中的最小权重$w(S)$。请注意，对于每对$(G,w)$，$\mathrm{s}(G,w) \le \mathrm{cs}(G,w)$由其定义。最近，有人问哪对$(G,w)$满足等式，并表明每个加权循环满足等式。本文给出了连通二部图$G$的一个完整列表，使得对$V(G)$上的每个权函数$w$，都有$\mathrm{s}(G,w)=\mathrm{cs}(G,w)$.

英文摘要：

Let $G$ be a graph, and let $w$ be a positive real-valued weight function on $V(G)$. For every subset $S$ of $V(G)$, let $w(S)=\sum_{v \in S} w(v).$ A non-empty subset $S \subset V(G)$ is a weighted safe set of $(G,w)$ if, for every component $C$ of the subgraph induced by $S$ and every component $D$ of $G-S$, we have $w(C) \geq w(D)$ whenever there is an edge between $C$ and $D$. If the subgraph of $G$ induced by a weighted safe set $S$ is connected, then the set $S$ is called a connected weighted safe set of $(G,w)$. The weighted safe number $\mathrm{s}(G,w)$ and connected weighted safe number $\mathrm{cs}(G,w)$ of $(G,w)$ are the minimum weights $w(S)$ among all weighted safe sets and all connected weighted safe sets of $(G,w)$, respectively. Note that for every pair $(G,w)$, $\mathrm{s}(G,w) \le \mathrm{cs}(G,w)$ by their definitions. Recently, it was asked which pair $(G,w)$ satisfies the equality and shown that every weighted cycle satisfies the equality. In this paper, we give a complete list of connected bipartite graphs $G$ such that $\mathrm{s}(G,w)=\mathrm{cs}(G,w)$ for every weight function $w$ on $V(G)$.

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。

摘要：

英文摘要：

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