关于应力传递函数
On the stress transit function
AI总结：
本文研究图的应力传递函数相关概念，给出应力区间性质、刻画s-极端顶点，构造差异任意大图，分析特殊图族不变量并证明分裂图中三者相等，且二部图中判断$sn(G)≤k$是NP完全问题。
摘要：
图$G$中顶点$u,v∈V(G)$之间的应力区间$S(u,v)$是所有位于每条最短$u,v$-路径上的顶点集合。若对任意$u,v∈U$都有$S(u,v)⊆U$，则集合$U⊆V(G)$是应力凸的。若$V(G)-v$是$G$中的应力凸集，则顶点$v∈V(G)$是s-极端的。图$G$的应力数$sn(G)$是满足$\bigcup_{u,v∈U}S(u,v)=V(G)$的集合$U$的最小基数。图$G$的应力壳数$sh(G)$是其应力凸壳为$V(G)$的集合的最小基数。本文给出了应力区间的许多基本性质，刻画了图$G$的s-极端顶点，并构造了s-极端顶点数、$sh(G)$和$sn(G)$之间差异任意大的图。然后研究了一些特殊图族（如图乘积、分裂图和块图）的这三个不变量，证明在任何分裂图$G$中，$sh(G)=sn(G)=|Ext_s(G)|$（其中$Ext_s(G)$是$G$的s-极端顶点集合）。最后证明对$k∈\text{ℕ}$，判断$sn(G)≤k$是NP完全问题，即使限制在二部图中。
英文摘要：
The stress interval $S(u,v)$ between $u,v\in V(G)$ is the set of all vertices in a graph $G$ that lie on every shortest $u,v$-path. A set $U \subseteq V(G)$ is stress convex if $S(u,v) \subseteq U$ for any $u,v\in U$. A vertex $v \in V(G)$ is s-extreme if $V(G)-v$ is a stress convex set in $G$. The stress number $sn(G)$ of $G$ is the minimum cardinality of a set $U$ where $\bigcup_{u,v \in U}S(u,v)=V(G)$. The stress hull number $sh(G)$ of $G$ is the minimum cardinality of a set whose stress convex hull is $V(G)$. In this paper, we present many basic properties of stress intervals. We characterize s-extreme vertices of a graph $G$ and construct graphs $G$ with arbitrarily large difference between the number of s-extreme vertices, $sh(G)$ and $sn(G)$. Then we study these three invariants for some special graph families, such as graph products, split graphs, and block graphs. We show that in any split graph $G$, $sh(G)=sn(G)=|Ext_s(G)|$, where $Ext_s(G)$ is the set of s-extreme vertices of $G$. Finally, we show that for $k \in \mathbb{N}$, deciding whether $sn(G) \leq k$ is NP-complete problem, even when restricted to bipartite graphs.
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