发表机构
Department of Mathematics, College of Engineering Trivandrum（特里凡得琅工程学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入连通互可见数$\mu_c(G)$，研究其界、极值图、正则图取值及块结构局部性，并给出识别算法及NP完全性证明。
AI 中文摘要
图$G$的顶点集合$S$称为连通互可见集，如果$S$中任意两个顶点都由一条内部顶点不在$S$中的最短路径连接，并且由$S$诱导的子图是连通的。我们引入了连通互可见数$\mu_c(G)$，定义为这样的集合的最大基数，并研究了其结构和算法性质。我们建立了基本界，推导了Nordhaus-Gaddum型不等式，并刻画了达到最小值和最大值的图。对于正则$(d,2,-\delta)$图，我们推导了$\mu_c(G)$的一般界，并确定了两个缺陷为$2$的三次图的精确值。我们进一步证明$\mu_c(G)$由$G$的块结构局部决定，即它等于$G$的各个块上对应值的最大值。最后，我们提出了一个多项式时间算法来识别连通互可见集，并证明了相关的判定问题是$\mathsf{NP}$-完全的，即使对于直径至多为$4$的连通二分图也是如此。
英文摘要
A set $S$ of vertices of a graph $G$ is a connected mutual-visibility set if every two vertices of $S$ are joined by a shortest path whose internal vertices lie outside $S$, and the subgraph induced by $S$ is connected. We introduce the connected mutual-visibility number $μ_c(G)$, defined as the maximum cardinality of such a set, and investigate its structural and algorithmic properties. We establish fundamental bounds, derive Nordhaus--Gaddum type inequalities, and characterise the graphs attaining the minimum and maximum possible values. For regular $(d,2,-δ)$-graphs, we derive general bounds on $μ_c(G)$ and determine its exact value for the two cubic graphs of defect $2$. We further show that $μ_c(G)$ is determined locally by the block structure of $G$, namely, it is equal to the maximum of the corresponding values over the blocks of $G$. Finally, we present a polynomial-time algorithm for recognising connected mutual-visibility sets and prove that the associated decision problem is $\mathsf{NP}$-complete, even for connected bipartite graphs of diameter at most $4$.
Comments15 pages, 2 figures
