Variety of mutual-visibility problems in hypercubes

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation Pub Date : 2024-12-02 DOI:10.1016/j.amc.2024.129218

Danilo Korže , Aleksander Vesel

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引用次数: 0

Abstract

Let G be a graph and

M \subseteq V (G)

. Vertices

x, y \in M

are M-visible if there exists a shortest

x, y

-path of G that does not pass through any vertex of

M ∖ {x, y}

. We say that M is a mutual-visibility set if each pair of vertices of M is M-visible, while the size of any largest mutual-visibility set of G is the mutual-visibility number of G. If some additional combinations for pairs of vertices

x, y

are required to be M-visible, we obtain the total (every

x, y \in V (G)

are M-visible), the outer (every

x \in M

and every

y \in V (G) ∖ M

are M-visible), and the dual (every

x, y \in V (G) ∖ M

are M-visible) mutual-visibility set of G. The cardinalities of the largest of the above defined sets are known as the total, the outer, and the dual mutual-visibility number of G, respectively.

We present results on the variety of mutual-visibility problems in hypercubes.

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超立方体中各种互可见性问题

设G为图，M⊥V(G)。顶点x，y∈M是M可见的，如果存在G的最短路径x，y不经过M的任何顶点{x，y}。如果M的每个顶点对都是M可见的，则M是一个互可见集，而G的任何最大的互可见集的大小是G的互可见数。如果要求顶点对x，y的一些附加组合是M可见的，则我们得到total（每个x，y∈V(G)）， outer（每个x∈M和每个y∈V(G)∈M）是M可见的，和G的对偶可见集（每个x，y∈V(G)∈M是M可见的）。上述定义集合中最大的集合的基数分别称为G的总可见数、外可见数和对偶可见数。我们给出了超立方体中各种互可见性问题的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied Mathematics and Computation 数学-应用数学

CiteScore

7.90

自引率

10.00%

发文量

755

审稿时长

36 days

期刊介绍： Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results. In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.