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<p>Let <i>M</i> be a subset of the vertex set of a graph <i>G</i>. We say that <i>M</i> is a mutual visibility set if, for every pair of vertices <i>v</i> and <i>u</i> in <i>M</i>, there exists a shortest path between them that avoids all other vertices in <i>M</i>. The mutual visibility number of <i>G</i> is the size of the largest mutual visibility set in <i>G</i>. This paper studies the mutual visibility number in the hierarchical product of two graphs. In particular, it provides exact results as well as upper and lower bounds on this number concerning the mutual visibility number of the second graph factor. Additionally, two optimization methods, integer linear programming and a genetic algorithm, are proposed for finding mutual visibility sets. Furthermore, an application of mutual visibility sets in the integration of power and communication networks is discussed.</p>

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Mutual visibility in graphs: hierarchical products, genetic algorithm and some applications

  • 期刊论文

  • 发表日期：

    2026年5月26日

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  • Zahra Hamed-Labbafian,
  • Mostafa Tavakoli,
  • Narjes Sabeghi,
  • Aleksander Vesel

摘要

Let M be a subset of the vertex set of a graph G. We say that M is a mutual visibility set if, for every pair of vertices v and u in M, there exists a shortest path between them that avoids all other vertices in M. The mutual visibility number of G is the size of the largest mutual visibility set in G. This paper studies the mutual visibility number in the hierarchical product of two graphs. In particular, it provides exact results as well as upper and lower bounds on this number concerning the mutual visibility number of the second graph factor. Additionally, two optimization methods, integer linear programming and a genetic algorithm, are proposed for finding mutual visibility sets. Furthermore, an application of mutual visibility sets in the integration of power and communication networks is discussed.