Title: On some mutual-visibility problems in graphs

Abstract: Given a connected graph \(G\), a set \(S\subset V(G)\) and two vertices \(x,y\in V(G)\), it is said that \(x,y\) are \(S\)-visible, if there exists a shortest \(x,y\)-path \(P\) in \(G\) such that \(P\cap S\subseteq \{x,y\}\). The set \(S\) is called a mutual-visibility set of \(G\) if any two vertices of \(S\) are \(S\)-visible. The cardinality of a largest mutual-visibility set of \(G\) is the mutual-visibility number \(\mu(G)\) of \(G\). This concept was recently introduced in [3] motivated by several applications of it in some computer science related models. Further on, a series of several other related investigations on the topic has been developed, like for instance, [1,2,4]. Among such works, there have appeared a few interesting variations of the main concept. They are as follows, for a given set \(S\subset V(G)\):

• total mutual-visibility set, if every \(u,v\in V(G)\) are \(S\)-visible,
• outer mutual-visibility set, if every \(u,v\in S\) are \(S\)-visible, and every \(u\in S\), \(v\in \overline{S}\) are \(S\)-visible,
• dual mutual-visibility set, if every \(u,v\in S\) are \(S\)-visible, and every \(u,v\in \overline{S}\) are \(S\)-visible.

References:

1. [1] S. Cicerone, G. Di Stefano, S. Klavˇzar, I.G. Yero, Mutual-visibility in strong products of graphs via total mutual-visibility, arXiv:2210.07835 [math.CO] (14 Oct 2022).
2. [2] S. Cicerone, G. Di Stefano, S. Klavˇzar, I.G. Yero, Variety of mutual-visibility problems in graphs, manuscript, (2023).
3. [3] G. Di Stefano, Mutual visibility in graphs, Appl. Math. Comput. 419 (2022) 126850.
4. [4] J. Tian, S. Klavˇzar, Graphs with total mutual-visibility number zero and total mutualvisibility in Cartesian products, arXiv:2212.07193 [math.CO] (14 Dec 2022)

Ismael Gonzalez Yero is an associate Professor at the Department of Mathematics, University of Cádiz, Spain. I obtained my Ph. D. in 2010 at Rovira i Virgili University, Spain. Since then my main research topics include domination in graphs and metric graph theory, areas in which I have published several articles in specialized journals. I have attended and delivered several talks in many international conferences, and have supervised a few Ph. D. students. I belong to the editorial boards of some scientific journals and have reviewed a large number of scientific articles from several of my colleagues.