Lower (total) mutual-visibility number in graphs

Abstract

Given a graph G, a set X of vertices in G satisfying that between every two vertices in X (respectively, in G) there is a shortest path whose internal vertices are not in X is a mutual-visibility (respectively, total mutual-visibility) set in G. The cardinality of a largest (total) mutual-visibility set in G is known under the name (total) mutual-visibility number, and has been studied in several recent works.In this paper, we propose two lower variants of these concepts, defined as the smallest possible cardinality among all maximal (total) mutual-visibility sets in G, and denote them by μ−(G) and μt−(G), respectively. While the total mutual-visibility number is never larger than the mutual-visibility number in a graph G, we prove that both differences μ−(G)−μt−(G) and μt−(G)−μ−(G) can be arbitrarily large. We characterize graphs G with some small values of μ−(G) and μt−(G), and prove a useful tool called the Neighborhood Lemma, which enables us to find upper bounds on the lower mutual-visibility number in several classes of graphs. We compare the lower mutual-visibility number with the lower general position number, and find a close relationship with the Bollobás-Wessel theorem when this number is considered in Cartesian products of complete graphs. Finally, we also prove the NP-completeness of the decision problem related to μt−(G).