RT journal article
T1 Lower (total) mutual-visibility number in graphs
A1 Brešar, B.
A1 González Yero, Ismael
A2 Matemáticas
K1 Mutual-visibility set
K1 Mutual-visibility number
K1 Total mutual-visibility set
K1 Computational complexity
AB 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)
PB Elsevier
SN 0096-3003
YR 2024
FD 2024
LK http://hdl.handle.net/10498/31409
UL http://hdl.handle.net/10498/31409
LA eng
DS Repositorio Institucional de la Universidad de Cádiz
RD 12-nov-2026