Solid-Resolving Sets on Directed Graphs

Abstract

A set $S \subseteq V(G)$ is a solid-resolving set of an undirected graph $G$ if each vertex can be distinguished from any vertex subset by their distances from the set $S$. Solid-resolving sets are a generalization of resolving sets, which have been studied extensively in the literature. We extend this concept to oriented graphs and directed graphs. In this setting, we first define solid-resolving sets for digraphs and then consider vertices that must be included in any solid-resolving set. We also study the solid-metric dimension, which is the minimum cardinality of a solid-resolving set of a given digraph. We determine the possible values of the solid-metric dimension for several common graph classes, including tournaments, directed trees, oriented cycles, complete bipartite graphs, as well as for Cartesian products of oriented paths and cycles. We also show via some examples that there is no trivial relation between the solid-metric dimension of a digraph and the solid-metric dimension of its underlying undirected graph. Furthermore, we show that the solid-metric dimension can increase or decrease by an arbitrary $k$ when one vertex is removed. Similarly, removing a single arc can decrease the solid-metric dimension by an arbitrary $k$; however, it can increase by at most 1. The arbitrary changes in solid-metric dimension indicate that the solid-metric dimension is a global, rather than a local, property.