Internally-Locating Dominating Sets in Graphs

Abstract

For a connected graph \( G \), a subset \( I \subseteq V(G) \) is a locating-dominating set if it is a dominating set and for every two distinct vertices \( x, y \in V(G) \setminus I \), \( N(x) \cap I \neq N(y) \cap I \). This paper introduces the concept of an internally-locating dominating set. Specifically, a nonempty set \( I \subseteq V(G) \) with \( |I| \geq 2 \) is an internally-locating set in a nontrivial connected graph \( G \) if and only if, for every \( u, v \in I \), \( N(u) \cap I \neq N(v) \cap I \). Thus, \( I \) is an internally-locating dominating set if it is both an internally-locating set and a dominating set. In addition, this paper identifies some properties of this concept, provides characterizations of certain special classes of graphs, including total graphs and shadow graphs with \( \Delta(G) = 2 \), with their corresponding internally-locating domination number, and cases where $\gamma_{li}(G) = 2$. Moreover, it provides a sufficient condition for \( \gamma(G) = \gamma_{li}(G) \), specifically when \( G \) is a corona product.