Locating fair domination in graphs

Abstract

Graphs considered here are simple, finite and undirected. A graph is denoted by [Formula: see text] and its vertex set by [Formula: see text] and edge set by [Formula: see text]. Many researchers were attracted by two concepts introduced in [P. J. Slater, Domination and location in acyclic graphs, Networks 17 (1987) 55–64; P. J. Slater, Dominating and reference sets in graphs, J. Math. Phys. Sci. 22 (1988) 445–455; Y. Caro, A. Hansberg and M. Henning, Fair domination in graphs, Discrete Math. 312 (2012) 2905–2914]. One is locating domination and the other is fair domination. A subset [Formula: see text] of [Formula: see text] is called a locating dominating set of [Formula: see text] if for any [Formula: see text] and both sets are non-empty. [Formula: see text] is called a fair dominating set of [Formula: see text] for any [Formula: see text]. In this paper, both properties are combined and locating fair domination is studied.