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Graphs considered here are simple, finite and undirected. A graph is denoted by G and its vertex set by V (G) and edge set by E (G). Many researchers were attracted by two concepts introduced in [P. J. Slater, Domination and location in acyclic graphs, Networks17 (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 D of V (G) is called a locating dominating set of G if for any u , v ∈ V − D , N (u) ∩ D ≠ N (v) ∩ D and both sets are non-empty. D is called a fair dominating set of G for any u , v ∈ D , | N (u) ∩ D | = | N (v) ∩ D | > 0. In this paper, both properties are combined and locating fair domination is studied. [ABSTRACT FROM AUTHOR]

In secure domination [A. P. Burger, M. A. Henning and J. H. Van Vuuren, Vertex covers and secure domination in graphs, Quaest Math. 31 (2008) 163–171; E. J. Cockayne, Irredundance, secure domination and maximum degree in trees, Discrete Math. 307(1) (2007) 12–17; E. J. Cockayne, O. Favaron and C. M. Mynhardt, Secure domination, weak Roman domination and forbidden subgraph, Bull. Inst. Combin. Appl. 39 (2003) 87–100; C. M. Mynhardt, H. C. Swart and E. Ungerer, Excellent trees and secure domination, Util. Math. 67 (2005) 255–267], a vertex outside has the chance of coming inside the dominating set by replacing an element of the set without affecting domination. This idea is combined with equitability. Secure equitable dominating set is introduced and studied. Other concepts like independence and rigid security are also studied in this paper. [ABSTRACT FROM AUTHOR]

Graphs considered here are simple, finite and undirected. A graph is denoted by G and its vertex set by V (G) and edge set by E (G). Kulli and Janakiram introduced the concept of the strong non-split domination number of a graph [The strong non-split domination number of a graph, Int. J. Manag. Syst.19 (2003) 2]. The concept of fair domination was introduced by Yaro Caro et al. in [Fair domination in graphs, Discrete Math.312 (2012) 2905–2914]. This paper is an outcome of the combination of these two concepts. [ABSTRACT FROM AUTHOR]