Secure Domination in Lict Graphs
Girish V. Rajasekharaiah, Usha P. Murthy · Open Journal of Mathematical Sciences · 2018
For any graph \\(G=(V,E)\\), lict graph \\(\\eta(G)\\) of a graph \\(G\\) is the graph whose vertex set is the union of the set of edges and the set of cut-vertices of \\(G\\) in which two vertices are adjacent if and only if the corresponding edges are adjacent or the corresponding members of \\(G\\) are incident. A secure lict dominating set of a graph \\(\\eta(G)\\) , is a dominating set \\(F \\subseteq V(\\eta(G))\\) with the property that for each \\(v_{1} \\in (V(\\eta(G))-F)\\), there exists \\(v_{2} \\in F\\) adjacent to \\(v_{1}\\) such that \\((F-\\lbrace v_{2}\\rbrace) \\cup \\lbrace v_{1} \\rbrace\\) is a dominating set of \\(\\eta(G)\\). The secure lict dominating number \\(\\gamma_{se}(\\eta(G))\\) of \\(G\\) is a minimum cardinality of a secure lict dominating set of \\(G\\). In this paper many bounds on \\(\\gamma_{se}(\\eta(G))\\) are obtained and its exact values for some standard graphs are found in terms of parameters of \\(G\\). Also its relationship with other domination parameters is investigated.