Secure domination in 4-regular planar and non-planar graphs with girth 3

Abstract

Let G(n) = (V, E) be a 4-regular graph of n vertices with girth 3. The set D C V is a secure dominating set of G(n) if for each u ∈ V\D, there exists a vertex v 6 D such that uv ∈ E and (D\{v}) U {u} ∗s a dominating set of G(n). The minimum cardinality of a secure dominating set in G(n) is a secure domination number γs(G(n)) of G(n). We investigate the secure domination number γs(G(n)) of G(n), secure connected domination number γsC(G(n)) of G(n) and secure total domination number γst(G(n)) of G(n). We prove that (Equation presented) for n ≥ 12. We further establish that 4-regular planar and non-planar graphs with girth 3 for these parameters. Nordhaus-Gaddum type results are also obtained for these parameters. © 2018 Utilitas Mathematica Publishing Inc. All rights reserved.