Secure connected domination in a graph

Abstract

Let G = (V (G), E(G)) be a connected simple graph. A connected dominating set S of V (G) is a secure connected dominating set of G if for each u ∈ V (G) \ S, there exists v ∈ S such that uv ∈ E(G) and the set (S \ {v} ∪ {u}) is a connected dominating set of G. The minimum cardinality of a secure connected dominating set of G, denoted by γ sc (G), is called the secure connected domination number of G. We characterized secure connected dominating set in terms of the concept of external private neighborhood of a vertex. Also, we give necessary and sufficient conditions for connected graphs to have secure connected domination number equal to 1 or 2. The secure connected domination numbers of graphs resulting from some binary operations are also investigated.

Secure doubly connected domination in the join of graphs

“…The converse is clear. Note that every secure connected dominating set (see [4] for detail) is a secure dominating set and for any non-complete graph G, γ s (G) ≥ 2. Thus, we have a remark.…”

Management challenges and work passion of school instructional leaders in the new Philippine setting

“…) for teachers and other school personnel and allowed them to explore their ideas in solving management problems, allowed the teachers to conduct community-based research particularly on the challenges brought about by the changes in the educational system," and "extended the programs for technology to students and parents to make them embrace its importance in today"s education. As Cabaro, et. al., (2014) suggested SILs and teachers must adapt their work methodology, by carrying out a training process that serves as a guide to assimilate the context caused by the coronavirus crisis.…”

Secure connected dominating sets in the join and composition of graphs

“…The following example will show that Corollary 2.3 is not the correct characterization. Before giving the correct characterization, we first give another result found in [1].…”

“…The connected domination, as a variant of domination, was introduced by Sampathkumar and Walikar in [4]. The concept of secure connected domination in graphs was initially studied in [1]. This paper will revisit the latter concept of domination.…”