Weakly Connected Hop Domination in Graphs Resulting from Some Binary Operations

Abstract

Let G = (V(G),E(G)) be a simple connected graph. A set S ⊆ V(G) is a weakly connected hop dominating set of G if for every q ∈ V \ S, there exists r ∈ S such that d_G(q,r) = 2, the subgraph weakly induced by S, denoted by ⟨S⟩w = ⟨NG[S],Ew⟩ where Ew = {qr ∈ E(G) : q ∈ S or r∈S } is connected and S is a dominating set of G. The minimum cardinality of a weakly connected hop dominating set of G is called weakly connected hop domination number and is denoted by γ_wh(G). In this paper, the authors show and explore the concept of weakly connected hop dominating set. The weakly connected hop dominating set of some special graphs, shadow of graphs, join, corona and Lexicographic product of two graphs are characterized. Also, the weakly connected domination number of the aforementioned graphs are determined.

Keywords: weakly connected set, hop dominating set, hop domination number, weakly connected hop dominating set, and weakly connected hop domination number