Domination in edge cycle graphs

Abstract

Let [Formula: see text] be a graph. A subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if every vertex not in [Formula: see text] is adjacent to some vertex in [Formula: see text] The domination number [Formula: see text] of [Formula: see text] is the minimum cardinality taken over all dominating sets of [Formula: see text] A dominating set [Formula: see text] is called a connected dominating set if the induced subgraph [Formula: see text] is connected. The minimum cardinality taken over all connected dominating sets of [Formula: see text] is called the connected domination number of [Formula: see text] and is denoted by [Formula: see text] In this paper, we give the exact values of domination parameters for edge cycle graphs [Formula: see text] where [Formula: see text] and [Formula: see text] Also, we obtain bounds for the domination parameters of edge cycle graphs [Formula: see text] where [Formula: see text] is an arbitrary graph and [Formula: see text] and characterize the extremal graphs. Moreover, Nordhaus–Gaddum-type results are presented for the domination parameters of edge cycle graphs.