A paired dominating set of a graph G = (V(G),E(G)) is a set D of vertices of G such that every vertex is adjacent to some vertex in D, and the subgraph of G induced by D contains a perfect matching. The upper paired domination number of G, denoted by Γpr(G) is the maximum cardinality of a minimal paired dominating set of G. A paired dominatin set of cardinality Γ pr(G) is called a Γpr(G) -set. The Γ -paired dominating graph of G, denoted by ΓPD(G), is the graph whose vertex set is the set of all Γ pr(G) -sets, and two Γpr(G) -sets are adjacentin ΓPD(G) if one can be obtained from the other by removing one vertex and adding another vertex of G. In this paper, we present the Γ-paired dominating graphs of some paths.
A paired dominating set of a graph \(G\) is a dominating set whose induced subgraph contains a perfect matching. The paired domination number, denoted by \(\gamma_{pr}(G)\), is the minimum cardinality of a paired dominating set of \(G\). A \(\gamma_{pr}(G)\)-set is a paired dominating set of cardinality \(\gamma_{pr}(G)\). The \(\gamma\)-paired dominating graph of \(G\), denoted by \(PD_{\gamma}(G)\), as the graph whose vertices are \(\gamma_{pr}(G)\)-sets. Two \(\gamma_{pr}(G)\)-sets \(D_1\) and \(D_2\) are adjacent in \(PD_{\gamma}(G)\) if there exists a vertex \(u\in D_1\) and a vertex \(v\notin D_1\) such that \(D_2=(D_1\setminus \{u\})\cup \{v\}\). In this paper, we present the \(\gamma\)-paired dominating graphs of cycles.
Let [Formula: see text] be a graph without isolated vertices. A total dominating set of [Formula: see text] is a set [Formula: see text] of vertices of [Formula: see text] such that every vertex of [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. A total dominating set [Formula: see text] is a paired dominating set of [Formula: see text] if the subgraph of [Formula: see text] induced by [Formula: see text] has a perfect matching. The minimum cardinality of a total dominating set (respectively, a paired dominating set) is called the total domination number (respectively, the paired domination number). This paper determines the total domination numbers and the paired domination numbers of windmill graphs.
A paired dominating set of a graph G is a dominating set whose induced subgraph has a perfect matching. The paired domination number γ pr (G) of G is the minimum cardinality of a paired dominating set. A paired dominating set D is a γ pr (G)-set if |D| = γ pr (G). The γ-paired dominating graph P D γ (G) of G is the graph whose vertex set is the set of all γ pr (G)-sets, and two γ pr (G)-sets D 1 and D 2 are adjacent in P D γ (G) if D 2 = (D 1 \ {u}) ∪ {v} for some u ∈ D 1 and v / ∈ D 1 . This paper determines the paired domination numbers of lollipop graphs, umbrella graphs, and coconut graphs. We also consider the γ-paired dominating graphs of those three graphs.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.