Γ-induced-paired Dominating Graphs of Paths

Abstract

Let D be a set of vertices of a graph G. Then D is called an induced-paired dominating set if every vertex in G is adjacent to some vertex in D and the subgraph induced by D contains only nonadjacent edges. The upper induced-paired domination number of G, denoted by $\Gamma_{ip}(G)$, is the maximum cardinality of a minimal induced-paired dominating set of G. An induced-paired dominating set of cardinality $\Gamma_{ip}(G)$ is called an $\Gamma_{ip}(G)$-set. We introduce the $\Gamma$-induced-paired dominating graph of G, denoted by $IPD_{\Gamma}$, to be the graph whose vertex set is the set of all $\Gamma_{ip}(G)$-sets, and two $\Gamma_{ip}(G)$-sets are adjacent in $IPD_{\Gamma}(G)$ if one can be obtained from the other by removing one vertex and adding another vertex of G. In this paper, we present the $\Gamma$-induced-paired dominating graphs of all paths.