Γ-independent dominating graphs of paths and cycles - Roongrat Samanmoo; Nantapath Trakultraipruk (Songklanakarin Journal of Science and Technology (SJST))
A set D of vertices in a graph G is an independent dominating set if D is a set of pairwise nonadjacent vertices of G such that every vertex of G not in D is adjacent to at least one vertex in D. An independent dominating set D is minimal if no proper subset of D is an independent dominating set. The upper independent domination number of a graph G, denoted by Γi(G), is the maximum cardinality of a minimal independent dominating set of G. An independent dominating set D is an Γi(G)-set if D is of size Γi(G). We introduce the Γ-independent dominating graph of G, denoted by IDΓ(G), as the graph whose vertex set is the set of all Γi(G)-sets, and two Γi(G)-sets are adjacent in IDΓ(G) if one can be obtained from the other by adding one vertex and removing another vertex of G. In this paper, we present the Γ-independent dominating graphs of all paths and cycles.