Independent semitotal domination excellent graphs

A dominating set [Formula: see text] of vertices of a graph [Formula: see text] with no isolated vertices is an independent semitotal dominating set of [Formula: see text] if it is an independent dominating set and to each [Formula: see text] there exists a [Formula: see text] at a distance of exactly two. The minimum and maximum cardinality of a independent semitotal dominating set are called the independent semitotal domination number [Formula: see text] and upper independent semitotal domination number [Formula: see text] of [Formula: see text], respectively. A graph [Formula: see text] is said to be [Formula: see text]-excellent if every vertex of [Formula: see text] is in some [Formula: see text]-set of [Formula: see text]. A graph [Formula: see text] is said to be [Formula: see text]-just excellent if for each [Formula: see text], there is a unique [Formula: see text]-set of [Formula: see text] containing [Formula: see text]. In this paper, we initiate a study of [Formula: see text]-excellent and [Formula: see text]-just excellent graphs.