Independent point-set dominating sets in graphs

In this paper, we study graphs which possess an independent point-set dominating set (in short, ipsd-set). We call such a graph as an ipsd-graph. We first provide general structural characterization of separable ipsd-graphs and thereafter, in our quest to characterize such graphs, we establish that girth of an ipsd-graph is at most 5. We further characterize ipsd-graphs with girth 5 and C5-free ipsd-graphs of girth 4. Then, we exhibit a class of ipsd-graphs with girth g(G)=4 containing C5 as an induced subgraph and in the process, we introduce a new graph equivalence relation termed as duplicated equivalence.