UNIQUE STRONG ISOLATE SUPER DOMINATION IN GRAPHS

Abstract

A dominating set "D" of "V(G)" in a graph "G = (V,E)" is called super dominating set if for every "v ∈ V(G) - D", there exists an external private neighbour of "v" with respect to "V(G) - D". A super dominating set "D" of a graph "G" is said to be an Isolate Super Dominating Set (ISD-set) of "G" if "⟨D⟩" has at least one isolated vertex. An ISD-set "D" is considered as the Unique Strong Isolate Super Dominating Set (USISD-set), if there exists exactly one isolated vertex "a ∈ D" such that "N₂(a) ∩ D = ∅", where "N₂(a) = {b : d(a,b) ≤ 2 and a ≠ b}". The unique strong isolate super domination number (USISD-number), denoted by "γ₀,sp^(U,S)(G)", is the minimum cardinality of a unique strong isolate super dominating set of "G". In this paper, we initiate a study on this parameter. We obtain basic properties of the unique strong isolate super dominating sets in graphs. Also we present upper and lower bounds for the unique strong isolate super domination number.