Generalized 3 – Complement of Set Domination

Abstract

Let G=(V,E) be a simple, undirected, finite nontrivial graph. A set SíV of vertices of a graph G = (V, E) is called a dominating set if every vertex víŽV is either an element of S or is adjacent to an element of S. A set SíV  is a set dominating set if for every set TíV-S,  there exists a non-empty set RíS such that the subgraph <RUT> is connected. The minimum cardinality of a set dominating set is called set domination number and it is denoted by  γ_s (G).Let P=(V₁,V₂,V₃) be a partition of V of order 3. Remove the edges between V_i and V_j  where i¹j (1£i,j£3) in G and join the edges between  V_i and V_j  which are not in G. The graph G₃^p thus obtained is called 3-complement of G with respect to ‘P'.