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Let G be a simple graph. A subset S Í V(G) is called a strong (weak) perfect dominating set of G if |Ns(u) ∩ S| = 1(|Nw(u) ∩ S| = 1) for every u ∊V(G) - S where Ns(u) = {v ∊ V(G) / uv  deg v ≥ deg u} (Nw(u) = {v ∊V(G) / uv  deg v ≤ deg u}. The minimum cardinality of a strong (weak) perfect dominating set of G is called the strong (weak) perfect domination number of G and is denoted by sp(G)( wp(G)). The strong perfect cobondage number bcsp(G) of a nonempty graph G is defined to be the minimum cardinality among all subsets of edges X E(G) for which sp (G + X) sp(G). If bcsp(G) does not exist, then bcsp(G) is defined as zero. In this paper study of this parameter is initiated.