The 2-rainbow bondage number in generalized Petersen graphs

Abstract: A 2-rainbow domination function of a graph G = (V, E) is a function f mapping each vertex v to a subset of {1, 2} such that ⋃ u∈N(v) f (u) = {1, 2} when f (v) = �, where N(v) is the open neighborhood of v. The weight of f is denoted by wf (G) = ∑ v∈V �f (v)�. The 2-rainbow domination number, denoted by r2(G), is the smallest wf (G) among all 2-rainbow domination functions f of G. The 2-rainbow bondage number, denoted by br2(G), is the minimum cardinality among all sets E ⊆ E such that γr2(G − E ) > γr2(G), where G − E � denotes the resulting graph after all edges in E are removed from G. In this paper, we show that br2(P(n, 2)) = 2 when n ⩾ 15 and n ≡ 3, 9 (mod 10), and br2(P(n, 2)) ⩽ 4 when n∕ ≡3, 9 (mod 10).