Restrained Italian domination in trees. Issue 3 (3rd July 2021) - Kim, Kijung
Abstract : Let G = ( V, E ) be a graph. A subset D of V is a restrained dominating set if every vertex in V ∖ D is adjacent to a vertex in D and to a vertex in V ∖ D . The restrained domination number, denoted by γ r ( G ), is the smallest cardinality of a restrained dominating set of G . A function f : V → 0, 1, 2 is a restrained Italian dominating function on G if (i) for each vertex v ∈ V for which f ( v ) = 0, it holds that ∑ u ∈ N G ( v ) f ( u ) ≥ 2, (ii) the subgraph induced by v ∈ V ∣ f ( v ) = 0 has no isolated vertices. The restrained Italian domination number, denoted by γ r I ( G ), is the minimum weight taken over all restrained Italian dominating functions of G . It is known that γ r ( G ) ≤ γ r I ( G ) ≤ 2 γ r ( G ) for any graph G . In this paper, we characterize the trees T for which γ r ( T ) = γ r I ( T ), and we also characterize the trees T for which γ r I ( T ) = 2 γ r ( T ) .