Generalization of the total outer-connected domination in graphs
Nader Jafari Rad, Lutz Volkmann · RAIRO - Operations Research · 2015
Let G = (V, E) be a graph without an isolated vertex.A set S ⊆ V is a total dominating set if S is a dominating set, and the induced subgraph G [S] does not contain an isolated vertex.The total domination number of G is the minimum cardinality of a total dominating set of G.A set D ⊆ V is a total outer-connected dominating set if D is a total dominating set, and the induced subgraph G[V -D] is connected.The total outer-connected domination number of G is the minimum cardinality of a total outer-connected dominating set of G.In this paper we generalize the total outer-connected domination number in graphs.Let k ≥ 1 be an integer.A set D ⊆ V is a total outer-k-connected component dominating set if D is a total dominating and the induced subgraph G[V -D] has exactly k connected component(s).The total outer-k-connected component domination number of G, denoted by γ k tc (G), is the minimum cardinality of a total outer-k-connected component dominating set of G.We obtain several general results and bounds for γ k tc (G), and we determine exact values of γ k tc (G) for some special classes of graphs G.