Generalization of the total outer-connected domination in graphs

Abstract

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 .