Abstract
A set of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex in such a set. We say that a total dominating set D is a total outer k-independent dominating set of G if the maximum degree of the subgraph induced by the vertices that are not in D is less or equal to k − 1 . The minimum cardinality among all total outer k-independent dominating sets is the total outer k-independent domination number of G. In this article, we introduce this parameter and begin with the study of its combinatorial and computational properties. For instance, we give several closed relationships between this novel parameter and other ones related to domination and independence in graphs. In addition, we give several Nordhaus–Gaddum type results. Finally, we prove that computing the total outer k-independent domination number of a graph G is an NP-hard problem.
Highlights
Theory of domination in graphs is one of the most important topics in graph theory
When k = 1, a TOkID set is a total outer-independent dominating set, that is, γt,oi t,oi a total dominating set D such that the subgraph induced by V ( G ) \ D is isomorphic to an empty graph
We have introduced and studied the total outer k-independent domination number of graphs
Summary
Theory of domination in graphs is one of the most important topics in graph theory. In the last few decades, the interest in this area has increased, due to its applications to different fields of science, such as linear algebra, communication networks, social sciences, computational complexity, algorithm design, complex ecosystems, optimization problems, among others (for example, see [1,2]). A TOkID set of cardinality total outer k-independent domination number of G and is denoted by γt,oi k ( G ) is a γk ( G )-set. When k = 1, a TOkID set is a total outer-independent dominating set, that is, γt,oi t,oi a total dominating set D such that the subgraph induced by V ( G ) \ D is isomorphic to an empty graph. This last concept was introduced in [8] and barely looked at in [9] under the name of total co-independent domination number.
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