Journal ArticleDOI
Total domination in graphs
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Results concerning the total domination number of G (the smallest number of vertices in a total dominating set) and the total domatic number ofG (the largest order of a partition of G into total dominating sets) are obtained.Abstract:
A set D of vertices of a finite, undirected graph G = (V, E) is a total dominating set if every vertex of V is adjacent to some vertex of D. In this paper we initiate the study of total dominating sets in graphs and, in particular, obtain results concerning the total domination number of G (the smallest number of vertices in a total dominating set) and the total domatic number of G (the largest order of a partition of G into total dominating sets).read more
Citations
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Journal ArticleDOI
A survey of selected recent results on total domination in graphs
TL;DR: This paper offers a survey of selected recent results on total domination in graphs and defines a set S of vertices in a graph G if every vertex of G is adjacent to some vertex in S.
Journal ArticleDOI
A survey of Nordhaus-Gaddum type relations
Mustapha Aouchiche,Pierre Hansen +1 more
TL;DR: In this paper, Nordhaus and Gaddum gave lower and upper bounds on the sum and product of the chromatic number of a graph and its complement, in terms of the order of the graph.
Journal ArticleDOI
Dominating sets for split and bipartite graphs
TL;DR: It is shown that the problem of finding a minimum cardinality dominating set is NP-complete for split graphs and bipartite graphs.
Journal ArticleDOI
Bibliography on domination in graphs and some basic definitions of domination parameters
TL;DR: The following bibliography on Domination in Graphs has been compiled over the past six years at Clemson University, where it is expected that this bibliography will continue to grow at a steady rate.
References
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Journal ArticleDOI
Towards a theory of domination in graphs
TL;DR: The domatic number of a graph is defined and studied and it is seen that the theory of domination resembles the well known theory of colorings of graphs.
Journal ArticleDOI
On domination and independent domination numbers of a graph
Robert B. Allan,Renu C. Laskar +1 more
TL;DR: Results are obtained for domination number and independent domination number of a graph G, where G does not have an induced subgraph isomorphic to K 1,3, and γ ( G ) = i ( G ).
Journal ArticleDOI
R-Domination in Graphs
TL;DR: A linear algorithm to find a minimum 1-basis (a minimum dominating set) when G is a tree and a linear algorithm that solves the problem for any forest is presented.
Journal ArticleDOI
Disjoint independent dominating sets in graphs
TL;DR: A new parameter, the maximum number of disjoint independent dominating sets in G, is studied and the class of graphs whose vertex sets partition intoindependent dominating sets is investigated.
Book ChapterDOI
Linear Separation of Dominating Sets in Graphs
Claude Benzaken,Peter L. Hammer +1 more
TL;DR: The class of finite undirected graphs G having the property that there exist real positive numbers associated to their vertices so that a set of vertices is dominating if and only if the sum of the corresponding weights exceeds a certain threshold θ is characterized by forbidden induced subgraphs.