A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...

In this note by considering a complete lattice L, we define thenotion of an L-Fuzzy hyperrelation on a given non-empty set X. Then wedefine the concepts of (POM)L-Fuzzy graph, hypergraph and subhypergroupand obtain some related results. In particular we construct the categories ofthe above mentioned notions, and give a (full and faithful) functor form thecategory of (POM)L-Fuzzy subhypergroups ...

A set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding ...

We give the first linear kernels for Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a topological minor. In other words, we give polynomial time algorithms that, for a given H-topological-minor free graph G and a positive integer k, output an H-topological-minor free graph G′ on O(k) vertices such that G has a (connected) dominating set of size k if ...

It was a long-standing open problem whether the minimum weight dominating set in unit disk graphs has a polynomial-time constant-approximation. In 2006, Ambühl et al solved this problem by presenting a 72-approximation for the minimum weight dominating set and also a 89-approximation for the minimum weight connected dominating set in unit disk graphs. In this paper, we improve their results by ...

Fuzzy graphs (FGs), broadly known as fuzzy incidence (FIGs), have been recognized being an effective tool to tackle real-world problems in which vague data and information are essential. Dominating sets (DSs) multiple applications diverse areas of life. In wireless networking, DSs used find efficient routes with ad hoc mobile networks. this paper, we extend the concept domination FGs FIGs show ...