The concept of vague graph was introduced by Ramakrishna in [12]. A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility, and compatibility to a system when compared with systems that are designed using fuzzy graphs. The main purpose of this paper is to introduce the concept of dominating set, perfect dominating set, minimal perfect dominating set and i...

A subset Q ⊆ V (G) is a dominating set of a graph G if each vertex in V (G) is either in Q or is adjacent to a vertex in Q. A dominating set Q of G is minimal if Q contains no dominating set of G as a proper subset. In this paper we study the number of minimal dominating sets in some classes of trees. Mathematics Subject Classification: 05C69

For an arbitrary undirected simple graph G with m edges, we give an algorithm with running time O(m|L|) to generate the set L of all minimal edge dominating sets of G. For bipartite graphs we obtain a better result; we show that their minimal edge dominating sets can be enumerated in time O(m|L|). In fact our results are stronger; both algorithms generate the next minimal edge dominating set wi...

We show that interval graphs on n vertices have at most 3 ≈ 1.4422 minimal dominating sets, and that these can be enumerated in time O∗(3n/3). As there are examples of interval graphs that actually have 3 minimal dominating sets, our bound is tight. We show that the same upper bound holds also for trees, i.e. trees on n vertices have at most 3 ≈ 1.4422 minimal dominating sets. The previous best...

A signed graph is a graph where the edges are assigned either positive ornegative signs. Net degree of a signed graph is the dierence between the number ofpositive and negative edges incident with a vertex. It is said to be net-regular if all itsvertices have the same net-degree. Laplacian energy of a signed graph is defined asε(L(Σ)) =|γ_1-(2m)/n|+...+|γ_n-(2m)/n| where γ_1,...,γ_n are the ei...

The collection of the vertex dominating sets of a graph defines a hypergraph on the set of vertices of the graph. However, there are hypergraphs H that are not the collection of the vertex dominating sets of any graph. This paper deals with the question of completing these hypergraphs H to the vertex dominating sets of some graphs G. We demonstrate that such graphs G exist and, in addition, we ...

A graph is said to be well-dominated if all its minimal dominating sets are of the same size. In this work, we introduce the notion of an irreducible dominating set, a variant of dominating set generalizing both minimal dominating sets and minimal total dominating sets. Based on this notion, we characterize the family of minimal dominating sets in a lexicographic product of two graphs and deriv...

A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by γt. The maximal size of an inclusionwise minimal total dominating set, the upper total domination num...

We show that all minimal dominating sets of a chordal bipartite graph can be generated in incremental polynomial, hence output polynomial, time. Enumeration of minimal dominating sets in graphs is equivalent to enumeration of minimal transversals in hypergraphs. Whether the minimal transversals of a hypergraph can be enumerated in output polynomial time is a well-studied and challenging questio...