A set S of vertices in a graph G is a 2-dominating set if every vertex of G not in S is adjacent to at least two vertices in S, and S is a 2-independent set if every vertex in S is adjacent to at most one vertex of S. The 2-domination number γ2(G) is the minimum cardinality of a 2-dominating set in G, and the 2-independence number α2(G) is the maximum cardinality of a 2-independent set in G. Ch...

A set S of vertices of a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by i(G), is the minimum cardinality of an independent dominating set of G. In this paper, some new classes of graphs with equal domination and independent domination numbers are presented and exa...

In this note, we prove several lower bounds on the domination number of simple connected graphs. Among these are the following: the domination number is at least two-thirds of the radius of the graph, three times the domination number is at least two more than the number of cut-vertices in the graph, and the domination number of a tree is at least as large as the minimum order of a maximal matc...

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs γt-critical. In this paper, we determine the domination and the total domination number in the Circulant graphs Cn〈1, 3〉, and then study γ-criticality...

Harary, F. , 1997 Graph Theory, Narosa Publishing House. Arumugam, S. , Joseph, J. P. 1999 On graphs with equal domination and connected domination numbers, Discrete Mathematics vol. 206, 45-49. Deo, N. 2005 Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall of India Private Limited. Saoud, M. , Jebran, J. 2009 Finding A Minimum Dominating Set by Transforming Domi...

Let G = (V,E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the ...

The domination polynomial of a graph G of order n is the polynomial D(G, x) = Pn i=γ(G) d(G, i)x , where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. In this paper, we obtain some properties of the coefficients of D(G, x). Also, by study of the dominating sets and the domination polynomials of specific graphs denoted by G′(m), we obtain a rela...

A directed dominating set in a directed graph D is a set S of vertices of V such that every vertex u ∈ V (D) \ S has an adjacent vertex v in S with v directed to u. The directed domination number of D, denoted by γ(D), is the minimum cardinality of a directed dominating set in D. The directed domination number of a graph G, denoted Γd(G), which is the maximum directed domination number γ(D) ove...

We study the monoid of global invariant types modulo domination-equivalence in context o-minimal theories. reduce its computation to problem proving that it is generated by classes 1-types. show this hold Real Closed Fields, where generators correspond convex subrings monster model. Combined with [C. Ealy, D. Haskell and J. Maríková, Residue field domination real closed valued fields, Notre Dam...