In this article we give a new definition of direct product of two arbitrary fuzzy graphs. We define the concepts of domination and total domination in this new product graph. We obtain an upper bound for the total domination number of the product fuzzy graph. Further we define the concept of total α-domination number and derive a lower bound for the total domination number of the product fuzzy ...

A broadcast on a graph G is a function f : V (G) → {0, 1, . . . , diamG} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V (G). The broadcast number of a graph is the minimum value of ∑ v∈V (G) f(v) among all broadcasts f with the property that each vertex of G is within distance f(v) from a vertex v with f(v) > 0. We characterize a class of trees with equal broadcast and domination n...

Almtract--We present a comprehensive survey of the theory of hypercube graphs. Basic properties related to distance, coloring, domination and genus are reviewed. The properties of the n-cube defined by its subgraphs are considered next, including thickness, coarseness, Hamiltonian cycles and induced paths and cycles. Finally, various embedding and packing problems are discussed, including the d...

In this paper, we survey some new results in four areas of domination in graphs, namely: (1) the toughness and matching structure of graphs having domination number 3 and which are “critical” in the sense that if one adds any missing edge, the domination number falls to 2; (2) the matching structure of graphs having domination number 3 and which are “critical” in the sense that if one deletes a...

Given a graph G = (V, E), the subdivision of an edge e = uv ∈ E(G) means the substitution of the edge e by a vertex x and the new edges ux and xv. The domination subdivision number of a graph G is the minimum number of edges of G which must be subdivided (where each edge can be subdivided at most once) in order to increase the domination number. Also, the domination multisubdivision number of G...

In this article we study some variants of the domination concept attending to the connectivity of the subgraph generated by the dominant set. This study is restricted to maximal outerplanar graphs. We establish tight combinatorial bounds for connected domination, semitotal domination, independent domination and weakly connected domination for any n-vertex maximal outerplaner graph.

In this paper, we introduce the concept of k-power domination which is a common generalization of domination and power domination. We extend several known results for power domination to k-power domination. Concerning the complexity of the k-power domination problem, we first show that deciding whether a graph admits a k-power dominating set of size at most t is NP-complete for chordal graphs a...

Dvořák [5] gave a bound on the minimum size of a distance r dominating set in the terms of the maximum size of a distance 2r independent set and generalized coloring numbers, thus obtaining a constant factor approximation algorithm for the parameters in any class of graphs with bounded expansion. We improve and clarify this dependence using an LP-based argument inspired by the work of Bansal an...

In this paper, we provide a new upper bound for the α-domination number. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper bound for the classical domination in graphs. We also prove similar upper bounds for the α-rate domination number, which combines the concepts of...