Several of the best known problems and conjectures in graph theory arise in studying the behavior of a graphical invariant on a graph product. Examples of this are Vizing’s conjecture, Hedetniemi’s conjecture and the calculation of the Shannon capacity of graphs, where the invariants are the domination number, the chromatic number and the independence number on the Cartesian, categorical and st...

Let G be a connected graph. A subset S ⊆ V (G) is strong resolving dominating set of if and for every pair vertices u, v ∈ (G), there exists vertex w such that u IG[v, w] or IG[u, w]. The smallest cardinality called the domination number G. In this paper, we characterize sets in lexicographic product graphs determine corresponding number.

Let γ(G) be the domination number of a graph G. It is shown that for any k ≥ 0 there exists a Cartesian graph bundle B φF such that γ(B φF ) = γ(B)γ(F )−2k. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing’s conjecture on strong graph bundles is shown not to be true by proving the inequ...

The existence of a constant time algorithm for solving different domination problems on the subclass of polygraphs, rotagraphs and fasciagraphs, is shown by means of path algebras. As these graphs include products (the Cartesian, strong, direct, lexicographic) of paths and cycles, we implement the algorithm to get formulas in the case of the domination numbers, the Roman domination numbers and ...

The power system monitoring problem asks for as few as possible measurement devices to be put in an electric power system. The problem has a graph theory model involving power dominating sets in graphs. The power domination number γP (G) of G is the minimum cardinality of a power dominating set. Dorfling and Henning [2] determined the power domination number of the Cartesian product of paths. I...