For a connected graph G = (V,E), a set S ⊆ E is called an edge-to-vertex geodetic set of G if every vertex of G is either incident with an edge of S or lies on a geodesic joining some pair of edges of S. The minimum cardinality of an edge-to-vertex geodetic set of G is gev(G). Any edge-to-vertex geodetic set of cardinality gev(G) is called an edge-to-vertex geodetic basis of G. A subset T ⊆ S i...

The cardinality of a maximum minimal dominating set of a graph is called its upper domination number. The problem of computing this number is generally NP-hard but can be solved in polynomial time in some restricted graph classes. In this work, we consider the complexity and approximability of the weighted version of the problem in two special graph classes: planar bipartite, split. We also pro...

For any $k in mathbb{N}$, the $k$-subdivision of graph $G$ is a simple graph $G^{frac{1}{k}}$, which is constructed by replacing each edge of $G$ with a path of length $k$. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $G$ has been introduced as a fractional power of $G$, denoted by ...

For two vertices u and v of a graph G, the closed interval I[u, v] consists of u, v, and all vertices lying in some u − v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u, v] for u, v ∈ S. If I[S] = V (G), then S is a geodetic set for G. The geodetic number g(G) is the minimum cardinality of a geodetic set. A set S of vertices in a graph G is uniform if the ...

Let K n denote the Cartesian product Kn Kn Kn, where Kn is the complete graph on n vertices. We show that the domination number of K n

In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination num...