Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that for every vertex with label 0, there is a vertex with label 2 at distance at most k from each other. The weight of a k-distance Roman dominating function f is the value ω(f) = ∑ v∈V f(v). The k-distance Roman domination number ...

The domination number of a graph G = (V,E) is the minimum cardinality of any subset S ⊂ V such that every vertex in V is in S or adjacent to an element of S. Finding the domination numbers of m by n grids was an open problem for nearly 30 years and was finally solved in 2011 by Goncalves, Pinlou, Rao, and Thomassé. Many variants of domination number on graphs, such as double domination number a...

The k-distance domination problem is to find a minimum vertex set D of a graph such that every vertex of the graph is either in D or within distance k from some vertex of D, where k is a positive integer. In the present paper, by using labeling method, a linear-time algorithm for k-distance domination problem on block graphs is designed.

A set D of vertices in a connected graph G is called a k-dominating set if every vertex in G − D is within distance k from some vertex of D. The k-domination number of G, γk(G), is the minimum cardinality over all k-dominating sets of G. A graph G is k-distance domination-critical if γk(G − x) < γk(G) for any vertex x in G. This work considers properties of k-distance domination-critical graphs...

We study graph searching games where a number of cops try to capture a robber that is hiding in a system of tunnels modelled as a graph. While the current position of the robber is unknown to the cops, each cop can see a certain radius d around his position. For the case d = 1 these games have been studied by Fomin, Kratsch and Müller [7] under the name domination games. We are primarily intere...

This paper discusses a distance guarding concept on triangulation graphs, which can be associated with distance domination and distance vertex cover. We show how these subjects are interconnected and provide tight bounds for any n-vertex maximal outerplanar graph: the 2d-guarding number, g2d(n) = ⌊ n 5 ⌋; the 2d-distance domination number, γ2d(n) = ⌊ n 5 ⌋; and the 2d-distance vertex cover numb...

A set S of vertices in a graph G = (V,E) is called a total k-distance dominating set if every vertex in V is within distance k of a vertex in S. A graph G is total k-distance domination-critical if γ t (G − x) < γ t (G) for any vertex x ∈ V (G). In this paper, we investigate some results on total k-distance domination-critical of graphs.

A set D ⊆ V of vertices is said to be a (connected) distance k-dominating set of G if the distance between each vertex u ∈ V − D and D is at most k (and D induces a connected graph in G). The minimum cardinality of a (connected) distance k-dominating set in G is the (connected) distance k-domination number of G, denoted by γk(G) (γ c k (G), respectively). The set D is defined to be a total k-do...