We consider new variants of the vertex/edge domination problems on graphs. A vertex is said to dominate itself and its all adjacent vertices, and similarly an edge is said to dominate itself and its all adjacent edges. Given an input graph G = (V,E) and an integer k, the k-Vertex (k-Edge) Maximum Domination (k-MaxVD and k-MaxED, respectively) is to find a subset DV ⊆ V of vertices (resp., DE ⊆ ...

We treat a variation of graph domination which involves partition ( V 1 , 2 ,..., k ) the vertex set G and each class i over distance d where all vertices edges may be used in process. Strict upper bounds extremal graphs are presented; results collected three handy tables. Further, we compare high number classes dominators needed.

In this paper, we continue the study of power domination in graphs (see SIAM J. Discrete Math. 15 (2002), 519–529; SIAM J. Discrete Math. 22 (2008), 554–567; SIAM J. Discrete Math. 23 (2009), 1382–1399). Power domination in graphs was birthed from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A set of vertices is defined to b...

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the k-tuple domination problem is to find a minimum sized vertex subset in a graph such that every vertex in the graph is dominated by at least k vertices in this set. The current paper studies k-tuple domination in graphs from an algorithmic point of view. In particular, we give a linear...

Let k be a positive integer and G = (V,E) be a graph of minimum degree at least k − 1. A function f : V → {−1, 1} is called a signed k-dominating function of G if ∑ u∈NG[v] f(u) ≥ k for all v ∈ V . The signed k-domination number of G is the minimum value of ∑ v∈V f(v) taken over all signed k-dominating functions of G. The signed total k-dominating function and signed total k-domination number o...

Let γ(G) be the domination number of a graph G, and let αk(G) be the maximum number of vertices in G, no two of which are at distance ≤ k in G. It is easy to see that γ(G) ≥ α2(G). In this note it is proved that γ(G) is bounded from above by a linear function in α2(G) if G has no large complete bipartite graph minors. Extensions to other parameters αk(G) are also derived.

The k-domination number γk(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is either in D or adjacent to at least k vertices in D. In 2010, the conjecture-generating computer program, Graffiti.pc, was queried for upperbounds on the 2-domination number. In this paper we prove new upper bounds on the 2-domination number of a g...

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...