Exact $2$-step domination in graphs

Step Domination in Graphs

Exact double domination in graphs

In a graph a vertex is said to dominate itself and all its neighbours. A doubly dominating set of a graph G is a subset of vertices that dominates every vertex of G at least twice. A doubly dominating set is exact if every vertex of G is dominated exactly twice. We prove that the existence of an exact doubly dominating set is an NP-complete problem. We show that if an exact double dominating se...

A Note on Exact n-Step Domination

We generalize to n steps the notion of exact 2-step domination introduced by Chartrand, et al in [2] and suggest a related minimization problem for which we nd a lower bound. A graph G is an exact n-step domination graph if there is some set of vertices in G such that each vertex in the graph is distance n from exactly one vertex in the set. We prove that such subsets have order at least blog2 ...

Independent domination in directed graphs

In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, tr...

On 2-rainbow domination and Roman domination in graphs

A Roman dominating function of a graph G is a function f : V → {0, 1, 2} such that every vertex with 0 has a neighbor with 2. The minimum of f (V (G)) = ∑ v∈V f (v) over all such functions is called the Roman domination number γR(G). A 2-rainbow dominating function of a graphG is a function g that assigns to each vertex a set of colors chosen from the set {1, 2}, for each vertex v ∈ V (G) such ...