A Roman dominating function (RDF) on a graph G = (V,E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of f is w(f) = ∑ v∈V f(v). The Roman domination number is the minimum weight of an RDF in G. It is known that for every graph G, the Roman domination number of G is bounded above...

The three domatic number problem asks whether a given undirected graph can be partitioned into at least three dominating sets, i.e., sets whose closed neighborhood equals the vertex set of the graph. Since this problem is NP-complete, no polynomial-time algorithm is known for it. The naive deterministic algorithm for this problem runs in time 3n, up to polynomial factors. In this paper, we desi...

We prove a new structural property regarding the “skyline” of uniform radius disks and use this to derive a number of new sequential and distributed approximation algorithms for well-known optimization problems on unit disk graphs (UDGs). Specifically, the paper presents new approximation algorithms for two problems: domatic partition and weighted minimum dominating set (WMDS) on UDGs, both of ...

For any integer k ≥ 1, a signed (total) k-dominating function is a function f : V (G) → {−1, 1} satisfying w∈N [v] f(w) ≥ k ( P w∈N(v) f(w) ≥ k) for every v ∈ V (G), where N(v) = {u ∈ V (G)|uv ∈ E(G)} and N [v] = N(v)∪{v}. The minimum of the values ofv∈V (G) f(v), taken over all signed (total) k-dominating functions f, is called the signed (total) k-domination number and is denoted by γkS(G) (γ...

A Roman dominating function of a graph G is a labeling f : V (G) −→ {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. The Roman domination subdivision number sdγR(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order t...

LetD = (V,A) be a finite and simple digraph. A Roman dominating function (RDF) on a digraph D is a labeling f : V (D) → {0, 1, 2} such that every vertex with label 0 has a in-neighbor with label 2. The weight of an RDF f is the value ω(f) = ∑ v∈V f(v). The Roman domination number of a digraph D, denoted by γR(D), equals the minimum weight of an RDF on D. In this paper we present some sharp boun...