Roman domination in graphs

Domination-Related Parameters in Rooted Product Graphs

“…Roman domination number was defined by Stewart in [18] and studied further by some researchers, for instance in [4]. Given a graph G = (V, E), a map f : V → {0, 1, 2} is a Roman dominating function for G if for every vertex v with f (v) = 0, there exists a vertex u ∈ N(v) such that f (u) = 2.…”

A note on the independent Roman domination in unicyclic graphs

“…Then we say that x is a private neighbor of a vertex y with f (y) = 2 if f is not an RDF for G−xy. Roman domination has been introduced by Cockayne et al [3] and has been studied for example in [7]. The study of independent Roman domination has been initiated in [1].…”

The Roman Domination Problem in Unit Disk Graphs

“…Some useful facts on Roman dominating functions were proved in [1], which will be used in our following discussions.…”

“…A set D ⊆ V is a dominating set if every vertex in V \D is adjacent to at least one vertex in D. The domination number, denoted γ(G), is the minimal cardinality of a dominating set in G. The minimum dominating set problem is to compute a dominating set of minimal cardinality for any given graph G. A Roman dominating function [1] of a graph G is defined as a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u) = 0 is adjacent to at least one vertex v with f (v) = 2. The weight of a Roman dominating function is f (V ) = v∈V f (v).…”

“…Minimum Roman domination problem is introduced in [1] as one of variants of classical dominating set problem. It comes with a nice story: The Roman Empire needs to defence itself by positioning some legions on the various parts of the Empire in such a way that either (1) a specific region v is also the location of at least one legion or (2) one region u neighboring v has two legions, so that u can afford sending off one army to the region v (in case of an attack) without loosing self-defence capabilities.…”