The differential and the roman domination number of a graph

Abstract

Let G = (V, E) be a graph of order n and let B(S) be the set of vertices in V \ S that have a neighbor in the vertex set S. The differential of a vertex set S is defined as ∂(S) = |B(S)| − |S| and the maximum value of ∂(S) for any subset S of V is the differential of G. A Roman dominating function of G is a function f : V → {0, 1, 2} such that every vertex u with f (u) = 0 is adjacent to a vertex v with f (v) = 2. The weight of a Roman dominating function is the value f (V ) = u∈V f (u). The minimum weight of a Roman dominating function of a graph G is the Roman domination number of G, written γR(G). We prove that γR(G) = n − ∂(G) and present several combinatorial, algorithmic and complexity-theoretic consequences thereof.

Lower bounds on the Roman and independent Roman domination numbers

“…It is worth mentioning that there is a known lower bound due Bermudo, Fernau, and Sigarreta [3] that is not on this list. In [3], the authors prove a very interesting relationship between Roman domination and the differential of a graph. For a set S, let B(S) be the set of vertices in V − S that have a neighbor in the set S. The differential of a set S is defined in [10] as ∂(S) = |B(S)| − |S|, and the maximum value of ∂(S) for any subset S of V is the differential of G, denoted ∂(G).…”

Approximation Algorithms Inspired by Kernelization Methods

“…If D can be partitioned as D = D 1 ∪ D 2 such that, for all x ∈ V \ D, there is a y ∈ D 2 ∩ N (x), then (D 2 , D 1 ) defines a Roman domination function f D1,D2 : V → {0, 1, 2} such that f D1,D2 (V ) = 2|D 2 |+|D 1 |. According to [10], |V | − f D1,D2 (V ) is also known as the differential of a graph (as introduced in [25]) if f D1,D2 (V ) is smallest possible. 4.…”

β-Differential of a Graph

“…In particular, several bounds for ∂(G) were given. The differential of a graph has also been investigated in [4,[6][7][8][9][10][11][12][13][14][15], and it was proved in [11] that ∂(G) + γ R (G) = n, where n is the order of the graph G and γ R (G) is the Roman domination number of G, so every bound for the differential of a graph can be used to get a bound for the Roman domination number. The differential of a set D was also considered in [16], where it was denoted by η(D), and the minimum differential of an independent set was considered in [17].…”