On the strong Roman domination number of graphs

Applied Mathematics and Computation, 2021

The Roman domination in graphs is well-studied in graph theory. The topic is related to a defensive strategy problem in which the Roman legions are settled in some secure cities of the Roman Empire. The deployment of the legions around the Empire is designed in such a way that a sudden attack to any undefended city could be quelled by a legion from a strong neighbour. There is an additional condition: no legion can move if doing so leaves its base city defenceless. In this manuscript we start the study of a variant of Roman domination in graphs: the triple Roman domination. We consider that any city of the Roman Empire must be able to be defended by at least three legions. These legions should be either in the attacked city or in one of its neighbours. We determine various bounds on the triple Roman domination number for general graphs, and we give exact values for some graph families. Moreover, complexity results are also obtained.

Proceedings - Mathematical Sciences, 2015

Let G = (V , E) be a graph and let f : V →{0, 1, 2} be a function. A vertex u is said to be protected with respect to f if f(u) > 0orf(u) = 0andu is adjacent to a vertex with positive weight. The function f is a co-Roman dominating function (CRDF) if: (i) every vertex in V is protected, and (ii) each v ∈ V with f(v) > 0 has a neighbor u ∈ V with f(u) = 0 such that the function f vu : V →{ 0, 1, 2}, defined by f vu (u) = 1, f vu (v) = f(v) − 1a n df vu (x) = f(x) for x ∈ V \{u, v} has no unprotected vertex. The weight of f is w(f) = v∈V f(v). The co-Roman domination number of a graph G, denoted by γ cr (G), is the minimum weight of a co-Roman dominating function on G. In this paper we initiate a study of this parameter, present several basic results, as well as some applications and directions for further research. We also show that the decision problem for the co-Roman domination number is NP-complete, even when restricted to bipartite, chordal and planar graphs.

Discussiones Mathematicae Graph Theory

Let G = (V, E) be a graph and let f : V (G) → {0, 1, 2} be a function. A vertex v is said to be protected with respect to f , if f (v) > 0 or f (v) = 0 and v is adjacent to a vertex of positive weight. The function f is a co-Roman dominating function if (i) every vertex in V is protected, and (ii) each v ∈ V with positive weight has a neighbor u ∈ V with f (u) = 0 such that the function f uv : V → {0, 1, 2}, defined by f uv (u) = 1, f uv (v) = f (v) − 1 and f uv (x) = f (x) for x ∈ V \ {v, u}, has no unprotected vertex. The weight of f is ω(f) = v∈V f (v). The co-Roman domination number of a graph G, denoted by γ cr (G), is the minimum weight of a co-Roman dominating function on G. In this paper, we give a characterization of graphs of order n for which co-Roman domination number is 2n 3 or n − 2, which settles Full PDF DMGT Page two open problem in [S. Arumugam, K. Ebadi and M. Manrique, Co-Roman domination in graphs, Proc. Indian Acad. Sci. Math. Sci. 125 (2015) 1-10]. Furthermore, we present some sharp bounds on the co-Roman domination number.

—A Strong Roman dominating function (SRDF) is a function f :V 0,1,2,3 satisfying the condition that every vertex u for which f u   0 is adjacent to at least one vertex v for which f v   3 and every vertex u for which f u  1 is adjacent to at least one vertex v for which f v   2 . The weight of an SRDF is the value        u V f V f u . The minimum weight of an SRDF on a graph G is called the Strong Roman domination numberof G . In this paper, we attempt to verify some properties on SRDF and moreover we present Strong Roman domination number for some special classes of graphs. Also we show that for a tree T with n  3 vertices, l leaves and s support vertices, we have   4 6n l s SR T     and we characterize all trees achieving this bound.

AKCE International Journal of Graphs and Combinatorics

A total Roman dominating function on a graph G is a function f : V ! f0, 1, 2g satisfying the conditions: (i) every vertex u with f(u) ¼ 0 is adjacent to at least one vertex v of G for which f(v) ¼ 2; (ii) the subgraph induced by the vertices assigned non-zero values has no isolated vertices. The minimum of f ðVðGÞÞ ¼ P v2V f ðvÞ over all such functions is called the total Roman domination number c tR ðGÞ: The total Roman domination stability number of a graph G with no isolated vertex, denoted by st c tR ðGÞ, is the minimum number of vertices whose removal does not produce isolated vertices and changes the total Roman domination number of G. In this paper we present some bounds for the total Roman domination stability number of a graph, and prove that the associated decision problem is NP-hard even when restricted to bipartite graphs or planar graphs.

Discrete Mathematics, 2004

A Roman dominating function 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 a Roman dominating function is the value f (V ) = u∈V f (u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper we study the graph theoretic properties of this variant of the domination number of a graph.

Motivated by the article in Scientific American [8], Michael A Henning and Stephen T. Hedetniemi explored the strategy of defending the Roman Empire. Cockayne defined Roman dominating function (RDF) on a Graph G = (V, E) to be 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. For a real valued function f : V → R the weight of f is ( ) ( ). v V w f f v ∈ = ∑ The Roman Domination Number (RDN) denoted by γR (G) is the minimum weight among all RDF in G. If V -D contains a Roman dominating function f 1 : V → {0, 1, 2}, where D is the set of vertices v for which f (v) > 0. Then f 1 is called inverse Roman dominating function (IRDF) on a graph G w.r.t. f. The inverse Roman domination number (IRDN) denoted by γ 1 R(G) is the minimum weight among all IRDF in G. In this paper we find few results of RDN and IRDN.

RAIRO - Operations Research, 2020

Let G = (V, E) be a simple graph with vertex set V and edge set E. A mixed Roman dominating function (MRDF) of G is a function f : V ∪ E → {0, 1, 2} satisfying the condition that every element x ∈ V ∪ E for which f (x) = 0 is adjacent or incident to at least one element y ∈ V ∪ E for which f (y) = 2. The weight of a mixed Roman dominating function f is ω(f) = x∈V ∪E f (x). The mixed Roman domination number γ * R (G) of G is the minimum weight of a mixed Roman dominating function of G. We first show that the problem of computing γ * R (G) is NP-complete for bipartite graphs and then we present upper and lower bounds on the mixed Roman domination number, some of them are for the class of trees.

Discrete Applied Mathematics, 2017

A double Roman dominating function (DRDF) on a graph G = (V , E) is a function f : V (G) → {0, 1, 2, 3} having the property that if f (v) = 0, then vertex v has at least two neighbors assigned 2 under f or one neighbor w with f (w) = 3, and if f (v) = 1, then vertex v must have at least one neighbor w with f (w) ≥ 2. The weight of a DRDF is the value f (V (G)) = ∑ u∈V (G) f (u). The double Roman domination number γ dR (G) is the minimum weight of a DRDF on G. First we show that the decision problem associated with γ dR (G) is NP-complete for bipartite and chordal graphs. Then we present some sharp bounds on the double Roman domination number which partially answer an open question posed by Beeler et al. (2016) in their introductory paper on double Roman domination. Moreover, a characterization of graphs G with small γ dR (G) is provided.