Total Roman domination in graphs

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.

A Roman dominating function of a graph G =( V, E)i sa function f : V →{ 0, 1, 2} such that every vertex x with f (x)=0 is adjacent to at least one vertex y with f (y) = 2. The weight of a Roman dominating function is defined to be f (V )= P x∈V f (x), and the minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper we answer an open problem mentioned in by showing that the Roman domination number of an interval graph can be computed in linear time. We also show that the Roman domination number of a cograph can be computed in linear time. Besides, we show that there are polynomial time algorithms for computing the Roman domination numbers of AT-free graphs and graphs with a d-octopus.

2020

Let G=(V,E) be a finite and simple graph of order n and maximum‎ ‎degree Δ(G)‎. ‎A strong Roman dominating function on a‎ ‎graph G is a function f‎:V (G)→{0‎, ‎1,… ,‎lceil‎ ‎ Δ(G)/2 rceil‎+ ‎1} satisfying the condition that every‎ ‎vertex v for which f(v)=0 is adjacent to at least one vertex u ‎for which‎ f(u) ≤ 1‎+ ‎lceil frac{1}{2}| N(u) ∩ V0| rceil‎, ‎where V0={v ∊ V | f(v)=0}. The minimum of the‎ values sumv∊ V f(v), ‎taken over all strong Roman dominating‎ ‎functions f of G‎, ‎is called the strong Roman domination‎ ‎number of G and is denoted by γStR(G)‎. ‎In this paper we‎ ‎continue the study of strong Roman domination number in graphs‎. ‎In‎ particular‎, ‎we present some sharp bounds for γStR(G) and‎ we determine the strong Roman domination number of some graphs‎.

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.

Journal of Physics: Conference Series

An Italian dominating function (or simply, IDF) on a graph G = (V, E) is a function f : V −→ {0, 1, 2} that satisfies the property that for every vertex v ∈ V , with f (v) = 0, u∈N (v) f (u) ≥ 2. The weight of an Italian dominating function f is defined as w(f) = f (V) = u∈V f (u). The minimum weight among all of the Italian dominating functions on a graph G is called the Italian domination number of G, and is denoted by γI (G). A double Roman dominating function (or simply, DRDF) is a function f : V −→ {0, 1, 2, 3} having the property that if f (v) = 0 for a vertex v, then v has at least two adjacent vertices assigned 2 under f or one adjacent vertex assigned 3 under f , and if f (v) = 1, then v has at least one neighbor with f (w) ≥ 2. The weight of a DRDF f is defined as the sum f (V) = v∈V f (v), and the minimum weight of a DRDF on G is the double Roman domination number of G, denoted by γdR(G). In this paper we show that γdR(G)/2 ≤ γI (G) ≤ 2γdR(G)/3, and characterize all trees T with γI (T) = 2γdR(T)/3.

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.

European Journal of Pure and Applied Mathematics

Let G be a connected graph. A function f : V (G) → {0, 1, 2, 3} is a double Roman dominating function of G if for each v ∈ V (G) with f(v) = 0, v has two adjacent vertices u and w for which f(u) = f(w) = 2 or v has an adjacent vertex u for which f(u) = 3, and for each v ∈ V (G) with f(v) = 1, v is adjacent to a vertex u for which either f(u) = 2 or f(u) = 3. The minimum weight ωG(f) = P v∈V (G) f(v) of a double Roman dominating function f of G is the double Roman domination number of G. In this paper, we continue the study of double Roman domination introduced and studied by R.A. Beeler et al. in [2]. First, we characterize some double Roman domination numbers with small values in terms of the domination numbers and 2-domination numbers. Then we determine the double Roman domination numbers of the join, corona, complementary prism and lexicographic product of graphs.

RAIRO - Operations Research

For a graph G = (V, E), a restrained double Roman dominating function is a function f : V → {0, 1, 2, 3} having the property that if f(v) = 0, then the vertex v must have at least two neighbors assigned 2 under f or one neighbor w with f(w) = 3, and if f(v) = 1, then the vertex v must have at least one neighbor w with f(w) ≥ 2, and at the same time, the subgraph G[V0] which includes vertices with zero labels has no isolated vertex. The weight of a restrained double Roman dominating function f is the sum f(V) = ∑v∈V f(v), and the minimum weight of a restrained double Roman dominating function on G is the restrained double Roman domination number of G. We initiate the study of restrained double Roman domination with proving that the problem of computing this parameter is NP-hard. Then we present an upper bound on the restrained double Roman domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. We study the restrained doub...

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.

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.

2020

A {em Roman dominating function} on a graph $G$ is a function $f:V(G)rightarrow {0,1,2}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A {em total Roman dominating function} is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has no isolated vertices. The weight of a total Roman dominating function $f$ is the value $Sigma_{uin V(G)}f(u)$. The {em total Roman domination number} of $G$, $gamma_{tR}(G)$, is the minimum weight of a total Roman dominating function in $G$.The {em total Roman domination subdivision number} ${rmsd}_{gamma_{tR}}(G)$ of a graph $G$ is the minimum number of edges that must besubdivided (each edge in $G$ can be subdivided at most once) inorder to increase the total Roman domination number. In this paper,we initiate the study of total Roman domination subdivisionnumber in graphs and we present sharp...

Discrete Mathematics, 2012

A Roman dominating function of a graph G is a function f : V (G) → {0, 1, 2} such that whenever f (v) = 0 there exists a vertex u adjacent to v with f (u) = 2. The weight of f is w(f) =  v∈V (G) f (v). The Roman domination number γ R (G) of G is the minimum weight of a Roman dominating function of G. This paper establishes a sharp upper bound on the Roman domination numbers of graphs with minimum degree at least 3. An upper bound on the Roman domination numbers of connected, big-claw-free and big-net-free graphs is also given.

Motivated by the article in Scientific American [7], 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 w(f ) = P v∈V f (v). The Roman domination number (RDN) denoted by γ R (G) is the minimum weight among all RDF in G. If V − D contains a roman dominating function f 1 : V → {0, 1, 2}. "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 IRDN.

Journal of Combinatorial Optimization, 2014

In this paper we continue the study of Roman dominating functions in graphs. A signed Roman dominating function (SRDF) on a graph G = (V , E) is a function f : V → {−1, 1, 2} satisfying the conditions that (i) the sum of its function values over any closed neighborhood is at least one and (ii) for every vertex u for which f (u) = −1 is adjacent to at least one vertex v for which f (v) = 2. The weight of a SRDF is the sum of its function values over all vertices. The signed Roman domination number of G is the minimum weight of a SRDF in G. We present various lower and upper bounds on the signed Roman domination number of a graph. Let G be a graph of order n and size m with no isolated vertex. We show that γ sR (G) ≥ 3 √ 2 √ n − n and that γ sR (G) ≥ (3n − 4m)/2. In both cases, we characterize the graphs achieving equality in these bounds. If G is a bipartite graph of order n, then we show that γ sR (G) ≥ 3 √ n + 1 − n − 3, and we characterize the extremal graphs.

Discrete Applied Mathematics, 2017

Based on the history that the Emperor Constantine decreed that any undefended place (with no legions) of the Roman Empire must be protected by a "stronger" neighbor place (having two legions), a graph theoretical model called Roman domination in graphs was described. A Roman dominating function for a graph G = (V, E), is a function f : V → {0, 1, 2} such that every vertex v with f (v) = 0 has at least a neighbor w in G for which f (w) = 2. The Roman domination number of a graph is the minimum weight, v∈V f (v), of a Roman dominating function. In this paper we initiate the study of a new parameter related to Roman domination, which we call strong Roman domination number and denote it by γ StR (G). We approach the problem of a Roman domination-type defensive strategy under multiple simultaneous attacks and begin with the study of several mathematical properties of this invariant. In particular, we first show that the decision problem regarding the computation of the strong Roman domination number is NP-complete, even when restricted to bipartite graphs. We obtain several bounds on such a parameter and give some realizability results for it. Moreover, we prove that for any tree T of order n ≥ 3, γ StR (T) ≤ 6n/7 and characterize all extremal trees.