On the signed strong total Roman domination number of graphs

Filomat, 2019

In this paper we continue the study of signed double Roman dominating functions in graphs. A signed double Roman dominating function (SDRDF) on a graph G = (V, E) is a function f : V(G) → {−1, 1, 2, 3} having the property that for each v ∈ V(G), f [v] ≥ 1, and if f (v) = −1, 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 leat one neighbor w with f (w) ≥ 2. The weight of a SDRDF is the sum of its function values over all vertices. The signed double Roman domination number γ sdR (G) is the minimum weight of a SDRDF on G. We present several lower bounds on the signed double Roman domination number of a graph in terms of various graph invariants. In particular, we show that if G is a graph of order n and size m with no isolated vertex, then γ sdR (G) ≥ 19n−24m 9 and γ sdR (G) ≥ 4 n 3 − n. Moreover, we characterize the graphs attaining equality in these two bounds.

Journal of Combinatorial Optimization, 2013

Let D be a finite and simple digraph with vertex set V (D) and arc set A(D). A signed Roman dominating function (SRDF) on the digraph D is a function f :

Discrete Applied Mathematics

Let G = (V, E) be a simple and finite graph with vertex set V (G), and let k ≥ 1 be an integer. A signed double Roman k-dominating function (SDRkDF) on a graph G is a function f : V (G) → {−1, 1, 2, 3} such that (i) every vertex v with f (v) = −1 is adjacent to at least two vertices assigned with 2 or to at least one vertex w with f (w) = 3, (ii) every vertex v with f (v) = 1 is adjacent to at least one vertex w with f (w) ≥ 2 and (iii) u∈N [v] f (u) ≥ k holds for any vertex v. The weight of an SDRkDF f is u∈V (G) f (u), and the minimum weight of an SDRkDF is the signed double Roman k-domination number γ k sdR (G) of G. In this paper, we initiate the study of the signed double Roman k-domination number in graphs and we present lower and upper bounds for γ k sdR (T). In addition we determine this parameter for some classes of graphs.

Tamkang Journal of Mathematics, 2016

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arc set $A(D)$. A twin signed Roman dominating function (TSRDF) on the digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the conditions that (i) $\sum_{x\in N^-[v]}f(x)\ge 1$ and $\sum_{x\in N^+[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ (resp. $N^+[v]$) consists of $v$ and all in-neighbors (resp. out-neighbors) of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ has an in-neighbor $v$ and an out-neighbor $w$ for which $f(v)=f(w)=2$. The weight of an TSRDF $f$ is $\omega(f)=\sum_{v\in V(D)}f(v)$. The twin signed Roman domination number $\gamma_{sR}^*(D)$ of $D$ is the minimum weight of an TSRDF on $D$. In this paper, we initiate the study of twin signed Roman domination in digraphs and we present some sharp bounds on $\gamma_{sR}^*(D)$. In addition, we determine the twin signed Roman domination number of some classes of digraphs.

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.

European Journal of Pure and Applied Mathematics, 2020

A perfect Roman dominating function on a graph G = (V (G), E(G)) is a function f : V (G) → {0, 1, 2} for which each u ∈ V (G) with f(u) = 0 is adjacent to exactly one vertex v ∈ V (G) with f(v) = 2. The weight of a perfect Roman dominating function f is the value ωG(f) = Pv∈V (G) f(v). The perfect Roman domination number of G is the minimum weight of a perfect Roman dominating function on G. In this paper, we study the perfect Roman domination numbers of graphs under some binary operation

Applicable Analysis and Discrete Mathematics, 2016

A Roman dominating function on a graph G is a function f : V (G) → {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 f is the sum, u∈V (G) f (u), of the weights of the vertices. The Roman domination number is the minimum weight of a Roman dominating function in G. A total Roman domination 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 vertex. The total Roman domination number is the minimum weight of a total Roman domination function on G. We establish lower and upper bounds on the total Roman domination number. We relate the total Roman domination to domination parameters, including the domination number, the total domination number and Roman domination number.

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.

Discussiones Mathematicae Graph Theory, 2016

Let k ≥ 1 be an integer, and G = (V, E) be a finite and simple graph. The closed neighborhood N G [e] of an edge e in a graph G is the set consisting of e and all edges having a common end-vertex with e. A signed Roman edge k-dominating function (SREkDF) on a graph G is a function f : E → {−1, 1, 2} satisfying the conditions that (i) for every edge e of G, x∈N [e] f (x) ≥ k and (ii) every edge e for which f (e) = −1 is adjacent to at least one edge e for which f (e) = 2. The minimum of the values e∈E f (e), taken over all signed Roman edge k-dominating functions f of G, is called the signed Roman edge k-domination number of G and is denoted by γ sRk (G). In this paper we establish some new bounds on the signed Roman edge k-domination number.

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.

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 [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.

Annals of the University of Craiova - Mathematics and Computer Science Series

Let D = (V, A) be a finite simple digraph. A signed double Roman dominating function (SDRD-function) on the digraph D is a function f : V (D) → {−1, 1, 2, 3} satisfying the following conditions: (i) x∈N − [v] f (x) ≥ 1 for each v ∈ V (D), where N − [v] consist of v and all in-neighbors of v, and (ii) if f (v) = −1, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor assigned 3, while if f (v) = 1, then the vertex v must have at least one in-neighbor assigned 2 or 3. The weight of a SDRD-function f is the value x∈V (D) f (x). The signed double Roman domination number (SDRD-number) γ sdR (D) of a digraph D is the minimum weight of a SDRD-function on D. In this paper we study the SDRD-number of digraphs, and we present lower and upper bounds for γ sdR (D) in terms of the order, maximum degree and chromatic number of a digraph. In addition, we determine the SDRD-number of some classes of digraphs.