Mustapha Chellali

Saad Dahlab University · Department of Mathematics

An independent Roman dominating function (IRD-function) on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the conditions that (i) every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$, and (ii) the set of all vertices assigned non-zero values under $f$ is independent. The weight of an IRD-f...

We continue the study of restrained double Roman domination in graphs. For a graph G=(V(G),E(G)), a double Roman dominating function f is called a restrained double Roman dominating function (RDRD function) if the subgraph induced by {v∈V(G)∣f(v)=0} has no isolated vertices. The restrained double Roman domination number (RDRD number) γrdR(G) is the...

The paired domination subdivision number of a graph G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the paired domination number of G. In this note, we show that the problem of computing the paired-domination subdivision number is NP-hard for bipartite graphs.

A subset S of vertices in a graph G=(V,E) is 2-independent if every vertex of S has at most one neighbor in S. The 2-independence number is the maximum cardinality of a 2-independent set of G. In this paper, we initiate the study of the 2-independence subdivision number sdβ2(G) defined as the minimum number of edges that must be subdivided (each ed...

A restrained Roman dominating function (RRD-function) on a graph \(G=(V,E)\) is a function \(f\) from \(V\) into \(\{0,1,2\}\) satisfying: (i) every vertex \(u\) with \(f(u)=0\) is adjacent to a vertex \(v\) with \(f(v)=2\); (ii) the subgraph induced by the vertices assigned 0 under \(f\) has no isolated vertices. The weight of an RRD-function is t...

A quasi-total Roman dominating function (QTRD-function) on G=(V,E) is a function f:V→{0,1,2} such that (i) every vertex x for which f(x) = 0 is adjacent to at least one vertex v for which f(v) = 2, and (ii) if x is an isolated vertex in the subgraph induced by the set of vertices with non-zero values, then f(x) = 1. The weight of a QTRD-function is...

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number γpr(G) is the minimum cardinality of a paired-dominating set of G. In this paper, we prove the conjecture of Goddard and Henning (2009) stating that if G is a connected cubic graph of order n and G is diff...

A double Roman dominating function (DRDF) on a graph G=(V,E) is a function f:V→{0,1,2,3} such that every vertex u with f(u) = 0 is adjacent to at least one vertex assigned a 3 or to at least two vertices assigned a 2, and every vertex v with f(v) = 1 is adjacent to at least one vertex assigned 2 or 3. The weight of a DRDF is the sum of its function...

Let [Formula: see text] be an integer and G a simple graph with vertex set V(G). Let f be a function that assigns labels from the set [Formula: see text] to the vertices of G. For a vertex [Formula: see text] the active neighbourhood AN(v) of v is the set of all vertices w adjacent to v such that [Formula: see text] A [k]-Roman dominating function...

For a connected graph G, let γ(G) and γc(G) denote the domination number and the connected domination number, respectively. Let H be a graph obtained from a triangle abc by adding a pendant edge at a and a pendant path of length 3 at each of b and c. In 2014, Camby and Schaudt conjectured that for any connected {P9,C9,H}-free graph G, γc(G)≤2γ(G)....

For positive integers j and k with \(j\le k,\) a subset \(S\subseteq V\) in a graph \(G=(V,E)\) is a [j, k]-set if every vertex in \(V-S\) is adjacent to at least j vertices in S but no more than k vertices in S. The [j, k]-domination number \(\gamma _{\left[ j,k\right] }(G)\) is the minimum cardinality of a [j, k]-set in G. The particular cases \(...

A set $S$ of vertices is a restrained dominating set of a graph $G=(V,E)$ if every vertex in $V\setminus S$ has a neighbor in $S$ and a neighbor in $V\setminus S$. The minimum cardinality of a restrained dominating set is the restrained domination number $\gamma_{r}(G)$. In this paper we initiate the study of the restrained reinforcement number $r_...

For a graph G=(V,E), an independent double Roman dominating function (IDRDF) is a function f:V→{0,1,2,3} having the property that: (i) every vertex v∈V with f(v) = 0 has a neighbor u with f(u) = 3 or at least two neighbors x and y such that f(x)=f(y)=2; (ii) every vertex v∈V with f(v) = 1 has at least one neighbor assigned a 2 or 3 under f; (iii) t...

A restrained Italian dominating function (RIDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying: (i) [Formula: see text] for every vertex [Formula: see text] with [Formula: see text], where [Formula: see text] is the set of vertices adjacent to [Formula: see text]; (ii) the subgraph induced by the vertices assigned 0 und...

A paired-dominating set of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number is the minimum cardinality of a paired-dominating set of G. The paired-domination subdivision number is the minimum number of edges that must be subdivided (each edge in G can be su...

Let k be a positive integer and G=(V,E) a graph. A [k]-Roman dominating function is a function f:V→{0,1,2,…,k+1} such that for every v∈V(G) with f(v)<k, f(AN[v])≥|AN(v)|+k, where AN(v) is the set of neighbors of v assigned a non-zero value under f and AN[v]=AN(v)∪{v}. When k=3, the function f is called a triple Roman dominating function (TRD-functi...

Let [Formula: see text] be a positive integer. A restrained [Formula: see text]-rainbow dominating function (RkRD-function) of a graph [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the set of all subsets of the set [Formula: see text] satisfying: (i) for any vertex [Formula: see text] with [Formula...

A double Roman dominating function (DRDF) on a graph \(G=(V,E)\) is a function \(f:V\rightarrow \{0,1,2,3\}\) having the property that if \(f(v)=0\), then 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 vertex v must have at least one neighbor w with \(f(w)\ge 2\). The weight o...

A maximal double Roman dominating function (MDRDF) on a graph G=(V,E) is a function f:V(G)→{0,1,2,3} such that (i) every vertex v with f(v)=0 is adjacent to least two vertices assigned 2 or to at least one vertex assigned 3, (ii) every vertex v with f(v)=1 is adjacent to at least one vertex assigned 2 or 3 and (iii) the set {w∈V|f(w)=0} is not a do...

In this short note, we disprove the conjecture of Jafari Rad and Volkmann that every γ-vertex critical graph is γR-vertex critical, where γ(G) and γR(G) stand for the domination number and the Roman domination number of a graph G, respectively.

In a graph G of minimum degree δ and maximum degree Δ, a subset S of vertices of G is j-independent, for some positive integer j, if every vertex in S has at most j−1 neighbors in S. The j-independence number βj(G) is the maximum cardinality of a j-independent set of G. We first establish an inequality between βj(G) and βΔ(G) for 1≤j≤δ−1. Then we c...

Let [Formula: see text] be a function on a graph [Formula: see text]. A vertex [Formula: see text] with [Formula: see text] is said to be undefended with respect to [Formula: see text] if it is not adjacent to a vertex [Formula: see text] with [Formula: see text]. A function [Formula: see text] is called a weak Roman dominating function (WRDF) if e...

Let $G=(V,E)$ be a graph of order $n$ and let $\gamma _{R}(G)$ and $\partial (G)$ denote the Roman domination number and the differential of $G,$ respectively. In this paper we prove that for any integer $k\geq 0$, if $G$ is a graph of order $n\geq 6k+9$, minimum degree $\delta \geq 2,$ which does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3...

For a graph \(\Gamma \), let \(\gamma (\Gamma ),\) \(\gamma _{t}(\Gamma )\), and \(\gamma _{tR2}(\Gamma )\) denote the domination number, the total domination number, and the total Roman \(\{2\}\)-domination number, respectively. It was shown in Abdollahzadeh Ahangar et al. (Discuss Math Graph Theory, in press) that for each nontrivial connected gr...

A domatic partition P of a graph G=(V,E) is a partition of V into classes that are pairwise disjoint dominating sets. Such a partition P is called b-maximal if no larger domatic partition P' can be obtained by gathering subsets of some classes of P to form a new class. The b-domatic number bd(G) is the minimum cardinality of a b-maximal domatic par...

Suppose [3]={0,1,2,3} and [3-]={-1,1,2,3}. An outer independent signed double Roman dominating function (OISDRDF) of a graph Γ is function l:V(Γ)→[3-] for which (i) each vertex t with l(t)=-1 is joined to at least two vertices labeled a 2 or to at least one vertex z with l(z)=3, (ii) each vertex t with l(t)=1 is joined to at least a vertex z with l...

In order to increase the paired-domination number of a graph G, the minimum number of edges that must be subdivided (where each edge in G can be subdivided no more than once) is called the paired-domination subdivision number sdγpr(G) of G. It is well known that sdγpr(G+e) can be smaller or larger than sdγpr(G) for some edge e∉E(G). In this note, w...

An outer-independent double Roman dominating function (OIDRDF) of a graph G is a function h : V ( G ) → { 0,1,2,3 } such that i) every vertex v with f ( v ) = 0 is adjacent to at least one vertex with label 3 or to at least two vertices with label 2, ii) every vertex v with f ( v ) = 1 is adjacent to at least one vertex with label greater than 1, a...

For a positive integer $k$, a subset $D$ of vertices in a digraph $\overrightarrow{G}$ is a $k$-dominating set if every vertex not in $D$ has at least $k$ direct predecessors in $D.$ The $k$-domination number is the minimum cardinality among all $k$-dominating sets of $\overrightarrow{G}$. The game $k$-domination number of a simple and undirected g...

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...

For a graph G with no isolated vertex, let γpr(G) and sdγpr(G) denote the paired-domination and paired-domination subdivision numbers, respectively. In this note, we show that if T is a tree of order n≥4 different from a healthy spider (subdivided star), then sdγpr(T)≤min{γpr(T)2+1,n2}, improving the (n−1)-upper bound that was recently proven.

A dominator coloring is a proper coloring of the vertices of a graph such that each vertex of the graph dominates all vertices of at least one color class (possibly its own class). The dominator chromatic number of a graph G is the minimum number of color classes in a dominator coloring of G . In this paper, we determine the exact value of the domi...

In this chapter, we present nine parameters, which are variations of Roman domination in graphs.

A subset S > V is a dominating set of G if every vertex in V n S has a neighbor in S, and it is a total dominating set if every vertex in V has a neighbor in S. The total domination number of G, t(G), is the minimum cardinality of a total dominating set of G. A vertex v of a graph G is said to ve-dominate every edge incident to v, as well as every...

Let G be a subcubic graph of order n and minimum degree at least 2. In this paper, we prove the conjecture of Bermudo and Fernau that if n≥23, then ∂(G)≥5n∕18, where ∂(G) is the differential of G. To do this, we use the Gallai-type result involving the Roman domination number γR(G) and ∂(G) by proving that, with the exception of thirteen graphs of...

An edge dominating set in a graph [Formula: see text] is a subset [Formula: see text] of [Formula: see text] such that each edge in [Formula: see text] is either in [Formula: see text] or adjacent to at least an edge in [Formula: see text]. The edge domination number [Formula: see text] is the minimum cardinality of an edge dominating set in [Formu...

Let [Formula: see text] be a graph, and let [Formula: see text] be an induced path centered at [Formula: see text]. An edge lift defined on [Formula: see text] is the action of removing edges [Formula: see text] and [Formula: see text] while adding the edge [Formula: see text] to the edge set of [Formula: see text]. In this paper, we initiate the s...

A set S of vertices of a connected graph G = (V, E) is a connected dominating set of G if every vertex of V − S is adjacent to at least one vertex of S and the subgraph induced by S is connected. In this chapter, we survey various results on the connected domination parameters obtained over the last 40 years.

This chapter is concerned with the concept Roman domination in graphs, which was introduced in 2004 by Cockayne, Dreyer, S.M. Hedetniemi, and S.T. Hedetniemi based on the strategies for defending the Roman Empire presented by Stewart (Sci Am 281:136–139, 1999) and ReVelle and Rosing (ReVelle CS, Rosing KE, Am Math Mon 107:585–594, 2000). Since then...

A set S of vertices is a perfect dominating set of a graph G if every vertex not in S is adjacent to exactly one vertex of S. The minimum cardinality of a perfect dominating set is the perfect domination number γp(G). A perfect Roman dominating function (PRDF) on a graph G=(V,E) is a function f:V→{0,1,2} satisfying the condition that every vertex u...

A Roman dominating function (RDF) of a graph G is a labeling \(f:V(G)\longrightarrow \{0,1,2\}\) such that every vertex with label 0 has a neighbor with label 2. The weight of an RDF is the sum of its functions values over all vertices, and the Roman domination number of G is the minimum weight of an RDF of G. The Roman domination subdivision numbe...

In this work, we continue to survey what has been done on the Roman domination. More precisely, we will present in two sections several variations of Roman dominating functions as well as the signed version of some of these functions. It should be noted that a first part of this survey comprising 9 varieties is published as a chapter book in “Topic...

A dominating set in a graph G is a set of vertices S ⊆ V ( G ) such that any vertex of V − S is adjacent to at least one vertex of S . A dominating setS of G is said to be a perfect dominating set if each vertex in V − S is adjacent to exactly one vertex in S. The minimum cardinality of a perfect dominating set is the perfect domination number γ p...

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 $f:V\cup E\rightarrow \{0,1,2\}$ satisfying the condition that every element $x\in V\cup E$ for which $f(x)=0$ is adjacent or incident to at least one element $% y\in V\cup E$ for which $f(y)=2$. The weight of a mixed...

A dominating set of $G=(V,E)$ is a subset $S$ of $V$ such that every vertex in $V-S$ has at least one neighbor in $S.$ A connected dominating set of $G$ is a dominating set whose induced subgraph is connected. The minimum cardinality of a connected dominating set is the connected domination number $\gamma _{c}(G)$. Let $\delta ^{\ast }(G)=\min \{\d...

An outer independent double Roman dominating function (OIDRDF) of a graph G is a function h from V(G) to {0, 1, 2, 3} for which each vertex with label 0 is adjacent to a vertex with label 3 or at least two vertices with label 2, and each vertex with label 1, is adjacent to a vertex with label greater than 1; and all vertices labeled by 0 is indepen...

In this paper, we survey results on the Roman domatic number and its variants in both graphs and digraphs. This fifth survey completes our works on Roman domination and its variations published in two book chapters and two other surveys.

A Roman {2}-dominating function (R2F) is a function f : V → {0, 1, 2} with the property that for every vertex v ∈ V with f(v) = 0 there is a neighbor u of v with f(u) = 2, or there are two neighbors x, y of v with f(x) = f(y) = 1. A total Roman {2}-dominating function (TR2DF) is an R2F f such that the set of vertices with f(v) > 0 induce a subgraph...

A perfect Italian dominating function (PIDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that for every vertex u with f(u) = 0, the total weight of f assigned to the neighbors of u is exactly two. The weight of a PIDF is the sum of its functions values over all vertices. The perfect Italian domination number of G, deno...

A vertex v of a graph G = ( V , E ) , ve-dominates every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S. The double vertex-edge domination number γ d v e ( G ) is the minimum cardinality of a double vertex...

A set [Formula: see text] of vertices in a graph [Formula: see text] is a [Formula: see text]-dominating set of [Formula: see text] if every vertex of [Formula: see text] is adjacent to at least two vertices in [Formula: see text] The [Formula: see text]-domination number is the minimum cardinality of a [Formula: see text]-dominating set of [Formul...

Let D be a finite and simple digraph with vertex set V(D). A signed dominating function (SDF) of D is a function f:V(D)⟶{-1,1} such that f(N-[v])=∑x∈N-[v]f(x)≥1 for every v∈V(D), where N-[v] consists of v and all vertices of D from which arcs go into v. The weight of an SDF is the sum of its function values over all vertices, and the minimum weight...

A double Roman dominating function (DRDF) on a graph G=(V,E) is a function f:V→{0,1,2,3} having the property that if f(v)=0, then 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 vertex v must have at least one neighbor w with f(w)≥2. The weight of a DRDF is the sum of its function valu...

In a graph, an edge dominates itself and all its adjacent edges. An edge dominating set in a graph G is a subset of edges that dominates every edge of G. In this paper, we characterize edges that are in all or in no minimum edge dominating sets in trees.

A set S={u1,u2,…,ut} of vertices of G is an efficient dominating set if every vertex of G is dominated exactly once by the vertices of S. Letting Ui denote the set of vertices dominated by ui, we note that {U1,U2,…Ut} is a partition of the vertex set of G and that each Ui contains the vertex ui and all the vertices at distance~1 from it in G. In th...

Given two real numbers \(b\ge a>0\), an (a, b)-Roman dominating function on a graph \(G=(V,E)\) is a function \(f:V\rightarrow \{0,a,b\}\) satisfying the condition that every vertex v for which \(f(v)=0\) is adjacent to a vertex u for which \(f(u)=b\). In the present paper, we design a linear-time algorithm to produce a minimum (a, b)-Roman dominat...

A signed double Roman dominating function (SDRDF) on a graph [Formula presented] is a function [Formula presented] such that (i) every vertex [Formula presented] with [Formula presented] is adjacent to at least two vertices assigned a 2 or to at least one vertex [Formula presented] with [Formula presented] (ii) every vertex [Formula presented] with...

A partition π={V1,V2,…,Vk} of the vertex set V of a graph G into k color classes Vi, with i∈{1,…,k}, is called a quorum coloring if, for every vertex v∈V, at least half of the vertices in the closed neighborhood N[v] of v have the same color as v. The maximum order of a quorum coloring of G is called the quorum coloring number of G and is denoted ψ...

A double Roman dominating function (DRDF) on a graph \(G=(V,E)\) is a function \(f:V(G)\rightarrow \{0,1,2,3\}\) such that (i) every vertex v with \(f(v)=0\) is adjacent to at least two vertices assigned a 2 or to at least one vertex assigned a 3, (ii) every vertex v with \(f(v)=1\) is adjacent to at least one vertex w with \(f(w)\ge 2.\) The weigh...

A total Roman dominating function on a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2 and the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The minimum weight of a total Roman dominating function on a graph G is called the total Roman dominati...

A Roman dominating function on a graph G = (V, E) is a function f:V (G) → {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman dominating function is the value w(f) = Σu∈V(G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination...

A 2-rainbow dominating function (2RDF) of a graph G = (V (G), E(G)) is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2} such that for every vertex v ∈ V (G) with f(v) = ∅ the condition ∪u∈N(v)f(u) = {1, 2} is fulfilled, where N(v) is the open neighborhood of v. A total 2-rainbow dominating function f of a graph wit...

For any integer k≥0, a set of vertices S of a graph G=(V,E) is k -cost-effective if for every v S, |N(v)∩(V S)|≥|N(v)∩S|+k. In this paper we study the minimum cardinality of a maximal k-cost-effective set and the maximum cardinality of a k-cost-effective set. We obtain Gallai-type results involving the k-cost-effective and global k-offensive allian...

For a graph G=(V,E), a Roman (2)-dominating function (R2DF) f:V→(0,1,2) has the property that for every vertex vεV with f(v)=0, either there exists an adjacent vertex, a neighbor uεN(v), with f(u)=2, or at least two neighbors x,yεN(v) having f(x)=f(y)=1. The weight of a R2DF is the sum f(V)=∑vεVf(v). A R2DF f=(V0,V1,V2) is called independent if V1∪...

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)....

A vertex v of a graph G = (V,E) is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set (double vertex-edge dominating set, respectively) if every edge of E is ve-dominated by at least one vertex (at least two vertices) of S. The minimum cardinality of a ve...

A vertex v of a graph $G=(V,E)$-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set $S\subseteq V$ is a vertex-edge dominating set (or simply, a ve-dominating set) if every edge of E is ve-dominated by at least one vertex of S. The minimum cardinality of a ve-dominating set of G is the vertex-edge domina...

A Roman dominating function (RDF) 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 function $f:V(G)\rightarrow \{0,1,2\}$ is an outer-independent Roman dominating function (OIRDF) on G if f is an RDF and $V_{0}$ is an i...

A Roman dominating function (RDF) on a graph G = (V,E) is a function f : V →{0, 1, 2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of a RDF is the sum f(V ) =∑v∈Vf(v), and the minimum weight of a RDF f is the Roman domination number γR(G). A subset S of V is a 2-...

Let G = (V, E) be a graph with vertex set V and edge set E. A vertex v is an element of V ve-dominates every edge incident to it as well as every edge adjacent to these incident edges. The vertex edge degree of a vertex v is the number of edges ve-dominated by v. Similarly, an edge e = uv ev-dominates the two vertices u and v incident to it, as wel...

For a graph , we consider placing a variable number of pebbles on the vertices of . A pebbling move consists of deleting two pebbles from a vertex and placing one pebble on a vertex adjacent to . We seek an initial placement of a minimum total number of pebbles on the vertices in , so that no vertex receives more than some positive integer pebbles...

A set of a graph is a liar’s dominating set if (1) for every vertex , and (2) for every pair of distinct vertices, . In this paper, we first provide a characterization of graphs with as well as the trees with . Then we present some bounds on the liar’s domination number, especially an upper bound for the ratio between the liar’s domination number a...

A neighborhood total dominating set in a graph G is a dominating set S of G with the property that the subgraph induced by N(S), the open neighborhood of the set S; has no isolated vertex. The neighborhood total domination number γnt(G) is the minimum cardinality of a neigh-borhood total dominating set of G. Arumugam and Sivagnanam in-troduced and...

A set S ⊆ V is a global dominating set of a graph G = (V,E) if S is a dominating set of G and G; where G is the complement graph of G. The global domination number ϒg(G) equals the minimum cardinality of a global dominating set of G. The square graph G² of a graph G is the graph with vertex set V and two vertices are adjacent in G² if they are join...

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number γt(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u)=0 is adjacent to...

In this paper, we initiate the study of a variant of Roman dominating functions. For a graph G=(V,E), a Roman (2)-dominating function f:V→(0,1,2) has the property that for every vertex v∈V with f(v)=0, either v is adjacent to a vertex assigned 2 under f, or v is adjacent to least two vertices assigned 1 under f. The weight of a Roman (2)-dominating...

A Roman dominating function (RDF) 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 RDF f is the value f(V(G)) = ∑u∈V(G)f(u). The Roman domination number, γR(G), of G is the minimum weight of a RDF on G. An RDF f =...

A 2-rainbow dominating function of a graph G is a function g that assigns to each vertex a set of colors chosen from the set (1,2) so that for each vertex v with g(v) = θ we have Uu∈N(v) g(u) = (1,2). The minimum of g(V(G)) = Σv∈v(G) |g(v)| over all such functions is called the 2-rainbow domination number γr2(G). A 2-rainbow dominating function g o...

A Roman dominating function (RDF) for a graph is a function satisfying the condition that every vertex u of G for which is adjacent to at least one vertex v of G for which . The weight of a Roman dominating function f is the sum , and the minimum weight of a Roman dominating function for G is the Roman domination number, of G. A maximal Roman domin...

A set of a graph is a dominating set of G if every vertex not in D is adjacent to at least one vertex in D. A secure dominating set S of a graph G is a dominating set with the property that each vertex is adjacent to a vertex such that is a dominating set. The secure domination number equals the minimum cardinality of a secure dominating set of G....

In this paper we study graph parameters related to vertex-edge domination, where a vertex dominates the edges incident to it as well as the edges adjacent to these incident edges. First, we present new relationships relating the ve-domination to some other domination parameters, answering in the affirmative four open questions posed in the 2007 PhD...

For a graph \(G=(V,E)\) , a Roman dominating function \(f:V\rightarrow \{0,1,2\}\) has the property that every vertex \(v\in V\) with \(f(v)=0\) has a neighbor \(u\) with \(f(u)=2\) . The weight of a Roman dominating function \(f\) is the sum \(f(V)=\sum \nolimits _{v\in V}f(v)\) , and the minimum weight of a Roman dominating function on \(G\) is t...

A Roman dominating function on a graph G is a function f : V(G) -> {0,1,2} satisfying the condition that every vertex u of G for which f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of a Roman dominating function is the value f(V(G)) = Sigma(u is an element of V)(G) f(u). The Roman domination number, gamma R(G), o...

Let k be a positive integer and G = (V(G),E(G)) a graph. A subset S of V(G) is a k-dominating set if every vertex of V(G) - S is adjacent to at least k vertices of S. The k-domination number gamma(k) (G) is the minimum cardinality of a k-dominating set of G. A graph G is called gamma(-)(k)-stable if gamma(k)(G - e) = gamma(k)(G) for every edge e of...

In this paper, we introduce and study a new coloring problem of a graph called the dominated coloring. A dominated coloring of a graph G is a proper vertex coloring of G such that each color class is dominated by at least one vertex of G. The minimum number of colors among all dominated colorings is called the dominated chromatic number, denoted by...

A 2-rainbow dominating function (2RDF) on a graph G = (V, E) is a function f from the vertex set V to the set of all subsets of the set {1, 2} such that for any vertex v ∈ V with f (v) = ∅ the condition u∈N (v) f (u) = {1, 2} is fulfilled. A 2RDF f is independent (I2RDF) if no two vertices assigned nonempty sets are adjacent. The weight of a 2RDF f...

A Roman dominating function (RDF) on a graph G is 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 of G for which f(v) = 2. The weight of a Roman dominating function is the sum f(V ) = Σv∈Vf(v), and the minimum weight of a Roman dominating function f is the Roman dominat...

We mainly study two related dominating functions, namely, the weak Roman and 2-rainbow dominating functions. We show that for all graphs, the weak Roman domination number is bounded above by the 2-rainbow domination number. We present bounds on the weak Roman domination number and the secure domination number in terms of the total domination number...

A set D of vertices in a graph G = (V (G), E(G)) is an open neighborhood locating-dominating set (OLD-set) for G if for every two vertices u, v of V (G) the sets N(u) ∩ D and N(v) ∩ D are non-empty and different. The open neighborhood locating-dominating number OLD(G) is the minimum cardinality of an OLD-set for G. In this paper we characterize gra...

A set D of vertices in a graph G=(V,E) is a total dominating set if every vertex of G is adjacent to some vertex in D. A total dominating set D of G is said to be weak if every vertex v∈V-D is adjacent to a vertex u∈D such that d_{G}(v)≥d_{G}(u). The weak total domination number γ_{wt}(G) of G is the minimum cardinality of a weak total dominating s...