A counterexample to a conjecture of Jafari Rad and Volkmann

Let $D$ be a simple digraph with vertex set $V$.
\nA Roman dominating function (RDF) on a digraph $D$ is a function $f: V\\rightarrow \\{0,1,2\\}$
\nsatisfying the condition that every vertex $v$ with $f(v)=0$ has an in-neighbor $u$ with $f(u)=2$.
\nThe weight of an RDF $f$ is the value $\\sum_{v\\in V}f(v)$.
\nThe Roman domination number of a digraph $D$ is the minimum weight of an RDF on $D$.
\nA set $\\{f_1,f_2,\\dots,f_d\\}$ of Roman dominating functions on $D$ with the property that $\\sum_{i=1}^df_i(v)\\le2$ for each $v\\in V$, is called a
\nRoman dominating family (of functions) on $D$. The maximum number of functions in a Roman dominating family on $D$ is the Roman domatic number of $D$, denoted by $d_{R}(D)$. In this paper we continue the investigation of the Roman domination number, and we
\ninitiate the study of the Roman domatic number in digraphs. We present some
\nbounds for $d_{R}(D)$. In addition, we determine the Roman domatic number of some digraphs.

Upper bounds on the [formula omitted]-domination number and the [formula omitted]-Roman domination number

<abstract><p>Let $ G $ be a graph with vertex set $ V(G) $. A function $ f:V(G)\rightarrow \{0, 1, 2\} $ is a Roman dominating function on $ G $ if every vertex $ v\in V(G) $ for which $ f(v) = 0 $ is adjacent to at least one vertex $ u\in V(G) $ such that $ f(u) = 2 $. The Roman domination number of $ G $ is the minimum weight $ \omega(f) = \sum_{x\in V(G)}f(x) $ among all Roman dominating functions $ f $ on $ G $. In this article we study the Roman domination number of direct product graphs and rooted product graphs. Specifically, we give several tight lower and upper bounds for the Roman domination number of direct product graphs involving some parameters of the factors, which include the domination, (total) Roman domination, and packing numbers among others. On the other hand, we prove that the Roman domination number of rooted product graphs can attain only three possible values, which depend on the order, the domination number, and the Roman domination number of the factors in the product. In addition, theoretical characterizations of the classes of rooted product graphs achieving each of these three possible values are given.</p></abstract>

The middle graph [Formula: see text] of a graph [Formula: see text] is the graph obtained by subdividing each edge of [Formula: see text] exactly once and joining all these newly introduced vertices of adjacent edges of [Formula: see text]. It is known that the decision problems for Italian domination number [Formula: see text], [Formula: see text]-rainbow domination number [Formula: see text] and Roman domination number [Formula: see text] are NP-complete. Recently, it was proved that [Formula: see text] for a graph [Formula: see text] of order [Formula: see text]. In this paper, we continue to study several domination and domatic numbers in middle graphs. We also obtain closed formulas for domination, secure domination and total domination numbers in middle graphs of rooted product graphs.

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, Σu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.

Let D=(V(D),A(D)) be a finite, simple digraph and k a positive integer. A function f:V(D)→{0,1,2,…,k+1} is called a [k]-Roman dominating function (for short, [k]-RDF) if f(AN−[v])≥|AN−(v)|+k for any vertex v∈V(D), where AN−(v)={u∈N−(v):f(u)≥1} and AN−[v]=AN−(v)∪{v}. The weight of a [k]-RDF f is ω(f)=∑v∈V(D)f(v). The minimum weight of any [k]-RDF on D is the [k]-Roman domination number, denoted by γ[kR](D). For k=2 and k=3, we call them the double Roman domination number and the triple Roman domination number, respectively. In this paper, we presented some general bounds and the Nordhaus–Gaddum bound on the [k]-Roman domination number and we also determined the bounds on the [k]-Roman domination number related to other domination parameters, such as domination number and signed domination number. Additionally, we give the exact values of γ[kR](Pn) and γ[kR](Cn) for the directed path Pn and directed cycle Cn.

Let [Formula: see text] be a graph. The weight of a function [Formula: see text] defined on the vertex set of [Formula: see text] is [Formula: see text]. A Roman dominating function (RDF) of [Formula: see text] is a function [Formula: see text] such that every vertex [Formula: see text] for which [Formula: see text] has a neighbor [Formula: see text] with [Formula: see text]. Many variations of RDF are available in the literature and among that a double Roman dominating function (DRDF) is a function [Formula: see text] having the property that if [Formula: see text], then vertex [Formula: see text] must have at least two neighbors assigned 2 under [Formula: see text] or one neighbor with [Formula: see text], and if [Formula: see text], then vertex [Formula: see text] must have at least one neighbor with [Formula: see text] and an Italian dominating function (IDF) on a graph [Formula: see text] is a function [Formula: see text], satisfying the property that if [Formula: see text] for a vertex [Formula: see text], then [Formula: see text] with [Formula: see text], that is, either [Formula: see text], with [Formula: see text], or at least two vertices [Formula: see text] with [Formula: see text]. The minimum weights among all RDF, DRDF and IDF on a graph [Formula: see text] are called the Roman domination number, the double Roman domination number and the Italian domination number of the graph [Formula: see text], respectively. In this paper, we find the double Roman domination number and Italian domination number of Kneser graphs.

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) = ?v?V f(v), and the minimum weight of a Roman dominating function f is the Roman domination number ?R(G). An RDF f is called an independent Roman dominating function (IRDF) if the set of vertices assigned positive values under f is independent. The independent Roman domination number iR(G) is the minimum weight of an IRDF on G. We show that for every nontrivial connected graph G with maximum degree ?, ?R(G)? ?+1/??(G) and iR(G) ? i(G) + ?(G)/?, where ?(G) and i(G) are, respectively, the domination and independent domination numbers of G. Moreover, we characterize the connected graphs attaining each lower bound. We give an additional lower bound for ?R(G) and compare our two new bounds on ?R(G) with some known lower bounds.

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. The weight of a RDF [Formula: see text] is the value [Formula: see text]. The Roman domination number, [Formula: see text], of [Formula: see text] is the minimum weight of a RDF on [Formula: see text]. An RDF [Formula: see text] is called an independent Roman dominating function (IRDF) if the set [Formula: see text] is an independent set. The independent Roman domination number, [Formula: see text], is the minimum weight of an IRDF on [Formula: see text]. In this paper, we study trees with independent Roman domination number twice their independent domination number, answering an open question.

Domination-type parameters are difficult to manage in Cartesian product graphs and there is usually no general relationship between the parameter in both factors and in the product graph. This is the situation of the domination number, the Roman domination number or the 2-domination number, among others. Contrary to what happens with the domination number and the Roman domination number, the 2-domination number remains unknown in cylinders, that is, the Cartesian product of a cycle and a path and in this paper, we will compute this parameter in the cylinders with small cycles. We will develop two algorithms involving the (min, +) matrix product that will allow us to compute the desired values of γ2( Cn□ Pm), with 3 ≤ n ≤ 15 and m ≤ 2. We will also pose a conjecture about the general formula for the 2-domination number in this graph class.

Italian domination in trees

Constructive characterizations concerning weak Roman domination in trees

Given a graph G, we consider the Italian domination number γI(G), the 2-rainbow domination number γr2(G) and the Roman domination number γR(G). It is known that γI(G) ≤ γr2(G) ≤ γR(G) holds for any graph G. In this paper, we prove that γI(M(G)) = γr2(M(G)) = γR(M(G)) = n for the middle graph M(G) of a graph G of order n, which gives an answer for an open problem posed by Chellali et al. [Discrete Appl. Math. 204 (2016) 22–28]. Moreover, we give a complete characterization of Roman domination stable middle graphs, 2-rainbow domination stable middle graphs and Italian domination stable middle graphs.

Restrained condition on double Roman dominating functions

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, Σu∈N(v) f(u) ≥ 2. The weight of an Italian dominating function f is defined as w(f) = f(V ) = Σu∈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) = Σv∈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.