Roman Domatic Number Research Articles

Restrained Roman and restrained Italian domatic numbers of graphs

Let G be a graph with vertex set V ( G ) . A Roman dominating function (RDF) on a graph G is a function f : V ( G ) ⟶ { 0 , 1 , 2 } satisfying the condition that each vertex u with f ( u ) = 0 has a neighbor v with f ( v ) = 2 . An Italian dominating function (IDF) is a function f : V ( G ) ⟶ { 0 , 1 , 2 } having the property that f ( N ( u ) ) ≥ 2 for every vertex u with f ( u ) = 0 , where N ( u ) is the neighborhood of u . If f is an RDF or an IDF on G , then let V 0 = { v ∈ V ( G ) : f ( v ) = 0 } . A restrained Roman (Italian) dominating function is an RDF (IDF) f having the property that the subgraph induced by V 0 does not have an isolated vertex. A set { f 1 , f 2 , … , f d } of distinct restrained Roman (Italian) dominating functions on G with the property that ∑ i = 1 d f i ( v ) ≤ 2 for each v ∈ V ( G ) is called a restrained Roman (Italian) dominating family (of functions) on G . The maximum number of functions in a restrained Roman (Italian) dominating family on G is the restrained Roman (Italian) domatic number of G , denoted by d r R ( G ) ( d r I ( G ) ). We initiate the study of the restrained Roman (Italian) domatic numbers, and we present different sharp bounds on d r R ( G ) and d r I ( G ) . In addition, we determine these parameters for some classes of graphs.

The double Roman domatic number of a digraph

The double Roman domatic number of a digraph

Total Roman domatic number of a graph

A total Roman dominating function on a graph [Formula: see text] is a labeling [Formula: see text] such that every vertex with label 0 has a neighbor with label 2 and the subgraph of [Formula: see text] induced by the set of all vertices of positive weight has no isolated vertex. A set [Formula: see text] of total Roman dominating functions on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text], is called a total Roman dominating family (of functions) on [Formula: see text]. The maximum number of functions in a total Roman dominating family on [Formula: see text] is the total Roman domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we investigate the properties of total Roman domatic number in graphs. In particular, we present some sharp bounds for [Formula: see text] and we determine the total Roman domatic number of some special graphs.

The signed total Roman domatic number of a digraph

Let [Formula: see text] be a finite simple digraph with vertex set [Formula: see text]. A signed total Roman dominating function (STRDF) on a digraph [Formula: see text] is a function [Formula: see text] such that (i) [Formula: see text] for every [Formula: see text], where [Formula: see text] consists of all inner neighbors of [Formula: see text], and (ii) every vertex [Formula: see text] for which [Formula: see text] has an inner neighbor [Formula: see text] for which [Formula: see text]. The weight of an STRDF [Formula: see text] is [Formula: see text]. The signed total Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of an STRDF on [Formula: see text]. A set [Formula: see text] of distinct STRDFs on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text] is called a signed total Roman dominating family (STRD family) (of functions) on [Formula: see text]. The maximum number of functions in an STRD family on [Formula: see text] is the signed total Roman domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we initiate the study of signed total Roman domatic number in digraphs and we present some sharp bounds for [Formula: see text]. In addition, we determine the signed total Roman domatic number of some classes of digraphs.

Twin signed Roman domatic numbers in digraphs

Let $D$ be a finite simple digraph with vertex set $V(D)$. A twin signed Roman dominating function 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 $f(v)=f(w)=2$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct twin signed Roman dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le 1$ for each $v\in V(D)$, is called a twin signed Roman dominating family (of functions) on $D$. The maximum number of functions in a twin signed Roman dominating family on $D$ is the twin signed Roman domatic number of $D$, denoted by $d_{sR}^*(D)$. In this paper, we initiate the study of the twin signed Roman domatic number in digraphs and we present some sharp bounds on $d_{sR}^*(D)$. In addition, we determine the twin signed Roman domatic number of some classes of digraphs.

On the signed total Roman domination and domatic numbers of graphs

A signed total Roman dominating function on a graph G is a function f:V(G)⟶{−1,1,2} such that ∑u∈N(v)f(u)≥1 for every vertex v∈V(G), where N(v) is the neighborhood of v, and every vertex u∈V(G) for which f(u)=−1 is adjacent to at least one vertex w for which f(w)=2. The signed total Roman domination number of a graph G, denoted by γstR(G), equals the minimum weight of a signed total Roman dominating function. A set {f1,f2,…,fd} of distinct signed total Roman dominating functions on G with the property that ∑i=1dfi(v)≤1 for each v∈V(G), is called a signed total Roman dominating family (of functions) on G. The maximum number of functions in a signed total Roman dominating family on G is the signed total Roman domatic number of G, denoted by dstR(G). In this paper we continue the investigation of the signed total Roman domination number, and we initiate the study of the signed total Roman domatic number in graphs. We present sharp bounds for γstR(G) and dstR(G). In addition, we determine the total signed Roman domatic number of some graphs.

Computing Roman domatic number of graphs

A Roman dominating function on a graph G=(V,E) is a mapping: V→{0,1,2} satisfying that every vertex v∈V with f(v)=0 is adjacent to some vertex u∈V with f(u)=2. A Roman dominating family (of functions) on G is a set {f1,f2,…,fd} of Roman dominating functions on G with the property that ∑i=1dfi(v)≤2 for all v∈V. The Roman domatic number of G, introduced by Sheikholeslami and Volkmann in 2010 [1], is the maximum number of functions in a Roman dominating family on G. In this paper, we study the Roman domatic number from both algorithmic complexity and graph theory points of view. We show that it is NP-complete to decide whether the Roman domatic number is at least 3, even if the graph is bipartite. To the best of our knowledge, this is the first computational hardness result concerning this concept. We also present an asymptotically optimal approximation threshold of Θ(log⁡n) for computing the Roman domatic number of a graph. Moreover, we determine the Roman domatic number of some particular classes of graphs, such as fans, wheels and complete bipartite graphs.

The signed Roman domatic number of a digraph

Let $D$ be a finite and simple digraph with vertex set $V(D)$.A {\em signed Roman dominating function} on the digraph $D$ isa function $f:V (D)\longrightarrow \{-1, 1, 2\}$ such that$\sum_{u\in N^-[v]}f(u)\ge 1$ for every $v\in V(D)$, where $N^-[v]$ consists of $v$ andall inner neighbors of $v$, and every vertex $u\in V(D)$ for which $f(u)=-1$ has an innerneighbor $v$ for which $f(v)=2$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct signedRoman dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le 1$ for each$v\in V(D)$, is called a {\em signed Roman dominating family} (of functions) on $D$. The maximumnumber of functions in a signed Roman dominating family on $D$ is the {\em signed Roman domaticnumber} of $D$, denoted by $d_{sR}(D)$. In this paper we initiate the study of signed Romandomatic number in digraphs and we present some sharp bounds for $d_{sR}(D)$. In addition, wedetermine the signed Roman domatic number of some digraphs. Some of our results are extensionsof well-known properties of the signed Roman domatic number of graphs.

The distance Roman domatic number of a graph

Let k be a positive integer, and let G be a simple graph with vertex set V(G). A k-distance Roman dominating function on G is a labeling f: V(G) → {0, 1, 2} such that every vertex with label 0 has a vertex with label 2 within distance k from each other. A set {f1, f2, …, fdd} of k-distance Roman dominating functions on G with the property that eqfi(v) ≤ 2 for each v ∊ V(G), is called a k-distance Roman dominating family (of functions) on G. The maximum number of functions in a k-distance Roman dominating family on G is the k-distance Roman domatic number of G, denoted by . In this paper we initiate the study of k-distance Roman domatic number in graphs and we present some sharp bounds for . In addition, we determine the k-distance Roman domatic number of some graphs.

The Roman k-domatic number of a graph

Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f: V (G) → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f 1, f 2, …, f d } of distinct Roman k-dominating functions on G with the property that Σ =1 f i (v) ≤ 2 for each v ∈ V (G), is called a Roman k-dominating family (of functions) on G. The maximum number of functions in a Roman k-dominating family on G is the Roman k-domatic number of G, denoted by d kR (G). Note that the Roman 1-domatic number d 1R (G) is the usual Roman domatic number d R (G). In this paper we initiate the study of the Roman k-domatic number in graphs and we present sharp bounds for d kR (G). In addition, we determine the Roman k-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.

The Roman domatic number of a graph

A 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. A set { f 1 , f 2 , … , f d } of Roman dominating functions on G with the property that ∑ i = 1 d f i ( v ) ≤ 2 for each v ∈ V ( G ) is called a Roman dominating family (of functions) on G . The maximum number of functions in a Roman dominating family on G is the Roman domatic number of G , denoted by d R ( G ) . In this work we initiate the study of the Roman domatic number in graphs and we present some sharp bounds for d R ( G ) . In addition, we determine the Roman domatic number of some graphs.