Roman Domination Research Articles

On the Roman domination subdivision number of a graph

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 number $$\mathrm {sd}_{\gamma _{R}}(G)$$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the Roman domination number of G. In this paper, we present a new upper bound on the Roman domination subdivision number by showing that for every connected graph G of order at least three, $$\begin{aligned} \mathrm {sd}_{\gamma _{R}}(G)\le 3+\min \{\deg _2(v)\mid v\in V\;\mathrm {and} \;d(v)\ge 2\}, \end{aligned}$$ where $$\deg _2(v)$$ is the number of vertices of G at distance 2 from vertex v. Moreover, we show that the decision problem associated with $$\mathrm {sd}_{\gamma _{R}}(G)$$ is NP-hard for bipartite graphs.

Complexity of Roman {2}-domination and the double Roman domination in graphs

For a simple, undirected graph a Roman {2}-dominating function (R2DF) has the property that for every vertex with f(v) = 0, either there exists a vertex with f(u) = 2, or at least two vertices with The weight of an R2DF is the sum The minimum weight of an R2DF is called the Roman {2}-domination number and is denoted by A double Roman dominating function (DRDF) on G is a function such that for every vertex if f(v) = 0, then v has at least two neighbors with or one neighbor w with f(w) = 3, and if f(v) = 1, then v must have at least one neighbor w with The weight of a DRDF is the value The minimum weight of a DRDF is called the double Roman domination number and is denoted by Given an graph G and a positive integer k, the R2DP (DRDP) problem is to check whether G has an R2DF (DRDF) of weight at most k. In this article, we first show that the R2DP problem is NP-complete for star convex bipartite graphs, comb convex bipartite graphs and bisplit graphs. We also show that the DRDP problem is NP-complete for star convex bipartite graphs and comb convex bipartite graphs. Next, we show that are obtained in linear time for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. Finally, we propose a -approximation algorithm for the minimum Roman {2}-domination problem and -approximation algorithm for the minimum double Roman domination problem, where Δ is the maximum degree of G.

Critical graphs with Roman domination number four

A Roman domination function on a graph G is a function satisfying the condition that every vertex u for which r(u) = 0 is adjacent to at least one vertex v for which r(v) = 2. The weight of a Roman domination function is the value The Roman domination number of G is the minimum weight of a Roman domination function on G. “Roman Criticality” often refers to the study of graphs where the Roman domination number decreases when adding an edge or removing a vertex of the graph. In this paper we add some condition to this notion of criticality and give a complete characterization of critical graphs with Roman Domination number

Algorithmic aspects of Roman domination in graphs

For a simple, undirected graph $$G = (V, E)$$ , a Roman dominating function (RDF) $$f{:}V \rightarrow \lbrace 0, 1, 2 \rbrace $$ has the property that, every vertex u with $$f(u) = 0$$ is adjacent to at least one vertex v for which $$f(v) = 2$$ . The weight of a RDF is the sum $$f(V) = \sum _{v \in V}f(v)$$ . The minimum weight of a RDF is called the Roman domination number and is denoted by $$\gamma _{R}(G)$$ . Given a graph G and a positive integer k, the Roman domination problem (RDP) is to check whether G has a RDF of weight at most k. The RDP is known to be NP-complete for bipartite graphs. We strengthen this result by showing that this problem remains NP-complete for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. We show that $$\gamma _{R}(G)$$ is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. The minimum Roman domination problem (MRDP) is to find a RDF of minimum weight in the input graph. We show that the MRDP for star convex bipartite graphs and comb convex bipartite graphs cannot be approximated within $$(1 - \epsilon ) \ln |V|$$ for any $$\epsilon > 0$$ unless $$NP \subseteq DTIME(|V|^{O(\log \log |V|)})$$ and also propose a $$2(1+\ln (\varDelta +1))$$ -approximation algorithm for the MRDP, where $$\varDelta $$ is the maximum degree of G. Finally, we show that the MRDP is APX-complete for graphs with maximum degree 5.

Constructive characterizations concerning weak Roman domination in trees

Given a graph G, we consider γr(G), γ{R2}(G), γr2(G) and γR(G) as the weak Roman domination number, the Roman {2}-domination number, the 2-rainbow domination number and the Roman domination number of G, respectively.It is known that γr(G)≤γ{R2}(G)≤γr2(G)≤γR(G) holds for any graph G. In connection with this, constructive characterizations of the trees T that satisfy the equalities above that are related with the weak Roman domination number are given in this work. That is, the trees T for which γr(T)=γ{R2}(T), γr(T)=γr2(T) and γr(T)=γR(T) are described.

Perfect double Roman domination of trees

For a graph G with vertex set V(G) and function f:V(G)→{0,1,2,3}, let Vi be the set of vertices assigned i by f. A perfect double Roman dominating function of a graph G is a function f:V(G)→{0,1,2,3} satisfying the conditions that (i) if u∈V0, then u is either adjacent to exactly two vertices in V2 and no vertex in V3 or adjacent to exactly one vertex in V3 and no vertex in V2; and (ii) if u∈V1, then u is adjacent to exactly one vertex in V2 and no vertex in V3. The perfect double Roman domination number of G, denoted γdRp(G), is the minimum weight of a perfect double Roman dominating function of G. We prove that if T is a tree of order n≥3, then γdRp(T)≤9n∕7. In addition, we give a family of trees T of order n for which γdRp(T) approaches this upper bound as n goes to infinity.

On the signed strong Roman domination number of graphs

Let [Formula: see text] be a finite and simple graph of order [Formula: see text] and maximum degree [Formula: see text]. A signed strong Roman dominating function on a graph [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) for every vertex [Formula: see text] of [Formula: see text], [Formula: see text], where [Formula: see text] is the closed neighborhood of [Formula: see text] and (ii) 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], where [Formula: see text]. The minimum of the values [Formula: see text], taken over all signed strong Roman dominating functions [Formula: see text] of [Formula: see text], is called the signed strong Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we continue the study signed strong Roman domination number of a graph and give several bounds for this parameter. Then, among other results, we determine the signed strong Roman domination number of special classes of graphs.

Further Results on the Total Roman Domination in Graphs

Let G be a graph without isolated vertices. A function f : V ( G ) → { 0 , 1 , 2 } is a total Roman dominating function on G if every vertex v ∈ V ( G ) for which f ( v ) = 0 is adjacent to at least one vertex u ∈ V ( G ) such that f ( u ) = 2 , and if the subgraph induced by the set { v ∈ V ( G ) : f ( v ) ≥ 1 } has no isolated vertices. The total Roman domination number of G, denoted γ t R ( G ) , is the minimum weight ω ( f ) = ∑ v ∈ V ( G ) f ( v ) among all total Roman dominating functions f on G. In this article we obtain new tight lower and upper bounds for γ t R ( G ) which improve the well-known bounds 2 γ ( G ) ≤ γ t R ( G ) ≤ 3 γ ( G ) , where γ ( G ) represents the classical domination number. In addition, we characterize the graphs that achieve equality in the previous lower bound and we give necessary conditions for the graphs which satisfy the equality in the upper bound above.

The signed Roman domination number of two classes graphs

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. A signed Roman dominating function (SRDF) of [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) [Formula: see text] for each [Formula: see text], where [Formula: see text] is the set, called closed neighborhood of [Formula: see text], consists of [Formula: see text] and the vertex of [Formula: see text] adjacent to [Formula: see text] (ii) 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 SRDF [Formula: see text] is [Formula: see text]. The signed Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of a SRDF of [Formula: see text]. In this paper, we determine the exact values of signed Roman domination number of spider and double star. Specially, one of them generalizes the known result.

Critical concept for double Roman domination in graphs

A double Roman dominating function (DRDF) on a graph [Formula: see text] is a function [Formula: see text] such that (i) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least two vertices assigned a [Formula: see text] or to at least one vertex assigned a [Formula: see text] and (ii) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text] The weight of a DRDF is the sum of its function values over all vertices. The double Roman domination number [Formula: see text] equals the minimum weight of a DRDF on [Formula: see text] The concept of criticality with respect to various operations on graphs has been studied for several domination parameters. In this paper, we study the concept of criticality for double Roman domination in graphs. In addition, we characterize double Roman domination edge super critical graphs and we will give several characterizations for double Roman domination vertex (edge) critical graphs.

Roman [formula omitted]-domination (double Italian domination)

For a graph G=(V,E) with V=V(G) and E=E(G), a Roman {3}-dominating function is a function f:V→{0,1,2,3} having the property that ∑u∈NG(v)f(u)≥3, if f(v)=0, and ∑u∈NG(v)f(u)≥2, if f(v)=1 for any vertex v∈G. The weight of a Roman {3}-dominating function f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a Roman {3}-dominating function on G is the Roman {3}-domination number of G, denoted by γ{R3}(G). We initiate the study of Roman {3}-domination and show its relationship to domination, Roman domination, Roman {2}-domination (Italian domination) and double Roman domination. Finally, we present an upper bound on the Roman {3}-domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we show that associated decision problem for Roman {3}-domination is NP-complete, even for bipartite graphs.

Characterization of trees with Roman bondage number 1

Let $G = (V, E)$ be a simple undirected graph. A Roman dominating function on $G$ is a function $f: V\to \{0, 1, 2\}$ satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to at least one vertex $v$ with $f(v) = 2$. The weight of a Roman dominating function is the value $f(G) = \sum_{u\in V} f(u)$. The Roman domination number of $G$ is the minimum weight of a Roman dominating function on $G$. The Roman bondage number of a nonempty graph $G$ is the minimum number of edges whose removal results in a graph with the Roman domination number larger than that of $G$. Rad and Volkmann [9] proposed a problem that is to determine the trees $T$ with Roman bondage number $1$. In this paper, we characterize trees with Roman bondage number $1$.

Representational Sovereignty in Moroccan Amazigh Documentary Film

Representational Sovereignty in Moroccan Amazigh Documentary Film Sheila Petty (bio) and Brahim Benbouazza (bio) For the Imazighen,1 who are the Indigenous peoples of North Africa, Indigenous language control and identity construction are the key pillars of human existence. After decades of repression through colonization, Imazighen are increasingly bringing to the fore their hidden histories, through means that ensure control of processes of self-representation, and they are working both locally and in the diaspora to reframe narratives that once erased their specific cultures by promoting Arab language and culture. Cleo Jay has written that "culture is an important arena for the Amazigh movement" and screen media plays a pivotal role in both the representation and retention of "endangered cultures. … torn between globalisation and conservatism" (69). Using a medium that has historically worked to obfuscate, exoticize, and orientalize Amazigh representations and voices by depicting them as adventurous, romantic, and non-European, Amazigh filmmakers take back their stories and reframe them from their sovereign points of view. As screen media researchers who have collaborated on a number of projects, we come from different backgrounds but share a common world-sense. As a Moroccan Amazigh, Brahim Benbouazza believes very strongly in the preservation of Amazigh identity and languages. For years, he participated in the struggle to have his language recognized as an official language in Morocco, which it finally was in 2011. As a descendant of Canadian settlers and a scholar of sub-Saharan African, Maghrebi, and Amazigh cinemas, Sheila Petty believes strongly that working in cultural contexts different from one's own necessitates the development of respectful listening skills and an openness to new experiences that are critical if one is to listen to, and understand, the goals of such artworks and artists. We both share the philosophy that applying eurocentric frames of analysis to non-eurocentric artworks [End Page 129] and texts leads to a context of mistrust and rejection of cultural and representational sovereignty. In her work on global Indigenous media, Pamela Wilson, building on previous work by Kristen Dowell (2013) and Michele Raheja (2010), proposed the term "indigenous representational sovereignty" to allow for the use of a culture's own particular logic to control and organize the process and product of artistic creation. Wilson maintains that "indigenous representational sovereignty" has evolved in different ways over time as various peoples have appropriated film and video technology and made it their own (97). In this essay, we draw on this foundational work on Indigenous screen media and build on this by looking at the theories of Amazigh scholars of culture and cinema to analyze two Moroccan Amazigh documentary films in light of their embedded representational sovereignty. In 2013, Filmmaker/historian Kamal Hachkar directed Tinghir-Jerusalem, Echoes from the Mellah, in which he travels to Tinghir, Morocco, in pursuit of his Jewish-Amazigh roots. This performative documentary contrasts sharply in style with Tala Hadid's more recent 2017 documentary, House in the Fields, in which she documents the life and rituals of a rural Amazigh community in the southwest region of Morocco's High Atlas Mountains. Themes of identity, language, land, and memory link the two films, as they depict Amazigh history and stories from Amazigh points of view. historical context and the amazigh renaissance By the 8th Century AD, Roman dominance of North Africa had waned leaving the region open to Arabo-Muslim penetration. Indigenous populations—the Imazighen (Free Men)—who had lived peacefully through major Amazigh dynasties such as the Almohades and the Almoravides, found their languages and identities slowly eroding through the centuries as Arabic took hold as the dominant language and Islam as the predominant religion (Pouessel 13–14). In 1912, France established a protectorate out of the majority of Moroccan territory and although the colonizer favored secularism, it generally allowed populations to follow their customary laws. After independence in 1956, however, Arab-Islamic nationalism dominated and the Arabic language and Islamic religion were considered pillars of the Moroccan state, which also served to justify the existence of the Moroccan king, a descendant of the prophet Mohammed (27). [End Page 130] In the early post-independence years, Arabic was enshrined in the Constitution as Morocco's official...

The “qanats” of the Los Alcores aquifer and the traditional water supply system to Seville

Due to the quality and the topographic position of its waters, the Los Alcores aquifer has been the main supply of running and drinking water for the city of Seville for many centuries. In the first century, a complex water supply system, of more than 17 km in length, was built to withdraw groundwaters in the southern part of the aquifer and whose main elements were an extensive network of “qanats”, a surface conduction and a reservoir in the city (castellum aquae). After the abandonment of this infrastructure, the surface conduction was reconstructed in the 12th century (known as Caños de Carmona), which was made up of a canal and, in the part closest to the city, by a brick aqueduct over arches. Since the end of the 19th century, the urban, industrial and the transport network development and the conduction of water from the reservoirs of the Sierra Morena produced, once again, the progressive abandonment and destruction of these infrastructures. However, a large part of the “qanat” network excavated in the Los Alcores aquifer has been preserved and recently the GEOS and AAES speleological groups have explored about 20 km of water mines. The characteristics of tunnels and vertical shafts, the construction techniques, the discovery of Roman mortar (opus signinum) and bricks (opus testaceum) and the Arab chronicles suggest that this network of “qanats” was made mainly during the Roman domination period.

On roman domination number of functigraph and its complement

Let G = ( V ( G ) , E ( G ) ) be a graph and f : V ( G ) → { 0 , 1 , 2 } be a function where for every vertex v ∈ V ( G ) with f ( v ) = 0 , there is a vertex u ∈ N G ( v ) , where f ( u ) = 2. The...

两类冠图的符号罗马控制数

两类冠图的符号罗马控制数

Extremal graphs for a bound on the Roman domination number

Extremal graphs for a bound on the Roman domination number

A linear algorithm for double Roman domination of proper interval graphs

A function [Formula: see text] is a double Roman dominating function on a graph [Formula: see text] if for every vertex [Formula: see text] with [Formula: see text] either there is a vertex [Formula: see text] with [Formula: see text] or there are distinct vertices [Formula: see text] with [Formula: see text] and for every vertex [Formula: see text] with [Formula: see text] there is a vertex [Formula: see text] with [Formula: see text]. The weight of a double Roman dominating function [Formula: see text] on [Formula: see text] is the value [Formula: see text]. The minimum weight of a double Roman dominating function on [Formula: see text] is called the double Roman domination number of [Formula: see text]. In this paper, we give an algorithm to compute the double Roman domination number of a given proper interval graph [Formula: see text] in [Formula: see text] time.

An Interval Graph with Alternate Cliiques of Size 3- Signed Roman Domination

An Interval Graph with Alternate Cliiques of Size 3- Signed Roman Domination

Algorithmic and complexity aspects of problems related to total Roman domination for graphs

A function $$f:V(G)\rightarrow \{0,1,2\}$$ is a Roman dominating function (RDF) if 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 $$f(V(G))=\sum _{u \in V}f(u)$$. The Roman domination number of a graph G, denoted by $$\gamma _{R}(G)$$, is the minimum weight of a Roman dominating function on G. A connected (respectively, total) Roman dominating function is an RDF f such that the vertices with non-zero labels under f induce a connected graph (respectively, a subgraph with no isolated vertex). The connected (respectively, total) Roman domination number of a graph G, denoted by $$\gamma _{cR}(G)$$ (respectively, $$\gamma _{tR}(G)$$) is the minimum weight of a connected (respectively, total) RDF of G. It this paper we first study the complexity issue of the problems posed in [H. Abdollahzadeh Ahangar, M. A. Henning, V. Samodivkin and I. G. Yero, Total Roman domination in graphs, Appl. Anal. Discret. Math. 10 (2016), 501–517], and show that the problem of deciding whether $$\gamma _{tR}(G)=2\gamma (G)$$, $$\gamma _{tR}(G)=2\gamma _t(G)$$ or $$\gamma _{tR}(G)=3\gamma (G)$$ is NP-hard even when restricted to chordal or bipartite graphs. Then, we give a linear algorithm that decides whether $$\gamma _{tR}(G)=2\gamma (G)$$, $$\gamma _{tR}(G)=2\gamma _t(G)$$ or $$\gamma _{tR}(G)=3\gamma (G)$$, if G is a tree or a unicyclic graph.