Roman Domination Research Articles

A game-theoretic approach to solving the Roman domination problem

A game-theoretic approach to solving the Roman domination problem

On the (total) Roman domination in Latin square graphs

<abstract><p>Latin square, also known as Latin square matrix, refers to a kind of $ n\times n $ matrix, in which there are exactly $ n $ different symbols and each symbol appears exactly once in each row and column. A Latin square graph $ \Gamma(L) $ is a simple graph associated with a Latin square $ L $. This paper studied the relationships between the (total) Roman domination number and (total) domination number of Latin square graph $ \Gamma(L) $. We showed that $ \gamma_{R}(\Gamma(L)) = 2\gamma(\Gamma(L)) $ or $ \gamma_{R}(\Gamma(L)) = 2\gamma(\Gamma(L))-1 $, and $ \gamma_{tR}(\Gamma(L))\ge \frac{8\gamma_t(\Gamma(L))}{5} $ for $ n\ge 2 $. In 2021, Pahlavsay et al. proved $ \gamma(\Gamma(L))\ge \lceil \frac{n}{2}\rceil $ and $ \gamma_t(\Gamma(L))\ge \lceil \frac{4n-2}{7}\rceil $ for $ n\ge 2 $. In this paper, we showed that $ \gamma_R(\Gamma(L))\ge 2\lceil \frac{n}{2}\rceil $ (equality holds if, and only if, $ \gamma(\Gamma(L)) = \lceil \frac{n}{2}\rceil $) and $ \gamma_t(\Gamma(L)) > \frac{4n}{7} $ for $ n\ge 2 $. Since $ \gamma_R(G)\le 2\gamma(G) $ and $ \gamma_{tR}(G)\le 2\gamma_t(G) $ for any graph $ G $, our results can deduce or improve Pahlavsay et al.'s results. Moreover, we characterized these Latin squares for $ \gamma_R(\Gamma(L)) = 2\lceil \frac{n}{2}\rceil $, which is equal to $ \gamma(\Gamma(L)) = \lceil \frac{n}{2}\rceil $.</p></abstract>

The Novel Concept of Roman Domination in Fuzzy Graph

Suppose = , is a graph. A Roman dominating function (RDF) of is a function: → {0,1,2} such that every vertex for which = 0 has a neighbor with = 2. The weight of an RDF is = Σ ∈ . The Roman domination number (RDN) of a graph , denoted by ℘, is the minimum weight of all possible RDFs. In this study, we define RDF on fuzzy graph (FG). Our purpose is to develop a notion of the RDF and also to present some basic definitions, notations, remarks, and proofs related to RDF on FGs.

Roman Domination in the Shadow Distance Graphs

Roman Domination in the Shadow Distance Graphs

The signed (total) roman domination problem on some classes of planar graphs – convex polytopes

The signed (total) roman domination problem on some classes of planar graphs – convex polytopes

TWO PONTOS CASTLES IN VEZIRKÖPRÜ, REG. SAMSUN: BÜYÜKKALE (SAGYLION) AND KÜÇÜKKALE

Vezirköprü, the district of Samsun, is an area of a continuous sedentism for millennia, from the Chalcolithic and Early Bronze Age until the present, due to its location at an important crossroads, with wide plains and numerous streams all around, and a landscape suitable for settled life. Being a village named Phazemon, in Strabo’s words, the site gained the identity of a city with the Roman domination in the region and took the name Neoklaudiopolis (Neapolis). The city has witnessed many wars and showed that it had an important strategic position. Mithridates VI Eupator, King of Pontus, was the biggest enemy of Roman domination in Anatolia back then. In Pontus’s coastal parts, castles have already been known, while those in the inner parts of the region have not been adequately researched. Büyükkale and Küçükkale, which are the scope of the paper, are among the latter. Both castles display many things of traditional Mithridatic fortification, with castle walls, rock tunnels, stairs, and tombs. The one in Büyükkale is identified with Sagylion of old written accounts. The castles under study have not yet been published elsewhere, and the paper is expected to contribute to the illumination of the construction activity of the Pontic Kingdom and the Hellenistic Period in Vezirköprü.

Hop Italian Domination in Graphs

Given a simple graph $G=(V(G),E(G))$, a function $f:V(G)\to \{0,1,2\}$ is a hop Italian dominating function if for every vertex $v$ with $f(v)=0$ there exists a vertex $u$ with $f(u)=2$ for which $u$ and $v$ are of distance $2$ from each other or there exist two vertices $w$ and $z$ for which $f(w)=1=f(z)$ and each of $w$ and $z$ is of distance $2$ from $v$. The minimum weight $\sum_{v\in V(G)}f(v)$ of a hop Italian dominating function is the hop Italian domination number of $G$, and is denoted by $\gamma_{hI}(G)$. In this paper, we initiate the study of the hop Italian domination. In particular, we establish some properties of the the hop Italian dominating function and explore the relationships of the hop Italian domination number with the hop Roman domination number \cite{Rad2,Natarajan} and with the $2$-hop domination number \cite{Canoy}. We study the concept under some binary graph operations. We establish tight bounds and determine exact values for their respective hop Italian domination numbers.

Geodetic Roman Dominating Functions in a Graph

Let $G$ be a connected graph. A function $f: V(G)\rightarrow \{0,1,2\}$ is a \textit{geodetic Roman dominating function} (or GRDF) if every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$ and $V_1 \cup V_2$ is a geodetic set in $G$. The weight of a geodetic Roman dominating function $f$, denoted by $\omega_{G}^{gR}(f)$, is given by $\omega_{G}^{gR}(f)=\sum_{v \in V(G)}f(v)$. The minimum weight of a GRDF on $G$, denoted by $\gamma_{gR}(G)$, is called the \textit{geodetic Roman domination number} of $G$. In this paper, we give some properties of geodetic Roman domination and determine the geodetic Roman domination number of some graphs.

Ancient history: A postcolonial view on Roman identity

Ancient history: A postcolonial view on Roman identity Prof Dr Felix K Maier, Professor for Ancient History at University of Zurich, explores the paradoxical dynamics of different identities in the multicultural Roman Empire. My history research project analyses the dynamics of different identities in the Roman Empire from around 50-150 AD. The Roman Empire is generally considered a ‘story of a success’ concerning the integration of the conquered peoples. The Romans surpassed other empires regarding temporal extension and maintained their power with little military presence. However, it was not only open rebellions that could have threatened Roman domination; it was also – and quite paradoxically – the successful integration of conquered peoples. This had to do with a couple of interdependent aspects.

Majority double Roman domination in graphs

A majority double Roman dominating function (MDRDF) 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 with 2 or to at least one vertex [Formula: see text] with [Formula: see text], (ii) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text] and (iii) [Formula: see text], for at least half of the vertices in [Formula: see text]. The weight of an MDRDF is the sum of its function values over all vertices. The majority double Roman domination number of a graph [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text]. In this paper, we introduce and study the majority double Roman domination number on some classes of graphs.

Solving the minimum-cost double Roman domination problem

Solving the minimum-cost double Roman domination problem

Hardness results of global roman domination in graphs

A Roman dominating function (RDF) of a graph G=(V,E) is a function f:V→{0,1,2} such that every vertex assigned the value 0 is adjacent to a vertex assigned the value 2. A global Roman dominating function (GRDF) of a graph G=(V,E) is a function f:V→{0,1,2} such that f is a Roman dominating function of both G and its complement G¯. The weight of f is f(V)=Σu∈Vf(u). The minimum weight of a GRDF in a graph G is known as global Roman domination number of G and is denoted by γgR(G). Minimum Global Roman Domination is to find a global Roman dominating function of minimum weight and Decide Global Roman Domination is the decision version of Minimum Global Roman Domination. In this paper, we show that Decide Global Roman Domination is NP-complete for bipartite graphs and chordal graphs. On the positive side, we give a polynomial-time algorithm for Minimum Global Roman Domination for threshold graphs. Further, we propose an O(ln|V|)-approximation algorithm for Minimum Global Roman Domination for a graph G=(V,E). We also show that Minimum Global Roman Domination cannot be approximated within a factor of (1−ϵ)ln|V| for any ϵ>0 unless P = NP. Next, we show that Minimum Global Roman Domination admits a constant approximation algorithm for bounded degree graphs. Finally, we show that Minimum Global Roman Domination is APX-complete for bounded degree graphs.

Unique Response Roman Domination: Complexity and Algorithms

Unique Response Roman Domination: Complexity and Algorithms

Total Perfect Roman Domination

A total perfect Roman dominating function (TPRDF) on a graph G=(V,E) is a function f from V to {0,1,2} satisfying (i) every vertex v with f(v)=0 is a neighbor of exactly one vertex u with f(u)=2; in addition, (ii) the subgraph of G that is induced by the vertices with nonzero weight has no isolated vertex. The weight of a TPRDF f is ∑v∈Vf(v). The total perfect Roman domination number of G, denoted by γtRp(G), is the minimum weight of a TPRDF on G. In this paper, we initiated the study of total perfect Roman domination. We characterized graphs with the largest-possible γtRp(G). We proved that total perfect Roman domination is NP-complete for chordal graphs, bipartite graphs, and for planar bipartite graphs. Finally, we related γtRp(G) to perfect domination γp(G) by proving γtRp(G)≤3γp(G) for every graph G, and we characterized trees T of order n≥3 for which γtRp(T)=3γp(T). This notion can be utilized to develop a defensive strategy with some properties.

On [k] -Roman domination in graphs

For an integer k ≥ 1 , let f be a function that assigns labels from the set { 0 , 1 , … , k + 1 } to the vertices of a simple graph G = ( V , E ) . The active neighborhood AN(v) of a vertex v ∈ V ( G ) with respect to f is the set of all neighbors of v that are assigned non-zero values under f. A [ k ] -Roman dominating function ( [ k ] -RDF) is a function f : V ( G ) → { 0 , 1 , 2 , … , k + 1 } such that for every vertex v ∈ V ( G ) with f(v) < k, we have ∑ u ∈ N [ v ] f ( u ) ≥ | A N ( v ) | + k . The weight of a [ k ] -RDF is the sum of its function values over the whole set of vertices, and the [ k ] -Roman domination number γ [ k R ] ( G ) is the minimum weight of a [ k ] -RDF on G. In this paper we determine various bounds on the [ k ] -Roman domination number. In particular, we show that for any integer k ≥ 2 every connected graph G of order n ≥ 3 , satisfies γ [ k R ] ( G ) ≤ ( 2 k + 1 ) 4 n , and we characterize the graphs G attaining this bound. Moreover, we show that if T is a nontrivial tree, then γ [ k R ] ( T ) ≥ k γ ( T ) + 1 for every integer k ≥ 2 and we characterize the trees attaining the lower bound. Finally, we prove the NP-completeness of the [ k ] -Roman domination problem in bipartite and chordal graphs.

Convex Roman Dominating Function in Graphs

Let $G$ be a connected graph. A function $f:V(G)\rightarrow \{0,1,2\}$ is a \textit{convex Roman dominating function} (or CvRDF) if every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$ and $V_1 \cup V_2$ is convex. The weight of a convex Roman dominating function $f$, denoted by $\omega_{G}^{CvR}(f)$, is given by $\omega_{G}^{CvR}(f)=\sum_{v \in V(G)}f(v)$. The minimum weight of a CvRDF on $G$, denoted by $\gamma_{CvR}(G)$, is called the \textit{convex Roman domination number} of $G$. In this paper, we determine the convex Roman domination numbers of some graphs and give some realization results involving convex Roman domination, connected Roman domination, and convex domination numbers.

Domination-related parameters in middle graphs

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.

New complexity results on Roman {2}-domination

The study of a variant of Roman domination was initiated by Chellali et al. [Discrete Appl. Math. 204 (2016) 22–28]. Given a graph G with vertex set V, a Roman {2}-dominating function f : V → {0, 1, 2} has the property that for every vertex v ∈ V with f(v) = 0, either there exists a vertex u adjacent to v with f(u) = 2, or at least two vertices x, y adjacent to v with f(x) = f(y) = 1. The weight of a Roman {2}-dominating function is the value f(V) = ∑v∈V f(v). The minimum weight of a Roman {2}-dominating function is called the Roman {2}-domination number and is denoted by γ{R2}(G). In this work we find several NP-complete instances of the Roman {2}-domination problem: chordal graphs, bipartite planar graphs, chordal bipartite graphs, bipartite with maximum degree 3 graphs, among others. A result by Chellali et al. [Discrete Appl. Math. 204 (2016) 22–28] shows that γ{R2}(G) and the 2-rainbow domination number of G coincide when G is a tree, and thus, the linear time algorithm for k-rainbow domination due to Brešar et al. [Taiwan J. Math. 12 (2008) 213–225] can be followed to compute γ{R2}(G). In this work we develop an efficient algorithm that is independent of k-rainbow domination and computes the Roman {2}-domination number on a subclass of trees called caterpillars.

New bounds on the outer-independent total double Roman domination number

A double Roman dominating function (DRDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying (i) if [Formula: see text] then there must be at least two neighbors assigned two under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text]; and (ii) if [Formula: see text] then [Formula: see text] must be adjacent to a vertex [Formula: see text] such that [Formula: see text]. A DRDF is an outer-independent total double Roman dominating function (OITDRDF) on [Formula: see text] if the set of vertices labeled [Formula: see text] induces an edgeless subgraph and the subgraph induced by the vertices with a non-zero label has no isolated vertices. The weight of an OITDRDF is the sum of its function values over all vertices, and the outer-independent total Roman domination number [Formula: see text] is the minimum weight of an OITDRDF on [Formula: see text]. In this paper, we establish various bounds on [Formula: see text]. In particular, we present Nordhaus–Gaddum-type inequalities for this parameter. Some of our results improve the previous results.

Several Roman domination graph invariants on Kneser graphs

This paper considers the following three Roman domination graph invariants on Kneser graphs: Roman domination, total Roman domination, and signed Roman domination. For Kneser graph $K_{n,k}$, we present exact values for Roman domination number $\gamma_{R}(K_{n,k})$ and total Roman domination number $\gamma_{tR}(K_{n,k})$ proving that for $n\geqslant k(k+1)$, $\gamma_{R}(K_{n,k}) =\gamma_{tR}(K_{n,k}) = 2(k+1)$. For signed Roman domination number $\gamma_{sR}(K_{n,k})$, the new lower and upper bounds for $K_{n,2}$ are provided: we prove that for $n\geqslant 12$, the lower bound is equal to 2, while the upper bound depends on the parity of $n$ and is equal to 3 if $n$ is odd, and equal to $5$ if $n$ is even. For graphs of smaller dimensions, exact values are found by applying exact methods from literature.