Perfect Roman Research Articles

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.

A quarry for the construction of a Roman camp next to the Celtiberian city of Deza during the Sertorian Wars (Soria, Spain)

Through an interdisciplinary investigation, a hastily exploited individual quarry of limestone is studied. The quarry was made all at once for the construction of a nearby military camp to improve the defence of a remarkable Celtiberian city during the Sertorian Wars. This is a perfect Roman military engineering project: on the one hand, the geological deposit was exploited selectively, differentiating the areas of extraction of large and small blocks of stone, according to the needs of the work. On the other hand, the material was transported by means of small roads along a carefully studied and laid out road of about 600 m in length and of clear Roman origin. The archaeological evidence shows the existence of a large camp that, according to previous surveys, adopts the known classical models of rectangular plans and the typology of Roman construction. In the camp environment, there are enough scattered remains that seem to point to a battlefield. All this indicates the importance of the primitive Celtiberian settlement, which is justified by an analysis of the territory by the existence of springs of high guarantee for human water supply, livestock and irrigation. The existence of silver mines is another added economic value. Due to the predominance in the monetary findings of the mint of Titiakos, the possibility is raised that the Celtiberian city was linked to the ethnic group of the Titos indicating their active participation during the Sertorian Wars.

Vertex-edge perfect Roman domination number

<abstract><p>A vertex-edge perfect Roman dominating function on a graph $ G = (V, E) $ (denoted by ve-PRDF) is a function $ f:V\left(G\right)\longrightarrow\{0, 1, 2\} $ such that for every edge $ uv\in E $, $ \max\{f(u), f(v)\}\neq0 $, or $ u $ is adjacent to exactly one neighbor $ w $ such that $ f(w) = 2 $, or $ v $ is adjacent to exactly one neighbor $ w $ such that $ f(w) = 2 $. The weight of a ve-PRDF on $ G $ is the sum $ w(f) = \sum_{v\in V}f(v) $. The vertex-edge perfect Roman domination number of $ G $ (denoted by $ \gamma_{veR}^{p}(G) $) is the minimum weight of a ve-PRDF on $ G $. In this paper, we first show that vertex-edge perfect Roman dominating is NP-complete for bipartite graphs. Also, for a tree $ T $, we give upper and lower bounds for $ \gamma_{veR}^{p}(T) $ in terms of the order $ n $, $ l $ leaves and $ s $ support vertices. Lastly, we determine $ \gamma_{veR}^{p}(G) $ for Petersen, cycle and Flower snark graphs.</p></abstract>

Kvintilijanovi principi odgoja i obrazovanja rimskih govornika / Quintilian’s principles of education of Roman orators

Social values, as well as the state of society at a certain time, are optimally reflected in the contents and methods of education of new generations. When it comes to Roman education, Marcus Fabius Quintilian belongs to those who placed the social ideal at the center of education and subordinated everything else to it. In the 1st century. D., distinguished himself as a teacher of rhetoric by theorizing about the education of orators and, also, by giving examples for practice. He accepted the task that was somehow imposed on him - to write a work based on many years of experience - how to raise a perfect orator. The Institutio Oratoria is a completely preserved work that determines the stages and methods of mastering the art of oratory from the earliest age of a child. As far as it is known, it is the only preserved systematic ancient pedagogical reflection. This paperwork tries to concisely present the principles that Quintilian considered fundamental in education, which resulted in a perfect Roman orator. The presentation of principles was done by analyzing historical sources and comparing them with contemporary pedagogical concepts.

Perfect Domination, Roman Domination and Perfect Roman Domination in Lexicographic Product Graphs

The aim of this paper is to obtain closed formulas for the perfect domination number, the Roman domination number and the perfect Roman domination number of lexicographic product graphs. We show that these formulas can be obtained relatively easily for the case of the first two parameters. The picture is quite different when it concerns the perfect Roman domination number. In this case, we obtain general bounds and then we give sufficient and/or necessary conditions for the bounds to be achieved. We also discuss the case of perfect Roman graphs and we characterize the lexicographic product graphs where the perfect Roman domination number equals the Roman domination number.

The Perfect Roman Domination Number of the Cartesian Product of Some Graphs

A perfect Roman dominating function on a graph G is a function f : V(G)⟶{0,1,2} for which every vertex v with f(v) = 0 is adjacent to exactly one neighbor u with f(u) = 2. The weight of f is the sum of the weights of the vertices. The perfect Roman domination number of a graph G, denoted by , is the minimum weight of a perfect Roman dominating function on G. In this paper, we prove that if G is the Cartesian product of a path Pr and a path Ps, a path Pr and a cycle Cs, or a cycle Cr and a cycle Cs, where r, s > 5, then .

Strong equality of Roman and perfect Roman Domination in trees

A Roman dominating function (RD-function) on a graph G = (V, E) is a function f : V → {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. An Roman dominating function f in a graph G is perfect Roman dominating function (PRD-function) if every vertex u with f(u) = 0 is adjacent to exactly one vertex v for which f(v) = 2. The (perfect) Roman domination number γR(G) (γpR(G)) is the minimum weight of an (perfect) Roman dominating function on G. We say that γpR(G) strongly equals γR(G), denoted by γpR(G) ≡ γR(G), if every RD-function on G of minimum weight is a PRD-function. In this paper we show that for a given graph G, it is NP-hard to decide whether γpR(G) = γR(G) and also we provide a constructive characterization of trees T with γpR(T) ≡ γR(T).

On the perfect differential of a graph

Let G be a graph of order n(G) and vertex set V(G). Given a set S ⊆ V(G), we define the perfect neighbourhood of S as the set Np (S) of all vertices in V(G)\\S having exactly one neighbour in S. The perfect differential of S is defined to be ∂p (S) = |Np (S)| − |S|. In this paper, we introduce the study of the perfect differential of a graph, which we define as ∂p (G) = max{∂p (S) : S ⊆ V(G)}. Among other results, we obtain general bounds on ∂p (G) and we prove a Gallai-type theorem, which states that ∂p (G) + γp R (G) = n(G), where γp R (G) denotes the perfect Roman domination number of G. As a consequence of the study, we show some classes of graphs satisfying a conjecture stated by Bermudo [Discrete Appl. Math. 232 (2017), 64-72].

More on Perfect Roman Domination in Graphs

A perfect Roman dominating function on a graph G = (V (G), E(G)) is a function f : V (G) → {0, 1, 2} for which each u ∈ V (G) with f(u) = 0 is adjacent to exactly one vertex v ∈ V (G) with f(v) = 2. The weight of a perfect Roman dominating function f is the value ωG(f) = Pv∈V (G) f(v). The perfect Roman domination number of G is the minimum weight of a perfect Roman dominating function on G. In this paper, we study the perfect Roman domination numbers of graphs under some binary operation

A characterization of perfect Roman trees

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 with f(u)=0 is adjacent to exactly one vertex v for which f(v)=2. The weight of a PRDF is the sum of its function values over all vertices, and the minimum weight of a PRDF of G is the perfect Roman domination number γRp(G). Obviously, for every graph G, γRp(G)≤2γp(G), and those graphs attaining the equality are called perfect Roman graphs. In this paper, we provide a characterization of perfect Roman trees.

On a Relation between the Perfect Roman Domination and Perfect Domination Numbers of a Tree

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 set S 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 ( G ) . A function f : V ( G ) → { 0 , 1 , 2 } is a perfect Roman dominating function (PRDF) on G if every vertex u ∈ V for which f ( u ) = 0 is adjacent to exactly one vertex v for which f ( v ) = 2 . The weight of a PRDF is the sum of its function values over all vertices, and the minimum weight of a PRDF of G is the perfect Roman domination number γ R p ( G ) . In this paper, we prove that for any nontrivial tree T, γ R p ( T ) ≥ γ p ( T ) + 1 and we characterize all trees attaining this bound.

Strong Equality of Perfect Roman and Weak Roman Domination in Trees

Let G = ( V , E ) be a graph and f : V ⟶ { 0 , 1 , 2 } be a function. Given a vertex u with f ( u ) = 0 , if all neighbors of u have zero weights, then u is called undefended with respect to f. Furthermore, if every vertex u with f ( u ) = 0 has a neighbor v with f ( v ) > 0 and the function f ′ : V ⟶ { 0 , 1 , 2 } with f ′ ( u ) = 1 , f ′ ( v ) = f ( v ) − 1 , f ′ ( w ) = f ( w ) if w ∈ V ∖ { u , v } has no undefended vertex, then f is called a weak Roman dominating function. Also, the function f is a perfect Roman dominating function if every vertex u with f ( u ) = 0 is adjacent to exactly one vertex v for which f ( v ) = 2 . Let the weight of f be w ( f ) = ∑ v ∈ V f ( v ) . The weak (resp., perfect) Roman domination number, denoted by γ r ( G ) (resp., γ R p ( G ) ), is the minimum weight of the weak (resp., perfect) Roman dominating function in G. In this paper, we characterize those trees where the perfect Roman domination number strongly equals the weak Roman domination number, in the sense that each weak Roman dominating function of minimum weight is, at the same time, perfect Roman dominating.

Note on the perfect Roman domination number of graphs

In this note, we list some propositions of the perfect Roman domination number of graphs and give the characterization of graphs G with special value of the perfect Roman domination. Furthermore, γRp(F)+γRp(F¯) is given. At the last, a linear time algorithm is shown to compute the perfect Roman domination number of a cograph.

Perfect Roman domination in graphs

A perfect Roman dominating function on a graph G is a function f:V(G)⟶{0,1,2} having the property that for every vertex u with f(u)=0, there exists exactly one vertex v such that uv∈E(G) and f(v)=2. The weight of f, denoted by w(f), is the value ∑v∈V(G)f(v). Given a graph G and a positive integer k, the perfect Roman domination problem is to decide whether there is a perfect Roman dominating function f on G such that w(f) is at most k. In this paper, we first show that the perfect Roman domination problem is NP-complete for chordal graphs, planar graphs, and bipartite graphs. Then we present polynomial time algorithms for computing a perfect Roman dominating function with minimum weight in block graphs, cographs, series-parallel graphs, and proper interval graphs.

On perfect Roman domination number in trees: complexity and bounds

A perfect Roman dominating function on a graph $$G =(V,E)$$ is a function $$f: V \longrightarrow \{0, 1, 2\}$$ satisfying the condition that every vertex u with $$f(u) = 0$$ is adjacent to exactly one vertex v for which $$f(v)=2$$ . The weight of a perfect Roman dominating function f is the sum of the weights of the vertices. The perfect Roman domination number of G, denoted by $$\gamma _{R}^{p}(G)$$ , is the minimum weight of a perfect Roman dominating function in G. In this paper, we first show that the decision problem associated with $$\gamma _{R}^{p}(G)$$ is NP-complete for bipartite graphs. Then, we prove that for every tree T of order $$n\ge 3$$ , with $$\ell $$ leaves and s support vertices, $$\gamma _R^P(T)\le (4n-l+2s-2)/5$$ , improving a previous bound given in Henning et al. (Discrete Appl Math 236:235–245, 2018).

Perfect roman domination in regular graphs

A perfect 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 exactly one vertex v for which f(v) = 2. The weight of a perfect Roman dominating function f is the sum of the weights of the vertices. The perfect Roman domination number of G, denoted ?pR(G), is the minimum weight of a perfect Roman dominating function in G. We show that if G is a cubic graph on n vertices, then ?pR(G) ? 3/4n, and this bound is best possible. Further, we show that if G is a k-regular graph on n vertices with k at least 4, then ?pR(G) ? (k2+k+3/k2+3k+1)n.

Perfect Roman domination in trees

A perfect 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 exactly one vertex v for which f(v)=2. The weight of a perfect Roman dominating function f is the sum of the weights of the vertices. The perfect Roman domination number of G, denoted γRp(G), is the minimum weight of a perfect Roman dominating function in G. We show that if G is a tree on n≥3 vertices, then γRp(G)≤45n, and we characterize the trees achieving equality in this bound.

XXXV. Some Observations on the Horns given by Henry I. to the Cathedral of Carlisle. By the Rev. Mr. Cole, F.A.S. In a Letter to the President

This curious and perfect Roman fibula, which I have the honour to send to you by my friend Mr. Lort for your inspection, was found at no great distance from Rome, about the year 1770, at a place called Rocca Priore, between Frescati and Palestrina, by a peasant, who sold it, together with some coins, to my friend Mr. Kerrich, fellow of Magdalen college, Cambridge, from whom I received it very lately.