Independent Roman Research Articles

Lower bounds on the Roman and independent Roman domination numbers

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.

Trees with independent Roman domination number twice the independent domination number

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.

Strong equality between the Roman domination and independent Roman domination numbers in trees

A Roman dominating function (RDF) 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. The weight of an RDF is the value f(V (G)) = P u2V (G) f(u). An RDF f in a graph G is independent if no two vertices assigned positive values are adjacent. The Roman domination number R(G) (respectively, the independent Roman domination number iR(G)) is the minimum weight of an RDF (respectively, independent RDF) on G. We say that R(G) strongly equals iR(G), denoted by R(G) ≡ iR(G), if every RDF on G of minimum weight is independent. In this paper we provide a constructive characterization of trees T with R(T) ≡ iR(T).

A note on the independent Roman domination in unicyclic graphs

A Roman dominating function (RDF) on a graph G = (V;E) is a function f : V ! f 0;1;2g 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 an RDF is the value f(V (G)) = P u2V (G) f(u): An RDF f in a graph G is independent if no two vertices as- signed positive values are adjacent. The Roman domination number R(G) (respectively, the independent Roman domination number iR(G)) is the minimum weight of an RDF (respectively, independent RDF) on G. We say that R(G) strongly equals iR(G); denoted by R(G) iR(G); if every RDF on G of minimum weight is independent. In this note we characterize all unicyclic graphs G with R(G) iR(G): We consider finite, undirected, and simple graphs G with vertex set V = V (G) and edge set E = E(G). The open neighborhood of a vertex v 2 V is N(v) = NG(v) = fu 2 V j uv 2 Eg and the degree of v, denoted by dG(v), is the cardinality of its open neighborhood. A vertex of degree one is called a leaf, and its neighbor is called a support vertex. If v is a support vertex; then v is called strong if v is adjacent to at least two leaves. For a graph G, let f : V (G)! f0;1;2g be a function, and let (V0;V1;V2) be the ordered partition of V = V (G) induced by f, where Vi =fv2 V (G) : f(v) = ig for i = 0;1;2. There is a 1 1 correspondence between the functions f : V (G)!f0;1;2g and the ordered partitions (V0;V1;V2) of V (G). So we will write f = (V0;V1;V2). A function f : V (G)! f0;1;2g is a Roman dominating function (RDF) on G if every vertex u of G for which f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of an RDF is the value f(V (G)) = P u2V (G) f(u): An RDF f in a graph G is independent if no two vertices assigned positive values

Aelius Gallus' Campaign and the Arab Trade in the Augustan Age

The paper throws fresh light on an important aspect of Augustus' Oriental policy. It is a reassessment of his attitude towards the role the South-Arabian States played in the East-West trade circuit on the one hand, and of his attempts to develop an independent Roman seaborne trade with India on the other. In order to understand the perspective of this undertaking, one has to analyse the far-reaching changes which took place in South Arabia just at that time. The earlier “caravan-kingdoms” had been in a state of decline and have gradually been replaced by the rule of tribal confederations and States of the Highlands. Aelius Gallus' campaign was a total failure showing that the Roman Oriental trade should not be based on military conquest and on a rather expensive direct control of the middlemen.