A total Roman dominating function on a graph G is a function f: V (G)! f0; 1; 2g such that for every vertex v 2 V (G) with f(v) = 0 there exists a vertex u 2 V (G) adjacent to v with f(u) = 2, and the subgraph induced by the set fx 2 V (G): f(x)  1g has no isolated vertices. The total Roman domination number of G, denoted tR(G), is the minimum weight! (f) = P v2V (G) f(v) among all total Roman dominating functions f on G. It is known that tR(G)  t2(G) + (G) for any graph G with neither isolated vertex nor components isomorphic to K2, where t2(G) and (G) represent the semitotal domination number and the classical domination number, respectively. In this paper we give a constructive characterization of the trees that satisfy the equality above.

In this paper, we investigate domination number as well as signed domination numbers of Cay(G: S) for all cyclic group G of order n, where n ϵ {pm, pq} and S = {k < n: gcd(k, n) = 1}. We also introduce some families of connected regular graphs 􀀀 such that S (􀀀 ) ϵ {2, 3, 4, 5}.

Let G = (V, E) be a simple, undirected and connected graph. A Roman dominating function (RDF) on the graph G is a function f: V → {0, 1, 2} such that each vertex v ∈ V with f(v) = 0 is adjacent to at least one vertex u ∈ V with f(u) = 2. A total Roman dominating function (TRDF) of G is a function f: V → {0, 1, 2} such that (i) it is a Roman dominating function, and (ii) the vertices with non-zero weights induce a subgraph with no isolated vertex. The total Roman dominating set (TRDS) problem is to minimize the associated weight, f(V ) = P u∈V f(u), called the total Roman domination number (γtR(G)). Similarly, a subset S ⊆ V is said to be a total dominating set (TDS) on the graph G if (i) S is a dominating set of G, and (ii) the induced subgraph G[S] does not have any isolated vertex. The objective of the TDS problem is to minimize the cardinality of the TDS of a given graph. The TDS problem is NP-complete for general graphs. In this paper, we propose a simple 10. 5-factor approximation algorithm for TRDS problem in UDGs. The running time of the proposed algorithm is O(|V | log k), where k is the number of vertices with weights 2. It is an improvement over the best-known 12-factor approximation algorithm with running time O(|V | log k) available in the literature. Next, we propose another algorithm for the TDS problem in UDGs, which improves the previously best-known approximation factor from 8 to 7. 79. The running time of the proposed algorithm is O(|V | + |E|).

This paper is a review of research conducted by the present author in five villages in Iran. Each village is located in a province where UNFP A has been active for sometime. The main aim was to study the role of male domination upon reproductive health and health seeking behaviors of women in those selected sites. Before presenting the research findings both key concepts of the research are introduced and main approaches to feminism are reviewed. In this regard Marxist Feminism, Radical Feminism, Socialist Feminism, Liberal Feminism and finally Black Feminism however very briefly, are discussed. The findings clearly show that Bio-medical approach which was the dominant approach in studying reproductive health for a very long period, have major shortcomings and are not able to arrive at a realistic understanding of the women condition especially in non-western societies. Moreover the research has revealed both the merits and the need for conducting socio-cultural research for a reliable knowledge concerning the women condition in general and reproductive health in particular. It is now clear that hardware approach toward development an approach which limits development to buildings, health facilities, equipments and alike ignores the most important factors needed for a genuine societal transformation namely cultural change which seems vital to transform societies. The main findings of the research also are presented.

Let $G$ be a simple, undirected graph. In this paper, we initiate the study of independent Roman $\{3\}$-domination. A function $g : V(G) \rightarrow \lbrace 0, 1, 2, 3 \rbrace$ having the property that $\sum_{v \in N_G(u)}^{} g(v) \geq 3$, if $g(u) = 0$, and $\sum_{v \in N_G(u)}^{} g(v) \geq 2$, if $g(u) = 1$ for any vertex $u \in V(G)$, where $N_G(u)$ is the set of vertices adjacent to $u$ in $G$, and no two vertices assigned positive values are adjacent is called an \textit{ independent Roman $\{3\}$-dominating function} (IR3DF) of $G$. The weight of an IR3DF $g$ is the sum $g(V) = \sum_{v \in V}g(v)$. Given a graph $G$ and a positive integer $k$, the independent Roman $\{3\}$-domination problem (IR3DP) is to check whether $G$ has an IR3DF of weight at most $k$. We investigate the complexity of IR3DP in bipartite and chordal graphs. The minimum independent Roman $\lbrace 3 \rbrace$-domination problem (MIR3DP) is to find an IR3DF of minimum weight in the input graph. We show that MIR3DP is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs. We also show that the domination problem and IR3DP are not equivalent in computational complexity aspects. Finally, we present an integer linear programming formulation for MIR3DP.

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} such that every vertex v with f(v) = 0 is adjacent to a vertex u with f(u) = 2. If f is an RDF on G, then let Vi = {v ,V (G): f(v) = i} for i ,{0,1,2}. An RDF f is called a restrained (total) Roman dominating function if the subgraph induced by V0 (induced by V1 , V2) has no isolated vertex. A total and restrained Roman dominating function is a total restrained Roman dominating function. The total restrained Roman domination number ɤ, trR(G) on a graph G is the minimum weight of a total restrained Roman dominating function on the graph G. We initiate the study of total restrained Roman domination number and present several sharp bounds on ɤ, trR(G). In addition, we determine this parameter for some classes of graphs.

A graph G with no isolated vertex is total domination vertex critical if for any vertex u of G that is not adjacent to vertex of degree one, the total domination number G-u of is less than the total domination number of G. We call these graphs total domination critical or just gt-critical. If such a graph G has total domination number k, we call k- gt -critical. We study some results on the characterization of total domination critical graphs of order D(G)+4.

‎‎A Roman dominating function (RDF) on a graph is a function satisfying the condition that every vertex with is adjacent to at least one vertex for which . The weight of an RDF is the sum of the weights of the vertices under . The Roman domination number, of is the minimum weight of an RDF in . The Roman domination polynomial of a graph of order is the polynomial , where is the number of RDFs of with weight . In this paper we prove properties of Roman domination polynomials and determine in several classes of graphs by new approaches. We also present bounds on the number of all Roman domination polynomials in a graph.