Signed total double Roman dominatıon numbers in digraphs

Let D = (V, A) be a finite simple digraph. A signed total double Roman dominating function (STDRD-function) on the digraph D is a function f : V (D) → {−1, 1, 2, 3} satisfying the following conditions: (i) P x∈N−(v) f(x) ≥ 1 for each v ∈ V (D), where N−(v) consist of all in-neighbors of v, and (ii) if f(v) = −1, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor assigned 3 under f, while if f(v) = 1, then the vertex v must have at least one in-neighbor assigned 2 or 3 under f. The weight of a STDRD-function f is the value P x∈V (D) f(x). The signed total double Roman domination number (STDRD-number) γtsdR(D) of a digraph D is the minimum weight of a STDRD-function on D. In this paper we study the STDRD-number of digraphs, and we present lower and upper bounds for γtsdR(D) in terms of the order, maximum degree and chromatic number of a digraph. In addition, we determine the STDRD-number of some classes of digraphs.Publisher's Versio

Restrained reinforcement number in graphs

A set $S$ of vertices is a restrained dominating set of a graph $G=(V,E)$ if every vertex in $V\setminus S$ has a neighbor in $S$ and a neighbor in $V\setminus S$. The minimum cardinality of a restrained dominating set is the restrained domination number $\gamma_{r}(G)$. In this paper we initiate the study of the restrained reinforcement number $r_{r}(G)$ of a graph $G$ defined as the cardinality of a smallest set of edges $F\subseteq E(\overline{G})$ for which $\gamma _{r}(G+F

Acute Flaccid Paralysis and Its Differential Diagnosis in in Kurdistan Province, Western Iran; an 11-Year Surveillance

Objective: The surveillance of acute flaccid paralysis (AFP) is a key strategy for monitoring the progress of poliomyelitis eradication and is a sensitive measure for detecting potential cases of poliomyelitis and poliovirus infection. This study was conducted to describe the characteristics of patients reported with AFP, and to evaluate the performance of the surveillance system in Kurdistan province, western Iran, using indicators recommended by the World Health Organization (WHO). Methods: This observational study was conducted from January 2000 to December 2010 at the Kurdistan Center for Disease Control and the Department of Pediatrics. All children who fulfilled the WHO definition for AFP were included in our study. The stool samples of all the children were sent for poliovirus isolation. All the patients were evaluated for 60 days after the onset of symptoms to identify the signs of residual weakness. Findings: One-hundred thirty nine children aged <15 years were reported to the Center for Diseases Control with AFP. In 138 (99%) stool samples no poliovirus was isolated. None of the patients was diagnosed as having acute poliomyelitis or polio-compatible paralysis. Guillain-Barré syndrome was the most frequent final diagnosis (79 cases) followed by Transverse Myelitis (7 cases) and Encephalitis (6 cases). By detecting 1.3 to 3.6 (mean 3.2) AFP cases per 100 000 population in Kurdistan during the study period, we achieved the WHO target for AFP surveillance. All performance indicators but one consistently met the WHO requirements and therefore demonstrated the effectiveness of the AFP surveillance program in Kurdistan. Conclusion: The effective surveillance system in Kurdistan and its evaluation may serve as a model for the surveillance of other infectious diseases

Roman game domination subdivision number of a graph

A {em Roman dominating function} on a graph $G = (V ,E)$ is a function $f : Vlongrightarrow {0, 1, 2}$ satisfying the condition that every vertex $v$ for which $f (v) = 0$ is adjacent to at least one vertex $u$ for which $f (u) = 2$. The {em weight} of a Roman dominating function is the value $w(f)=sum_{vin V}f(v)$. The Roman domination number of a graph $G$, denoted by $gamma_R(G)$, equals the minimum weight of a Roman dominating function on G. The Roman game domination subdivision number of a graph $G$ is defined by the following game. Two players $mathcal D$ and $mathcal A$, $mathcal D$ playing first, alternately mark or subdivide an edge of $G$ which is not yet marked nor subdivided. The game ends when all the edges of $G$ are marked or subdivided and results in a new graph $G'$. The purpose of $mathcal D$ is to minimize the Roman dominating number $gamma_R(G')$ of $G'$ while $mathcal A$ tries to maximize it. If both $mathcal A$ and $mathcal D$ play according to their optimal strategies, $gamma_R(G')$ is well defined. We call this number the {em Roman game domination subdivision number} of $G$ and denote it by $gamma_{Rgs}(G)$. In this paper we initiate the study of the Roman game domination subdivision number of a graph and present sharp bounds on the Roman game domination subdivision number of a tree

Safety of a Trivalent Inactivated Influenza Vaccine in Health Care Workers in Kurdistan Province, Western Iran; A Longitudinal Follow-up Study

We studied the safety of a trivalent inactivated surface antigen (split virion, inactivated) influenza vaccine, Begrivac® (Novartis Company), widely used in health care workers in Kurdistan. A longitudinal follow-up study was performed in Sanandaj city, west of Iran, recruiting 936 people. A questionnaire was completed for each participant, and all symptoms or abnormal physical findings were recorded. In part 1 of the study, the post-vaccination complaints were headache (5.3%), fever (7.9%), weakness (9.6%), chills (10.1%), sweating (10.5%), arthralgia (20.2%), and malaise (21.5%). Swelling of the injection site was seen in 267 (30.3%) participants, and pruritus of the injection site was seen in 290 (32.9%) participants. Redness and induration were also reported in 42.5% of the participants. Local reactions were mainly mild and lasted for 1-2 days. No systemic reactions were reported in the second part of the study. None of the participants experienced any inconvenience. We concluded that local adverse reactions after the trivalent inactivated split influenza vaccine, Begrivac®, in health care workers were far more common than expected. Continuous surveillance is needed to assess the potential risks and benefits of newly produced influenza vaccines

On The Total Roman Domination in Trees

A total Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the following conditions: (i) every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2 and (ii) the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The weight of a total Roman dominating function f is the value f(V (G)) = Σu∈V(G)f (u). The total Roman domination number γtR(G) is the minimum weight of a total Roman dominating function of G. Ahangar et al. in [H.A. Ahangar, M.A. Henning, V. Samodivkin and I.G. Yero, Total Roman domination in graphs, Appl. Anal. Discrete Math. 10 (2016) 501–517] recently showed that for any graph G without isolated vertices, 2γ(G) ≤ γtR(G) ≤ 3γ(G), where γ(G) is the domination number of G, and they raised the problem of characterizing the graphs G achieving these upper and lower bounds. In this paper, we provide a constructive characterization of these trees

Restrained {2}\{2\}-domination in graphs

A restrained $\{2\}$-dominating function (R$\{2\}$-DF) on a graph $G=(V,E)$ is a function $f:V\rightarrow\{0,1,2\}$ such that : \textrm{(i)} $f(N[v])\geq2$ for all $v\in V,$ where $N[v]$ is the set containing $v$ and all vertices adjacent to $v;$ \textrm{(ii)} the subgraph induced by the vertices assigned 0 under $f$ has no isolated vertices. The weight of an R$\{2\}$-DF is the sum of its function values over all vertices, and the restrained $\{2\}$-domination number $\gamma_{r\{2\}}(G)$ is the minimum weight of an R$\{2\}$-DF on $G.$ In this paper, we initiate the study of the restrained $\{2\}$-domination number. We first prove that the problem of computing this parameter is NP-complete, even when restricted to bipartite graphs. Then we give various bounds on this parameter. In particular, we establish upper and lower bound on the restrained $\{2\}$-domination number of a tree $T$ in terms of the order, the numbers of leaves and support vertices

Independent Roman bondage of graphs

An independent Roman dominating function (IRD-function) on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the conditions that (i) every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$, and (ii) the set of all vertices assigned non-zero values under $f$ is independent. The weight of an IRD-function is the sum of its function values over all vertices, and the independent Roman domination number $i_{R}(G)$ of $G$ is the minimum weight of an IRD-function on $G$. In this paper, we initiate the study of the independent Roman bondage number $b_{iR}(G)$ of a graph $G$ having at least one component of order at least three, defined as the smallest size of set of edges $F\subseteq E(G)$ for which $i_{R}(G-F)>i_{R}(G)$. We begin by showing that the decision problem associated with the independent Roman bondage problem is NP-hard for bipartite graphs. Then various upper bounds on $b_{iR}(G)$ are established as well as exact values on it for some special graphs. In particular, for trees $T$ of order at least three, it is shown that $b_{iR}(T)\leq3,$ while for connected planar graphs the upper bounds are in terms of the maximum degree with refinements depending on the girth of the graph