iii Acknowledgments I want to thank my thesis advisor Jörg Rothe for all of his support during the past four years. In the first place, I am deeply grateful to him for giving me the chance to be part of his research team. Without his great efforts, I would never have had the chance to work in the scientific community. Many inspiring and valuable discussions with him initiated fruitful ideas tha...

A restrained Roman dominating function (RRD-function) on a graph \(G=(V,E)\) is \(f\) from \(V\) into \(\{0,1,2\}\) satisfying: (i) every vertex \(u\) with \(f(u)=0\) adjacent to \(v\) \(f(v)=2\); (ii) the subgraph induced by vertices assigned 0 under has no isolated vertices. The weight of an RRD-function sum its value over whole set vertices, and domination number minimum \(G.\) In this paper...

‎Let $f$ be a proper $k$-coloring of a connected graph $G$ and‎ ‎$Pi=(V_1,V_2,ldots,V_k)$ be an ordered partition of $V(G)$ into‎ ‎the resulting color classes‎. ‎For a vertex $v$ of $G$‎, ‎the color‎ ‎code of $v$ with respect to $Pi$ is defined to be the ordered‎ ‎$k$-tuple $c_{{}_Pi}(v)=(d(v,V_1),d(v,V_2),ldots,d(v,V_k))$‎, ‎where $d(v,V_i)=min{d(v,x):~xin V_i}‎, ‎1leq ileq k$‎. ‎If‎ ‎distinct...

background: considering the negative consequences of using physical restraints, we conducted this study to identify patients who are more frequently restrained in a psychiatric emergency ward as an initial step to limit the use of restraint to the minimum possible. methods: this was a retrospective case control study conducted in iran psychiatric hospital in tehran, iran. we reviewed the files ...

A restrained Italian dominating function (RID-function) on a graph G=(V,E) is f:V→{0,1,2} satisfying: (i) f(N(u))≥2 for every vertex u∈V(G) with f(u)=0, where N(u) the set of vertices adjacent to u; (ii) subgraph induced by assigned 0 under f has no isolated vertices. The weight an RID-function sum its value over whole vertices, and domination number minimum G. In this paper, we initiate study ...

A subset D of the vertex set V (G) of a graph G is called point-set dominating, if for each subset S ⊆ V (G) − D there exists a vertex v ∈ D such that the subgraph of G induced by S ∪ {v} is connected. The maximum number of classes of a partition of V (G), all of whose classes are point-set dominating sets, is the point-set domatic number dp(G) of G. Its basic properties are studied in the paper.

A subset D of V (G) is called an equitable dominating set of a graph G if for every v ∈ (V − D), there exists a vertex u ∈ D such that uv ∈ E(G) and |deg(u) − deg(v)| 6 1. The minimum cardinality of such a dominating set is denoted by γe(G) and is called equitable domination number of G. In this paper we introduce the equitable edge domination and equitable edge domatic number in a graph, exact...

For a nonempty graph G = (V, E), a signed edge-domination of G is a function f : E(G) → {1,−1} such that ∑e′∈NG [e] f (e′) ≥ 1 for each e ∈ E(G). The signed edge-domatic number of G is the largest integer d for which there is a set { f1, f2, . . . , fd} of signed edge-dominations of G such that ∑d i=1 fi (e) ≤ 1 for every e ∈ E(G). This paper gives an original study on this concept and determin...

Let G = (V,E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex in V is adjacent to a vertex in S and every vertex of V −S is adjacent to a vertex in V −S. The total restrained domination number of G, denoted by γtr(G), is the minimum cardinality of a total restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We sho...