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A 2-rainbow dominating function (2RDF) of a graph G = (V (G), E(G)) is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2} such that for every vertex v ∈ V (G) with f(v) = ∅ the condition ∪u∈N(v)f(u) = {1, 2} is fulfilled, where N(v) is the open neighborhood of v. A total 2-rainbow dominating function f of a graph with no isolated vertices is a 2RDF with the additional condition that the subgraph of G induced by {v ∈ V (G) | f(v) ≠∅} has no isolated vertex. The total 2-rainbow domination number, γtr2(G), is the minimum weight of a total 2-rainbow dominating function of G. In this paper, we establish some sharp upper and lower bounds on the total 2-rainbow domination number of a tree. Moreover, we show that the decision problem associated with γtr2 (G) is NP-complete for bipartite and chordal graphs.

A total Roman dominating function on a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2 and the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The minimum weight of a total Roman dominating function on a graph G is called the total Roman domination number of G. The total Roman reinforcement number rtR (G) of a graph G is the minimum number of edges that must be added to G in order to decrease the total Roman domination number. In this paper, we investigate the proper- ties of total Roman reinforcement number in graphs, and we present some sharp bounds for rtR (G). Moreover, we show that the decision problem for total Roman reinforcement is NP-hard for bipartite graphs.

A restrained Italian dominating function (RIDF) on a graph G = (V , E) is a function f : V → { 0 , 1 , 2 } satisfying: (i) f (N (u)) ≥ 2 for every vertex u ∈ V (G) with f (u) = 0 , where N (u) is the set of vertices adjacent to u ; (ii) the subgraph induced by the vertices assigned 0 under f has no isolated vertices. The weight of an RIDF is the sum of its function values over the whole set of vertices, and the restrained Italian domination number γ r I (G) is the minimum weight of an RIDF on G. In this paper, we initiate the study of the restrained Italian bondage number b r I (G) of a graph G with no isolated vertices defined as the smallest size of set of edges F ⊆ E (G) for which γ r I (G − F) > γ r I (G). We begin by showing that the decision problem associated with the restrained Italian bondage problem is NP-hard. Then basic properties of the restrained Italian bondage number are presented. Finally, some sharp bounds for b r I (G) are also established. [ABSTRACT FROM AUTHOR]

The modified Sombor index of a graph G is defined as the sum of weights 1 / d u 2 + d v 2 over all edges u v of G , where d u and d v are the degrees of the vertices u and v in G , respectively. In this paper, we give a sharp lower bound on the modified Sombor index of unicyclic graphs with given diameter in terms the number of vertices. [ABSTRACT FROM AUTHOR]

Let k ≥ 1 be an integer and G be a simple graph with vertex set V (G). Let f be a function that assigns label from the set { 0 , 1 , 2 , ... , k + 1 } to the vertices of a graph G. For a vertex v ∈ V (G) , the active neighborhood of v , denoted by AN (v) , is the set of vertices w ∈ N G (v) such that f (w) ≥ 1. A [ k ] -RDF is a function f : V (G) → { 0 , 1 , 2 , ... , k + 1 } satisfying the condition that for any vertex v ∈ V (G) with f (v) < k , f (N G [ v ]) ≥ | AN (v) | + k. The weight of a [ k ] -RDF is ω (f) = Σ v ∈ V (G) f (v). The [ k ] -Roman domination number γ [ k R ] (G) of G is the minimum weight of an [ k ] -RDF on G. The case k = 4 is called quadruple Roman domination number. In this paper, we first establish an upper bound for quadruple Roman domination number of graphs with minimum degree two, and then we derive a Nordhaus–Gaddum bound on the quadruple Roman domination number of graphs. [ABSTRACT FROM AUTHOR]

Let D be a finite and simple digraph with vertex set V (D). A double Roman dominating function (DRDF) on digraph D is a function f : V (D) → { 0 , 1 , 2 , 3 } such that every vertex with label 0 has an in-neighbor with label 3 or two in-neighbors with label 2 and every vertex with label 1 have at least one in-neighbor with label at least 2. The weight of a DRDF f is the value ω (f) = ∑ v ∈ V (D) f (v). A DRDF f on D with no isolated vertex is called a total double Roman dominating function if the subgraph of D induced by the set { v ∈ V (D) | f (v) ≠ 0 } has no isolated vertex. In this paper, we initiate the study of the total double Roman domination number in digraphs and show its relationship to other domination parameters. In particular, we present some bounds for the total double Roman domination number and we determine the total double Roman domination number of some classes of digraphs. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) , a double Roman dominating function is a function f : V (G) → { 0 , 1 , 2 , 3 } having the property that if f (v) = 0 , then vertex v must have at least two neighbors assigned 2 under f or one neighbor with f (w) = 3 , and if f (v) = 1 , then vertex v must have at least one neighbor with f (w) ≥ 2. The weight of a double Roman dominating function f is the value w (f) = ∑ v ∈ V (G) f (v). The double Roman domination number of a graph G , denoted by γ d R (G) , equals the minimum weight of a double Roman dominating function on G. The double Roman domination subdivision number sd γ d R (G) of a graph G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the double Roman domination number. In this paper, we first show that the decision problem associated with sd γ d R (G) is NP-hard and then establish upper bounds on the double Roman domination subdivision number for arbitrary graphs. [ABSTRACT FROM AUTHOR]

Let G be a finite and simple graph with vertex set V (G). Let f be a function that assigns label from the set { 0 , 1 , 2 , 3 , 4 , 5 } to the vertices of a graph G. For a vertex v ∈ V (G) , the active neighborhood of v , denoted by AN (v) , is the set of vertices w ∈ N G (v) such that f (w) ≥ 1. A quadruple Roman dominating function (QRDF) is a function f : V (G) → { 0 , 1 , 2 , 3 , 4 , 5 } satisfying the condition that for any vertex v ∈ V (G) with f (v) < 4 , f (N G [ v ]) ≥ | AN (v) | + 4. The weight of a QRDF is ω (f) = Σ v ∈ V (G) f (v). The quadruple Roman domination number γ [ 4 R ] (G) of G is the minimum weight of a QRDF on G. In this paper, we investigate the properties of the quadruple Roman domination number of graphs, present bounds on γ [ 4 R ] (G) and give exact values for some graph families. In addition, complexity results are also obtained. [ABSTRACT FROM AUTHOR]

For a graph G = (V,E), a triple Roman domination function is a function f : V (G) → {0, 1, 2, 3, 4} having the property that for any vertex v ∈ V (G), if f(v) < 3, then f(AN[v]) ≥ |AN(v)|+3, where AN(v) = {w ∈ N(v) | f(w) ≥ 1} and AN[v] = AN(v)∪{v}. The weight of a triple Roman dominating function f is the value ω(f) = Σv∈V (G) f(v). The triple Roman domination number of G, denoted by γ[3R](G), equals the minimum weight of a triple Roman dominating function on G. The triple Roman domination subdivision number sdγ[3R] (G) of a graph G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the triple Roman domination number. In this paper, we first show that the decision problem associated with sdγ[3R] (G) is NP-hard and then establish upper bounds on the triple Roman domination subdivision number for arbitrary graphs. [ABSTRACT FROM AUTHOR]

Background and Objective: Tonsillectomy is one of the most common surgeries in children, which is associated with pain and bleeding after the operation. This study was conducted to investigate the effect of lidocaine-epinephrine administration in controlling complications after tonsillectomy. Methods: This clinical trial was conducted on 109 patients aged 5 to 18, who had referred to Kowsar Hospital in Sanandaj for tonsillectomy. After dividing the patients randomly, 40 mg of 1% lidocaine and 5 µg of epinephrine (a volume of 5 ml) were injected into the tonsillar bed in the intervention group before the surgery, but in the patients of the control group, surgery was started without lidocaine-epinephrine administration. Blood pressure and heart rate of the patients before, during and after surgery, and pain intensity based on Wong-Baker Faces Pain Scale at minutes 30 and 60 and 6, 12 and 24 hours after surgery, and the amount of bleeding during and after surgery were recorded and compared by calculating the volume of suction and blood pads as well as the length of stay in recovery. Findings: The mean amount of bleeding in the intervention group was 65.5±47.1 ml and in the control group was 113.7±45.7 ml (p=0.0001). The mean pain intensity and hemodynamic changes at minutes 30 and 60, and 6, 12 and 24 hours after surgery in the intervention group were lower than the control group and showed statistically significant differences (p=0.0001). The mean time of discharge from recovery in the intervention group was 0.35±23.2 minutes and in the control group was 50.4±17.5 minutes (p=0.0001). Conclusion: Based on the results of this study, it seems that the administration of lidocaineepinephrine in the tonsil bed before tonsillectomy can reduce the amount of pain, bleeding, and hemodynamic changes during and after surgery, and the patient can recover sooner. [ABSTRACT FROM AUTHOR]

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 ƒ : V (D) → {-1; 1; 2; 3} satisfying the following conditions: (i) Σx∈N-(v) ƒ(x) ≥ 1 for each v ∈ V (D), where N-(v) consist of all in-neighbors of v, and (ii) if ƒ(v) = -1, then the vertex v must have at least two in-neighbors assigned 2 under ƒ or one in-neighbor as igned 3 under ƒ, while if ƒ(v) = 1, then the vertex v must have at least one in-neighbor assigned 2 or 3 under ƒ. The weight of a STDRD-function ƒ is the value ∑x∈V(D) ƒ(x). The signed total double Roman domination number (STDRD-number) γsdRt(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 γsdRt(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. [ABSTRACT FROM AUTHOR]

Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set V (G). A signed Roman k-dominating function (SRkDF) on a graph G is a function f : V (G) → {−1, 1, 2} such that (i) every vertex v with f (v) = −1 is adjacent to at least one vertex u with f (u) = 2, (ii) Σu ϵN[v]f (u) ≥ k holds for any vertex v. The weight of a SRkDF f is ΣuϵV (G)f (u), and the minimum weight of a SRkDF is the signed Roman k-domination number (G) of G. In this paper, we investigate the (G) signed Roman k-domination number of graphs, and we establish some bounds on (G). In the case that T is a tree, we present lower and upper bounds on (T) for k ∈ {3, 4} and classify all extremal trees. [ABSTRACT FROM AUTHOR]

A total Roman dominating function on a graph G is a labeling f : V (G) → { 0 , 1 , 2 } such that every vertex with label 0 has a neighbor with label 2 and the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. A set { f 1 , f 2 , ... , f d } of total Roman dominating functions on G with the property that ∑ i = 1 d f i (v) ≤ 2 for each v ∈ V (G) , is called a total Roman dominating family (of functions) on G. The maximum number of functions in a total Roman dominating family on G is the total Roman domatic number of G , denoted by d tR (G). In this paper, we investigate the properties of total Roman domatic number in graphs. In particular, we present some sharp bounds for d tR (G) and we determine the total Roman domatic number of some special graphs. [ABSTRACT FROM AUTHOR]

A double Roman dominating function (DRDF) on a graph G = (V;E) is a function f : V ! f0; 1; 2; 3g having the property that if f(u) = 0, then a vertex u has at least two neighbors assigned 2 under f or one neighbor w with f(w) = 3, and if f(u) = 1, then a vertex u must have at least one neighbor w with f(w) 2. A total double Roman dominating function (TDRDF) on a graph G with no isolated vertex is a DRDF f on G with the additional property that the subgraph of G induced by the set fv 2 V : f(v) 6= 0g has no isolated vertices. The weight of a total double Roman dominating function f is the value, f(V) = u2V (G)f(u). The total double Roman domination number tdR(G) is the minimum weight of a TDRDF on G. A subset S of V is a 2-independent set of G if every vertex of S has at most one neighbor in S. The maximum cardinality of a 2-independent set of G is the 2-independence number fi2(G). In this paper, we show that if T is a tree, then tdR(T) 2fi2(T), and we characterize all trees attaining the equality. [ABSTRACT FROM AUTHOR]