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1

On signed arc total domination in digraphs

Asgharsharghi L., Khodkar A., Sheikholeslami S. M.

Opuscula Mathematica

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2018

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Vol. 38, no. 6

779--794

EN

Let D = (V, A) be a finite simple digraph and N(uv) = {u'v' ≠ uv | u = u' or v = v'} be the open neighbourhood of uv in D. A function ƒ : A → { — 1, +1} is said to be a signed arc total dominating function (SATDF) of D if [formula] holds for every arc uv ∈ A. The signed arc total domination number [formula] is defined as [formula]. In this paper we initiate the study of the signed arc total domination in digraphs and present some lower bounds for this parameter.

2

Bounds on the inverse signed total domination numbers in graphs

Atapour M., Norouzian S., Sheikholeslami S. M., Volkmann L.

Opuscula Mathematica

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2016

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Vol. 36, no. 2

145--152

EN

Let G = (V, E) be a simple graph. A function ƒ : V→ {- 1,1} is called an inverse signed total dominating function if the sum of its function values over any open neighborhood is at most zero. The inverse signed total domination number of G, denoted by [formula], equals to the maximum weight of an inverse signed total dominating function of G. In this paper, we establish upper bounds on the inverse signed total domination number of graphs in terms of their order, size and maximum and minimum degrees.

3

Signed star (k, k)-domatic number of a graph

Sheikholeslami S. M., Volkmann L.

Opuscula Mathematica

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2014

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Vol. 34, no. 3

609--620

EN

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function ƒ: E(G) →{−1, 1} is said to be a signed star k-dominating function on [formula] for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. A set {f1, f2, . . . , fd} of signed star k-dominating functions on G with the property that [formula] for each e ∈ E(G) is called a signed star (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed star (k, k)-dominating family on G is the signed star (k, k)-domatic number of G, denoted by [formula]. In this paper we study properties of the signed star (k, k)-domatic number [formula]. In particular, we present bounds on [formula], and we determine the signed (k, k)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213–218] for the signed star domatic number.

4

Signed star {k}-domatic number of a graph

Atapour M., Sheikholeslami S. M., Volkmann L.

Fasciculi Mathematici

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2013

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Nr 51

33--43

EN

Let G be a simple graph without isolated vertices with vertex set V(G) and edge set E(G) and let k be a positive integer. A function f : E(G) —> {±1, ±2,..., ±k} is said to be a signed star {k}-dominating function on G if Σe∈E(v) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. The signed star {k}-domination number of a graph G is y{k}ss(G) = min{ Σe∈Ef(v) | f is a SS{k}DF on G}. A set {f1, f2,..., fd} of distinct signed star {k}-dominating functions on G with the property that …[wzór] for each e ∈ E(G), is called a signed star {k}-dominating family (of functions) on G. The maximum number of functions in a signed star {k}-dominating family on G is the signed star {k}-domatic number of G, denoted by d{k}SS(G). In this paper we study the properties of the signed star {k}- domination number y{k}SS(G) and signed star {k}-domatic number d{k}SS(G). In particular, we determine the signed star {k}-domination number of some classes of graphs. Some of our results extend these one given by Xu [7] for the signed star domination number and Atapour et al. [1] for the signed star domatic number.

5

Trees whose 2-domination subdivision number is 2

Atapour M., Sheikholeslami S. M., Khodkar A.

Opuscula Mathematica

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2012

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Vol. 32, no. 3

423-437

EN

A set S of vertices in a graph G = (V,E) is a 2-dominating set if every vertex of V \ S is adjacent to at least two vertices of S. The 2-domination number of a graph G, denoted by γ2(G), is the minimum size of a 2-dominating set of G. The 2-domination subdivision number sdγ2 (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 2-domination number. The authors have recently proved that for any tree T of order at least 3, 1 ≤ sdγ2 (T ) ≤ 2. In this paper we provide a constructive characterization of the trees whose 2-domination subdivision number is 2.