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1
Content available On minimum intersections of certain secondary dominating sets in graphs
Kosiorowska Anna, Michalski Adrian, Włoch Iwona
Opuscula Mathematica
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2023
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Vol. 43, no. 6
813--827
EN
In this paper we consider secondary dominating sets, also named as (1,k)-dominating sets, introduced by Hedetniemi et al. in 2008. In particular, we study intersections of the (1, 1)-dominating sets and proper (1, 2)-dominating sets. We introduce (1,2)-intersection index as the minimum possible cardinality of such intersection and determine its value for some classes of graphs.
2
Content available remote On Combining the Methods of Link Residual and Domination in Networks
Turacı Tufan
Fundamenta Informaticae
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2020
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Vol. 174, nr 1
43--59
EN
The concept of vulnerability is very important in network analysis. Several existing parameters have been proposed in the literature to measure network vulnerability, such as domination number, average lower domination number, residual domination number, average lower residual domination number, residual closeness and link residual closeness. In this paper, incorporating the concept of the domination number and link residual closeness number, as well as the idea of the average lower domination number, we introduce new graph vulnerability parameters called the link residual domination number, denoted by γLR(G), and the average lower link residual domination number, denoted by γLRaν(G) , for any given graph G. Furthermore, the exact values and the upper and lower bounds for any graph G are given, and the exact results of well-known graph families are computed.
3
Content available remote Combining the Concepts of Residual and Domination in Graphs
Turacı Tufan, Aytaç Aysun
Fundamenta Informaticae
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2019
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Vol. 166, nr 4
379--392
EN
Let G = (V (G), E(G)) be a simple undirected graph. The domination and average lower domination numbers are vulnerability parameters of a graph. We have investigated a refinement that involves the residual domination and average lower residual domination numbers of these parameters. The lower residual domination number, denoted by γvkR(G), is the minimum cardinality of dominating set in G that received from the graph G where the vertex vk and all links of the vertex vk are deleted. The residual domination number of graphs G is defined as [formula]. The average lower residual domination number of G is defined by [formula]. In this paper, we define the residual domination and the average lower residual domination numbers of a graph and we present the exact values, upper and lower bounds for some graph families.
4
Content available Review of Methods and Algorithms for Modelling Transportation Networks Based on Graph Theory
Guze S.
Zeszyty Naukowe Akademii Morskiej w Gdyni
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2018
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nr 107
25--39
EN
One of the best ways of modelling a transport network is to use a graph with vertices and edges. They represent nodes and arcs of such network respectively. Graph theory gives dozens of parameters or characteristics, including a connectivity, spanning trees or the different types of domination number and problems related to it. The main aim of the paper is to show graph theory methods and algorithms helpful in modelling and optimization of a transportation network. Firstly, the descriptions of basic notations in graph theory are introduced. Next, the concepts of domination, bondage number, edge-subdivision and their implementations to the transportation network description and modeling are proposed. Moreover, the algorithms for finding spanning tree or maximal flow in networks are presented. Finally, the possible usage of distinguishing concepts to exemplary transportation network is shown. The conclusions and future directions of work are presented at the end of the paper.
PL
Jednym z najlepszych sposobów modelowania sieci transportowej jest użycie grafu z wierzchołkami i krawędziami. Reprezentują one odpowiednio węzły i łuki takiej sieci. Teoria grafów daje możliwość użycia dziesiątek parametrów lub charakterystyk, w tym spójności, drzew spinających lub różnych typów liczb dominowania i związanych z tym problemów. Głównym celem artykułu jest przedstawienie metod i algorytmów teorii grafów pomocnych w modelowaniu i optymalizacji sieci transportowej. Po pierwsze, wprowadzono opisy podstawowych pojęć w teorii grafów. Następnie zaprezentowano koncepcje domino­wania, liczby zniewolenia czy podziału krawędzi grafu oraz ich implementacji do opisu i modelowania sieci transportowej. Ponadto przedstawiono algorytmy do wyszukiwa­nia drzewa opinającego i maksymalnego przepływu w sieciach. Wreszcie pokazano możliwe sposoby wykorzystania wyróżnionych koncepcji do przykładu sieci transportowej. Na zakoń­czenia przedstawiono wnioski i przyszłe kierunki prac.
5
Content available On domination multisubdivision number of unicyclic graphs
Raczek J.
Opuscula Mathematica
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2018
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Vol. 38, no. 3
409--425
EN
The paper continues the interesting study of the domination subdivision number and the domination multisubdivision number. On the basis of the constructive characterization of the trees with the domination subdivision number equal to 3 given in [H. Aram, S.M. Sheikholeslami, O. Favaron, Domination subdivision number of trees, Discrete Math. 309 (2009), 622-628], we constructively characterize all connected unicyclic graphs with the domination multisubdivision number equal to 3. We end with further questions and open problems.
6
Content available The graph theory approach to analyze critical infrastructures of transportation systems
Guze S.
Journal of Polish Safety and Reliability Association
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2014
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Vol. 5, No. 2
57--62
EN
The main aim of the paper is to use algorithms and parameters of graph theory as tool to analyze the transpiration systems. To realize this goal the well-known information about graph theory algorithms and parameters will be introduced and described. The possible application of graph theory algorithms and parameters to analyze the critical infrastructures of exemplary transportation system will be shown.
7
Content available remote On Fully Split Lacunary Polynomials in Finite Fields
Bibak K., Shparlinski I. E.
Bulletin of the Polish Academy of Sciences. Mathematics
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2011
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Vol. 59, no 3
197--202
EN
We estimate the number of possible degree patterns of k-lacunary polynomials of degree t< p which split completely modulo p. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.
8
Content available On the global offensive alliance number of a tree
Bouzefrane M., Chellali M.
Opuscula Mathematica
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2009
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Vol. 29, no. 3
223-228
EN
For a graph G = (V, E), a set S ⊆ V is a dominating set if every vertex in V - S has at least a neighbor in S. A dominating set S is a global offensive alliance if for every vertex v in V - S, at least half of the vertices in its closed neighborhood are in S. The domination number ϒ(G) is the minimum cardinality of a dominating set of G and the global offensive alliance number ϒo(G) is the minimum cardinality of a global offensive alliance of G. We first show that every tree of order at least three with l leaves and s support vertices satisfies ϒo(T) ≥ (n - l + s + 1)/3 and we characterize extremal trees attaining this lower bound. Then we give a constructive characterization of trees with equal domination and global offensive alliance numbers.
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