Domination Invariants and Their Properties тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Голмохаммади Хамидреза

• Голмохаммади Хамидреза
• кандидат наук
• 2025

Оглавление диссертации кандидат наук Голмохаммади Хамидреза

Contents.

List of figures

Introduction

1. Definitions

§1.1 General concepts

§1.2 Domination and its variations in graphs

§1.3 Domination Partitions

§1.4 Introduction to coalitions in graphs

§1.5 Coalition graphs

2. Total coalitions in graphs

§2.1 Introduction to total coalition

§2.2 Upper bounds and exact values with 5(G) =

§2.3 Upper bounds and exact values with 5(G) =

3. Connected coalitions in graphs

§3.1 Introduction to connected coalition

§3.2 Graphs with pendant edges

§3.3 Connected coalition Number of Trees

§3.4 Graphs G with CC(G) = n and CC(G) = n

4. Independent coalitions in graphs

§4.1 Preliminaries

§4.2 Graphs G with IC(G) e [0,n(G),n(G) - 1}

§4.3 Independent coalition graphs

§4.4 Independent coalition graphs of paths

§4.5 Independent coalition graphs of cycles

5. Coalitions in cubic graphs

§5.1 Coalition of cubic graphs of order at most

§5.2 Total coalition of cubic graphs of order at most

§5.3 On cubic graphs having the maximal coalition number

6. On coalition graphs

§6.1 On coalition graphs of cycles

§6.2 Coalition graphs defined by C\5

§6.3 Coalition graphs of cycles of order less than

7. Conclusion and future research 99 Bibliography

List of publications

List of figures

1 Queens dominating the chessboard

2 A c-partition w of P9 and the coalition graph cg(p9,k)

3 The connected graph (2K2 + K1) + K1 belonging to T

4 The case that T[Vi] is connected

5 The edge-dominated matrix £ for C6

6 The graph C6

7 The graph X5; so n = 5 in the figure

8 The graphs F%, i e [5]

9 Cubic graphs of order

10 Cubic graphs of order

11 Cubic graphs of order

12 Petersen graph P

13 Cubic graphs of order 16 with C(G) = 9 and their coalition partitions (without single-vertex sets)

14 Cubic graphs of order 16 with C(G) = 9 and their coalition partitions (without single-vertex sets) (continue)

15 A family of cubic graphs with C(G) =

16 Coalition graphs defined by cycles

17 All possible c-graphs C5 derived by vertex partitions of C6, C7, and

cQ

18 All possible c-graphs C5 derived by vertex partitions of C12

19 Placement coalitions in C12 generated by cycle

Введение диссертации (часть автореферата) на тему «Domination Invariants and Their Properties»

Introduction

While domination in graphs was first formally defined by Berge in 1958, the origin of domination is attributed to back to defense strategies used by the Roman Empire in the fourth century AD, as well as a variety of chess problems in the mid-to-late 1800s. In a graph G a dominating set is a set S of vertices of G such that every vertex outside S is adjacent to at least one vertex in S. The objective is to determine the minimum number of vertices required to dominate the whole graph. Domination in graphs quickly gained popularity since its introduction leading to over 1200 papers published on domination in graphs by the end of 1990. In 1998, Haynes, Hedetniemi, and Slater recognized the requirement for a comprehensive survey of the literature on domination in graphs, and they published the initial two books on domination [34, 35]. Domination in graphs is nowadays a very well known topic in the area of graph theory. The development of domination has continued, and today more than 5000 papers have been published in this field. Much of the interest in domination theory in graphs is significantly influenced by its applications in broad areas of study, such as genetics, chemistry, bioinformatics, facility location, social networking, transportation, and so on. Over time, motivated by real-life challenges, different types of adaptations of dominating sets have emerged, leading to a broad spectrum of related concepts. Depending on the need, numerous variations of domination including total, connected, independent dominating sets, and so on, produced by imposing additional conditions. The recent progress in domination theory can be found in the books [31, 32, 33]. The concept of coalition in graphs has attracted much interest of late. Motivated by the idea that the union of two sets can have a property that neither set has, Haynes et al. [26] introduced this concept as a mathematical model in graph theory based on the dominating set in graphs. A coalition is generally described as a temporary alliance of two or

more parties to work together toward a common vision. Coalitions are frequently formed in governments when no political party achieves an absolute majority after an election, but a collaborative agreement between two or more political parties is sufficient to created a majority and set up a government, with members of the coalition serving on a cabinet. Indeed, coalitions can be an effective strategy for changing the policies in business, government, and other relevant sectors that are needed to solve the problem or achieve the goal. For instance, in the industrial organizations the strategic coalitions allow industrial businesses to boost the development of joint ventures, joint production, and research facilities [6]. Haynes et al. [26] explored the coalition number of a graph C(G) and established several upper and lower bounds for the coalition number of a graph in terms of its order, minimum and maximum degree. They also obtained the coalition numbers of all paths and all cycles. In addition, in the paper [27], they established a tight upper bound on the coalition number of any graph G in terms of the maximum degree of G. Since studying the properties of graphs associated with a c-partition is a natural approach, in [29], Haynes et al. initiated the study of the coalitions graphs. They focused on coalition graphs of trees, paths, and cycles. They proved that there is a finite number of coalition graphs of paths and cycles, respectively, and identified all of them. Moreover, they characterized the infinite family of coalition graphs of trees. They studied coalition graphs further in [28, 30]. To pursue the investigation of coalitions in graphs, Bakhshesh, Henning and Pradhan in [3] characterized all graphs G of order n with 5(G) = 1 and C(G) = n. Moreover, they characterized all trees T of order n with C(T) = n — 1. Due to the fact that special attention has been paid to cubic graphs, and the dominating set and its variants have been examined extensively for cubic graphs [1, 13, 15, 18, 43, 44, 45, 50], recently, Alikhani, Golmohammadi and Konstantinova studied the coalitions in cubic graphs [P4, P9]. In particular, they determined coalition number of cubic graphs of order at most 10. In order to expand the study of coalitions, various aspects of coalitions based on different

types of dominating sets have recently been studied. By imposing restrictions on a coalition partition, various coalition parameters can be generated, including total, connected, independent coalition numbers, and so on. Golmohammadi et al. studied the concepts of total coalition, connected coalition, independent coalition and strong coalition in [P1, P2, P3, P10, 24]. It is worth mentioning that the total coalition in graphs has subsequently been investigated in [4, P7, P8, 37]. Furthermore, the concept of independent coalition has also been studied in [47].

Within the framework of the dissertation, we have provided many new results for coalition and its variants including total coalition, connected coalition, and independent coalition.The novelty of the outcomes which are obtained in the dissertation contributes to the further development of the domination theory. In this sense, it is our goal to fill some gaps throughout this dissertation. The results are based on [P1, P2, P3, P4, P5, P6, P7, P8, P9, P10].

In Chapter 1, basic definitions and the relevant notation will be used throughout the dissertation are provided. We use [16, 52] as references for terminology and notation. In Chapter 2, we introduce and discuss one of the important modifications to the coalition concept is the total coalition. After introducing the total coalitions, we show that not all graphs admit a total coalition partition. We investigate the relationship between the total coalition number, symbolized as Ct(G), and total domatic number dt(G). We establish lower and upper bounds on the total coalition number. We also determine the total coalition numbers of all paths and all cycles. In Chapter 3, our objective is to initiate an examination into the notion of connected coalitions in graphs. We first provide a thorough characterization of all graphs possessing a connected coalition partition. We present some bounds on the connected coalition number CC(G). We determine the connected coalition number of graphs with at least one pendant edge. Furthermore, we derive the connected coalition number of trees. Finally, two polynomial-time algorithm to find graphs G with CC(G) e {n(G),n(G) — 1} are presented. In Chapter 4,

we focus on the independent coalitions in graphs. In the first section of this chapter, we consider graphs with extremal independent coalition number IC(G). We describe a large family of graphs with IC(G) = 0 and we characterize graphs G with IC(G) e {n(G),n(G) — 1}. In the second section, we explore the independent coalition graph associated with the independent c-partition of a graph G and study the properties of the independent coalition graph of paths and cycles. In Chapter 5, we proceed with the investigation of coalition and total coalition in regular graphs. In particular, we obtain of coalition and total coalition numbers of cubic graphs of order at most 10. In the first part, we demonstrate that C(G) e {6, 7,8} for cubic graph of order at most 10. In the second part, the total coalition numbers of cubic graphs of order at most 10 are given by either 4, or 5, or 6. Additionally, in the last part, we answer a problem posed in [P4]. Indeed, we construct an infinite family having the maximum coalition number. In Chapter 6, we turn our attention to coalition graph of cycles. We identify the the shortest cycle having the maximum number of coalition graphs. In fact, we show that C15 is the shortest cycle satisfying this property. In the last chapter, we conclude by mentioning some other directions of further research. The main results of the dissertation are as follows.

1) We demonstrate that every graph G with no isolated vertex has a total coalition partition [Theorem 2.3, Chapter 2].

2) We show that if G is a graph with maximum degree A(G), then for any Cf (G)-partition Y and for any X e Y, the number of total coalitions formed by X is at most A(G) [Theorem 2.10, Chapter 2].

3) We prove that for any graph G, CC(G) = 0 if and only if G e T [Theorem 3.10, Chapter 3].

4) We demonstrate that a graph G is an ICG of some path if and only if G e {Pi,P4,P5, 2Pi, 2P2, 2P3.P1 u P2,P2 u P3} [Theorem 4.10, Chapter 4].

5) We estimate that the coalition number of cubic graphs of order 10 is at most 8 [Theorem 5.7, Chapter 5].

6) We estimate that the total coalition number of cubic graphs of order 10 is at most 6 [Theorem 5.15, Chapter 5].

7) We find an infinite family of cubic graphs having the maximum coalition number [Proposition 5.17, Chapter 5].

8) We prove that the cycle Ck does not define the coalition graph C5, where k e {6, 7,9} [Theorem 6.4, Chapter 6].

9) We prove that the cycle Ck does not define the coalition graph 3K2, where k e {8,10,11,13,14} [Theorem 6.5, Chapter 6].

10) We show that the cycle C\2 does not define the coalition graph C5 [Theorem 6.6, Chapter 6].

Finally, it is worth mentioning that the results of this dissertation presented at various scientific events, such as the following.

1) XXIII International Conference Mathematical Optimization Theory and Operations Research (MOTOR 2024), June 30 - July 06, 2024, Omsk, Russia [https://motor24.oscsbras.ru].

2) 22nd International Conference Mathematical Optimization Theory and Operations Research (MOTOR 2023), July 2 - 8, 2023, Ekaterinburg, Russia [https://motor2023.uran.ru].

3) International (54th National) Youth School-Conference, February 6-10, 2023, Ekaterinburg, Russia [https://sopromat.imm.uran.ru].

4) Weekly seminars on Graph Theory, Sobolev Institute of Mathematics, March 7, 14 and 21, 2023, Novosibirsk, Russia.

5) Weekly seminars of Department of Mathematical Sciences, Yazd University, February 8, 2023, Yazd, Iran.

6) Weekly seminars on Graph Theory, Sobolev Institute of Mathematics, September 29, 2022, Novosibirsk, Russia.

Заключение диссертации по теме «Другие cпециальности», Голмохаммади Хамидреза

Conclusion and future research

In this thesis, several aspects of coalitions in graphs and their properties are examined. The main achievements of the thesis are described below.

1

2

3

4

6

9 10 11

The sufficient condition for the existence of a total coalition partition of a graph G is examined.

The number of total coalitions involving any set in a total coalition partition of G is estimated.

The upper bounds on the total coalition number in terms of maximum degree A(G) are established.

The total coalition numbers of all paths and all cycles are calculated.

The relationship between the connected coalition number CC(G) and the connected domatic number dc(G) is established.

The necessary and sufficient conditions for the existence of a connected coalition partition of a graph G are examined.

All trees T satisfying CC(T) = 2 are characterized.

A large family of graphs with IC(G) = 0 is described and graphs G with IC (G) £ {n(G),n(G) — 1} are characterized.

The independent coalition graphs of paths are characterized.

The coalition numbers of cubic graphs of order 10 are calculated.

The total coalition numbers of cubic graphs of order 10 are calculated.

5

12) An infinite family of cubic graphs having maximum coalition number is constructed.

13) The shortest cycle having the maximum number of coalition graphs is identified.

To wrap up, we would like to point out that the coalition has established new perspectives on domination theory, and it appears that a competition is coming up soon to publish a wide range of research articles on this topic. In order to expand the study of coalitions, we have outlined some unresolved problems and potential research directions that are suggested by our research.

Problem 7.1. There are different kinds of dominating sets which have been explored, such as Roman domination, k-tuple domination, k-tuple total domination, restrained domination, locating domination, and so on. For more details on domination, see [7, 8, 10, 17, 38]. In order to develop future research, we propose new perspectives on coalitions, including Roman coalition, -tuple coalition, -tuple total coalition, restrained coalition, and locating coalition.

Problem 7.2. Study relations between coalition number and several other classical parameters in graph theory like the domination, chromatic, covering, packing, independence, matching numbers.

Problem 7.3. Establishing upper bounds for the coalition number of a graph in terms of diameter and girth.

Problem 7.4. Study Nordhaus-Gaddum-type results for coalition number of graphs.

Problem 7.5. What is the coalition number of graph operations, such as corona, Cartesian product, join, lexicographic, and so on?

Problem 7.6. Establishing upper bounds for the total coalition number of a graph in terms of diameter and girth.

Problem 7.7. Study Nordhaus-Gaddum-type results for total coalition number of graphs.

Problem 7.8. What is the total coalition number of graph operations, such as corona, Cartesian product, join, lexicographic, and so on?

Problem 7.9. What is the effects on Ct(G) when G is modified by operations on vertex and edge of G?

Problem 7.10. Characterize all connected cubic graphs G with C(G) =Ct(G).

Problem 7.11. There are 85 connected cubic graphs of order 12. Compute the total coalition number of connected cubic graphs of order 12.

Problem 7.12. Similar to the coalition graph, it is natural to define and study the connected coalition graph for a given graph G with respect to the connected coalition partition $. We define it as follows. Corresponding to any connected coalition partition $ = {V\, V2,..., Vk} of a given graph G, a connected coalition graph CCG(G, $) is associated with a one-to-one correspondence between the vertices of CCG(G, $) and the sets V\,V2,... ,Vk of $. Two vertices of CCG(G, $) are adjacent if and only if their corresponding sets in $ form a connected coalition.

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List of publications

[P1] S. Alikhani, D. Bakhshesh, H. Golmohammadi, Total coalitions in graphs, Quaest. Math. (2024) https://doi.org/10.2989/16073606.2024.2365365

[P2] S. Alikhani, D. Bakhshesh, H. Golmohammadi, S. Klavzar, On independent coalition in graphs and independent coalition graphs, Discuss. Math. Graph Theory (2024)

https://doi.org/10.7151/dmgt.2543

[P3] S. Alikhani, D. Bakhshesh, H. Golmohammadi, E.V. Konstantinova, Connected coalitions in graphs, Discuss. Math. Graph Theory. 44 (2024) 1551-1566.

[P4] S. Alikhani, H.R. Golmohammadi, E.V. Konstantinova, Coalition of cubic graphs of order at most 10, Commun. Comb. Optim. 9 (2024) 437-450.

[P5] A. A. Dobrynin, H. Golmohammadi, On cubic graphs having the maximum coalition number, Sib. Elektron. Math. Izv. 2 (2024) 356-362.

[P6] A. A. Dobrynin, H. Golmohammadi, The shortest cycle having the maximal number of coalition graphs, Discrete Math. Lett. 14 (2024) 21-26.

[P7] H. Golmohammadi, Total coalitions of cubic graphs of at most order 10, Commun. Comb. Optim (2024)

https://doi.org/10.22049/cco.2024.29015.1813

[P8] H. Golmohammadi, Total coalitions in cubic graphs, XXIII International Conference Mathematical Optimization Theory and Operations Research (MOTOR 2024), June 30 - July 06, 2024, Omsk, Russia, Abstracts, pp. 38.

[P9] H. Golmohammadi, E.V. Konstantinova, S. Alikhani, Some results on the coalition number of cubic graphs of order at most 10, 22nd International Conference Mathematical Optimization Theory and Operations Research (MOTOR 2023), July 2 - 8, 2023, Ekaterinburg, Russia, Abstracts, pp. 34.

[P10] H. Golmohammadi, S. Alikhani, D. Bakhshesh, Introduction to total coalitions in graphs, International (54th National) Youth School-Conference, February 6-10, 2023, Ekaterinburg, Russia, Abstracts, pp. 39-40.