Ključne reči: graf; petougaoni kaktus-lanac; dominantni skup; dominacijski broj

Keywords: graph; pentagonal cactus-chain; dominant set; dominance number

Sažetak

Graf kao matematički objekat zauzima posebno mesto u nauci. Teorija grafova nalazi sve veću primenu u mnogobrojnim sferama poslovanja, kao i naučnim oblastima. U ovom radu analizirani su petougaoni kaktus-lanci koji predstavljaju posebnu vrstu grafa sastavljenog od petougaonih ciklusa u kojima dva susedna ciklusa imaju zajednički samo jedan čvor. Cilj istraživanja jeste određivanje dominantnog skupa i dominacijskog broja na orto i meta petougaonim kaktus-lancima. Metode: Kada se odgovarajuća odredišta tretiraju kao čvorovi grafa, a veze među njima kao grane u grafu, dobija se potpuna struktura grafa na koju se primenjuju zakonitosti teorije grafova. Temena petougla su tretirana kao čvorovi grafa, a stranice kao grane u grafu. Primenom matematičkih metoda određena je dominacija na jednom petouglu, zatim na dva petougla sa zajedničkim čvorom, a nakon toga na orto i meta petougaonim kaktus-lancima. Rezultati: Istraživanja su pokazala da je dominacijski broj na orto lancu Oℎ dužine h ≥2 jednak vrednosti izraza ⌈ 3ℎ 2 ⌉, dok je na meta lancu Mℎ jednak vrednosti izraza h + 1, što je dokazano u radu. Zaključak: Rezultati pokazuju da su dominantni skupovi i dominacijski brojevi na orto i meta petougaonim kaktus-lancima određeni i eksplicitno iskazani matematičkim izrazima. Takođe, upućuju na mogućnost njihove primene u oblastima nauke, kao i u sferama poslovanja u kojima se pojavljuju ove strukture.

Abstract

Introduction/purpose: A graph as a mathematical object occupies a special place in science. Graph theory is increasingly used in many spheres of business and scientific fields. This paper analyzes pentagonal cactus chains, a special type of graphs composed of pentagonal cycles in which two adjacent cycles have only one node in common. The aim of the research is to determine the dominant set and the dominance number on ortho and meta pentagonal cactus chains. Methods: When the corresponding destinations are treated as graph nodes and the connections between them as branches in the graph, the complete structure of the graph is obtained, to which the laws of graph theory are applied. The vertices of the pentagon are treated as nodes of the graph and the sides as branches in the graph. By applying mathematical methods, the dominance was determined on one pentagon, then on two pentagons with a common node, and then on ortho and meta pentagonal cactus chains. Results: The research has shown that the dominance number on the ortho chain 𝑂ℎ of the length h ≥ 2 is equal to the value of the expression ⌈ 3ℎ 2 ⌉ while on the meta chain 𝑀ℎ it is equal to the value of the expression h+1, which was proven in this paper. Conclusion: The results show that the dominant sets and the dominance numbers on ortho and meta pentagonal cactus chains are determined and explicitly expressed by mathematical expressions. They also point to the possibility of their application in the fields of science as well as in the spheres of business in which these structures appear.

Introduction

Mathematical apparatus and mathematical methods are used in almost all fields of science, both natural (Ghergu & Radulescu, 2011; Veličković et al, 2020) and social (Vladimirovich & Vasilyevich-Chernyaev, 2021). A graph as a mathematical object occupies a special place in science (Bakhshesh, 2022; Hajian & Rad, 2021; Hernández Mira et al, 2021). It is used in medicine, genetics, chemistry, etc. All structural formulas of covalently bound compounds are graphs. Chemical elements are represented by graphs where atoms are vertices and chemical bonds are lines in the graph (Balaban, 1985). A graphical representation of chemical structures provides a visual insight into molecular bonds and chemical properties of molecules. The QSPR study has shown that many of chemical properties of molecules are closely related to theoretical graphical invariants called molecular descriptors (Mihalić & Trinajstić, 1992). The theoretical graphical invariant is also the dominance number, which is the simplest variant of the k-dominance number that is used many times in mathematics (Zmazek & Žerovnik, 2003).

A graph is usually denoted by G, a set of its vertices (nodes) by V(G) and a set of its branches (lines) by E(G).

A set D that is a subset of the set V(G) is called a k-dominant set in the graph G if for each vertex outside the set D there is at least one vertex in the set D such that the distance between them is less than or equal to k. The number of elements of the smallest k-dominant set is called the kdominance number and is denoted by y_{k}. If k = 1, the 1-dominance number is called the dominance number and is denoted by γ and the 1-dominant set is called the dominant set.

A cactus graph is a connected graph in which no line (branch) is in more than one cycle. The study of cactus graphs began in the middle of the 20th century. In his work (Husimi, 1950) Husimi uses these graphs in studies of cluster integrals. Riddell (Riddell, 1951) uses them in the theory of condensation. They were later used in the theory of electrical and communication networks (Zmazek & Zerovnik, 2005) as well as in chemistry (Sharma et al, 1997; Gupta et al, 2001; Gupta et al, 2002).

It is known that many chemical compounds have a pentagonal shape in their configuration. Among them are cycloalkanes, which are very common compounds in the nature. The five-membered and six-membered cycloalkanes, cyclopentane (Figure 1) and cyclohexane, which contain 5 and 6 ring carbon atoms, respectively, are very stable and their structures appear in many biological molecules.

Their ring structures are also included in the composition of steroids. A large number of steroids are synthesized in laboratories and used in the treatment of cancer, arthritis, various allergies and other diseases (Balaban & Zeljković, 2021). Pentagonal forms in combination with hexagonal forms are present in many compounds, among which are heterocyclic compounds: morphine, benzofuran, dibenzothiophene and others.

In this paper, we analyze the k-dominance of pentagonal cactus chains. Hexagonal cactus chains were investigated in papers (Farrell, 1987; Vukičević & Klobučar, 2007). Afterwards, the papers (Majstorovic et al, 2012; Klobučar & Klobučar, 2019) determined the dominance number on a uniform hexagonal cactus chain, the dominance number on an arbitrary hexagonal network, and the total and double total dominance number on a hexagonal network. The K-dominance on rhomboidal cactus chains (Carević et al, 2020) as well as on the icosahedral-hexagonal network (Carević, 2021) was also investigated.

Pentagonal cactus-chains

The pentagonal cactus-chain G is a graph consisting of a cycle with 5 vertices. A vertex that is common to two or three pentagons is called a cutvertex. If each pentagon in the graph G has at most 2 cut-vertices and each cut-vertex is divided between exactly 2 pentagons, the graph G is called a pentagonal cactus-chain. ^{}

With G_{ℎ} we will denote a pentagonal cactus-chain of the length h and G_{ℎ }= 𝑃^{1}𝑃^{2} ... 𝑃^{ℎ} where 𝑃^{i} are successive pentagons in the chain (Figure 2).

Figure 2 Pentagonal cactus-chain of the length 7 Рис. 2 – Пятиугольная кактус-цепочка длиной 7 Слика 2 – Петоугаони кактус-ланац дужине 7

Denote by x and y the vertices in the graph G and by d(x, y) the distance between them, where the distance between two vertices is equal to the number of branches located from one vertex to another. Denote by 𝑃_{i} the minimum distance between the pentagons p^{i} and p^{i+2}:

\(p_i = min \{d(x,y): x \in P^{i} \wedge y \in P^{i+2},i = 1, 2, ... , h - 2\}\)

Then 𝑃_{i} is the distance between the pentagons 𝑃^{i} and 𝑃^{i+2}.

With the exception of the first and last pentagons in the cactus chain, which have one cut-vertex, all other pentagons have two cut-vertices, and they are called inner pentagons.

In the pentagonal cactus chain G_{ℎ}, we distinguish between ortho and meta inner pentagons. An inner pentagon is called an ortho pentagon if its cut-vertices are adjacent, and a meta pentagon if the distance between its cut-vertices is d = 2.

A pentagonal cactus chain is uniform if all its inner pentagons are of the same type. A chain G_{ℎ} is called an ortho-chain, and is denoted by O_{ℎ} if all its inner pentagons are ortho-pentagons (Figure 3).

Figure 3 Ortho cactus-chain 𝑂_{5} Рис. 3 – Орто-кактус-цепочка 𝑂_{5} Слика 3 – Орто кактус-ланац О_{5}

To determine the dominant set on the uniform pentagonal cactus chains O_{ℎ} and M_{ℎ}, it will be necessary to point out certain vertices in the cactus chain. That is why it is necessary to mark them. In the ortho pentagon p^{i} the cut-vertices are adjacent and we will denote them by V_{i} and V_{i+1}. The other vertices in 𝑃^{i} it will be denoted by \(x_{1}^{i}, x_{2}^{i}\) and \(x_{3}^{i}\) (Figure 5):

Figure 5 Marking vertices in the ortho pentagon Рис. 5 – Обозначение вершин в ортогональном пятиугольнике Слика 5 – Означавање чворова у орто петоуглу

In the meta pentagon 𝑃^{i} the cut-vertices are at a distance d = 2 and we will denote them by V_{2i −1} and V_{2i +1}. With V_{2i} we will denote the vertex to which it applies d(𝑉_{2𝑖−1}, 𝑉_{2𝑖}) = d(𝑉_{2𝑖}, 𝑉_{2𝑖+1}) = 1.

Figure 6 Marking vertices in the meta pentagon Рис. 6 – Обозначение вершин в мета-пятиугольнике Слика 6 – Означавање чворова у мета петоуглу

The other two nodes in the pentagon 𝑃^{i} will be denoted \(x_{1}^{i}\) and \(x_{2}^{i},\) (Figure 6):

Research results

In this section, we consider 1-dominance on ortho and meta pentagonal cactus chains. We will first consider the dominance of one pentagon and two adjacent pentagons in the ortho and meta chain of cacti.

Lemma 3.1. The dominance number for the pentagon is γ = 2.

Proof: Let us denote the vertices of the pentagon by 𝑥_{1}, 𝑥_{2}, 𝑥_{3}, 𝑥_{4}, 𝑥_{5} (Figure 7):

Figure 7 Dominant set on a pentagon Рис. 7 – Доминирующее множество на пятиугольнике Слика 7 – Доминантни скуп на петоуглу

One pentagon vertex dominates two adjacent vertices. Let us take the vertex 𝑥_{1}. It dominates the vertices 𝑥_{2} and 𝑥_{5}. As the pentagon has 5 vertices, domination over the other two vertices 𝑥_{3} and 𝑥_{4} is necessary. We conclude that one of the remaining two vertices must belong to the dominant set on the pentagon. Let it be the vertex 𝑥_{3}. Thus, the set D = {𝑥_{1}, 𝑥_{3}} is the dominant set for a given pentagon but it is not the only dominant set whose cardinality is equal to 2. They are also sets that contain any two non-adjacent pentagon vertices. Let us prove that any of the mentioned two-membered sets is the minimum dominant set on the pentagon. Assuming that there is a dominant set of less cardinality D', it would have to contain only one vertex and one vertex cannot dominate the remaining 4 vertices of the pentagon. Thus, the minimum dominant set on a pentagon is a two-membered set, so the dominance number for the pentagon is 𝛾 = 2.

Lemma 3.2. The dominance number for two pentagons with one cut-vertex is 𝛾 = 3.

Proof: Let us denote the vertices of two pentagons by one common vertex with 𝑥_{1}, 𝑥_{2}, . . . , 𝑥_{9} (Figure 8):

Figure 8 Dominant set for two pentagons with a cut-vertex Рис. 8 – Доминирующее множество для двух пятиугольников с пересекающейся вершиной Слика 8 – Доминантни скуп за два петоугла са пресеченим чвором

Let 𝑥_{1} be the cut-vertex of the given pentagons 𝑃^{1} and 𝑃^{2}. Based on Lemma 3.1. the pentagon 𝑃^{1} excluding the vertex 𝑥_{1} must have another dominant vertex that is not adjacent to the vertex 𝑥_{1}. Let it be the vertex 𝑥_{3}. Also by applying Lemma 3.1. the pentagon 𝑃^{2} excluding the vertex 𝑥_{1} must have another dominant vertex that is not adjacent to the vertex 𝑥_{1}. Let it be the vertex 𝑥_{7}. Thus the nodes 𝑥_{1}, 𝑥_{3} and 𝑥_{7} dominate over the nodes 𝑥_{2}, 𝑥_{4}, 𝑥_{5}, 𝑥_{6}, 𝑥_{8} and 𝑥_{9} so the dominant set for the pentagons 𝑃^{1}𝑃^{2 }is the set D = {𝑥_{1},𝑥_{3}, 𝑥_{7}}. Analogous to the consideration in Lemma 3.1. the set D is not the only three-membered set that is dominant on 𝑃^{1}𝑃^{2}but there is no dominant set of less cardinality. Suppose that there is a dominant set D' whose cardinality is equal to 2. Let D' contain one vertex from each pentagon, for example 𝐷′ = {𝑥_{1},𝑥_{3}}.The vertices 𝑥_{1} and 𝑥_{3} would then dominate over the remaining 7 vertices in 𝑃^{1}𝑃^{2} and this is impossible. The vertex 𝑥_{1} as a common vertex for both pentagons dominates over two neighboring vertices in both pentagons, so it dominates over 4 vertices in 𝑃^{1}𝑃^{2}.The vertex 𝑥_{3}, or any other vertex not adjacent to the vertex 𝑥_{1 }dominates two adjacent vertices. So, the total sum of vertices covered by dominance is 4 + 2 = 6 and that is less than 7. Thus, 2 vertices cannot dominate the remaining 7 vertices in 𝑃^{1}𝑃^{2}. We conclude that the minimum dominant set for 𝑃^{1}𝑃^{2 }is a three-membered set and 𝛾 = 3.

Let us consider the dominance on pentagonal ortho and meta cactus chains of arbitrary length.

Theorem 3.1.\(y(O_h) = ⌈\frac{3h}{2}⌉\) for each \(h≥ 2 ˄ h \in N\).

Proof: We observe a pentagonal ortho cactus-chain 𝑂_{ℎ} = 𝑃^{1}𝑃^{2 }... 𝑃^{ℎ} (Figure 9) and a set:

Figure 9 Minimum dominant set for 𝑂_{ℎ} Рис. 9 – Минимально доминирующее множество для 𝑂_{ℎ} Слика 9 – Минимални доминантни скуп за О_{ℎ}

Let us prove that \(D_{O_{h}}\) is the dominant set of minimum cardinality for a pentagonal ortho cactus-chain 𝑂_{ℎ }= 𝑃^{1}𝑃^{2}… 𝑃^{ℎ}.

Let us divide the ortho-chain 𝑂_{ℎ} into subchains 𝑃^{2𝑖−1}𝑃^{2𝑖}, i = 1, 2, ... , ⌊\(\frac{h}{2}\)⌋ (Figure 10) and the last pentagon 𝑃^{ℎ} if h is an odd number.

Figure 10 Subchain of the ortho-chain 𝑂_{ℎ} Рис. 10 – Подцепочка орто-цепочки 𝑂_{ℎ} Слика 10 – Подланац орто ланца 𝑂_{ℎ}

Based on Lemma 3.2. the set 𝐴_{𝑖 }= {\(x_{2}^{2i-1}, x_{2}^{2i}, V_{2i}\)} for i = 1, 2, ... , ⌊\(\frac{h}{2}\)⌋ is the dominant set of minimum cardinality for the subchain 𝑃^{2𝑖−1}𝑃^{2𝑖}. An ortho-chain of the length h for h = 2k, k ϵ N is composed of \(\frac{h}{2}\) subchains 𝑃^{2𝑖−1}𝑃^{2𝑖}, i = 1, 2, ... ,\(\frac{h}{2}\) (Figure 9A), so the set

\(D_1 = \bigcup_{i= 1}^{k}A_i, \ for \ k= \frac{h}{2}\)

is a dominant set for the ortho-chain O_{ℎ}. Therefore, it is

In case h is an even number, \(\frac{h}{2}= ⌈\frac{h}{2}⌉\) then we conclude that it is 𝐷_{1}= 𝐷_{2}.

So, the set \(D_{O_{h}}\)= { \(𝑥_2^𝑖\), i = 1, h} ∪ { 𝑉_{2𝑖}, i = 1, \(⌈\frac{h}{2}⌉\)} is the dominant set for the ortho-chain 𝑂_{ℎ} when h is even or odd number.

Also, in the case where h is an even number, \(\frac{3h}{2}= ⌈\frac{3h}{2}⌉ \). So, 𝛾(O_{h}) ≤\(⌈\frac{3h}{2}⌉\) when h is even or odd number. Prove that the set \(D_{O_{h}}\) is the dominant set of minimal cardinality. Each subchain 𝑃^{2𝑖−1}𝑃^{2𝑖} contains 3 dominant nodes based on Lemma 3.2. Based on this, we conclude that each dominant set on the chain 𝑂_{ℎ} contains more than 3 or exactly 3 dominant nodes in each subchain 𝑃^{2𝑖−1}𝑃^{2𝑖} and more than 2 or exactly 2 dominant nodes in the last pentagon if h is an odd number, based on Lemma 3.1. So, we conclude that it is 𝛾(O_{h}) ≥\(\frac{h}{2}\)∙ 3 in case h is an even number, and 𝛾(O_{h}) ≥\(\frac{h}{2}\)∙ 3 + 2 in case h is an odd number. When we combine both cases, we get that 𝛾(O_{h}) ≥ \(⌈\frac{3h}{2}⌉\).

It follows from 𝛾(𝑂_{ℎ}) ≤ \(⌈\frac{3h}{2}⌉\) and 𝛾(𝑂_{ℎ}) ≥ \(⌈\frac{3h}{2}⌉\) that it is 𝛾(𝑂_{ℎ}) = \(⌈\frac{3h}{2}⌉\).

Corollary 3.1.\(D_{O_{h}}\subset D_{O_{h+1}}\) for each \(h ≥ 2 ˄ h \in N\).

Theorem 3.2.\(y(M_h) = h + 1\) for each \(h ≥ 2 ˄ h \in N\).

Proof: We observe a pentagonal meta cactus-chain 𝑀_{ℎ} = 𝑃^{1}𝑃^{2}... 𝑃^{ℎ }(Figure 11) and set:

Figure 11 Minimum dominant set for 𝑀_{ℎ} Рис. 11 – Минимально доминирующее множество для 𝑀_{ℎ} Слика 11 – Минимални доминантни скуп за М_{ℎ}

Let us prove that \(D_{M_h}\) is the dominant set of minimum cardinality for a pentagonal meta cactus-chain 𝑀_{ℎ }= 𝑃^{1}𝑃^{2}… 𝑃^{ℎ}. Based on Lemma 3.1. each pentagon has a dominant set made up of two non-adjacent vertices. Thus, the set { 𝑉_{2𝑖−1}, 𝑉_{2𝑖+1}} is dominant for the pentagon 𝑃^{𝑖} for each i = 1, h. By merging the dominant sets of all pentagons in the chain, we get a set that is dominant for the whole chain. But, each pentagon 𝑃^{𝑖} has a common vertex with the pentagon 𝑃^{𝑖+1} for each i = 1, h ‒ 1. Common vertices should not be repeated in the dominant set. So, the set

Thus, the set \(D_{M_h}= \{ V_{2i-1}, i = 1, h + 1\}\) is the dominant set for the meta-chain 𝑀_{ℎ} for each \(h \in N\) and \(h ≥2\). Let us prove that \(D_{M_h}\) is the dominant set of minimal cardinality. Suppose that there is a set S of less cardinality that is dominant on the meta-chain 𝑀_{ℎ}. The set S would then have one node less than the set \(D_{M_h}\). Let it be a vertex 𝑉_{2𝑖+1} for any i = 1, h. Then the pentagon 𝑃^{𝑖} would have only one dominant node 𝑉_{2𝑖−1}. Based on Lemma 3.1. that is not possible. We conclude that \(D_{M_h}\) is the minimum dominant set for 𝑀_{ℎ} so it is 𝛾(𝑀_{ℎ}) = h + 1.

Corollary 3.2.\(D_{M_h} ⊂ D_{M_h+1}\) for each \(h ≥ 2 ˄ h \in N\).

Conclusion

In this paper, we have shown the arrangement of vertices in dominant sets on uniform ortho and meta pentagonal cactus chains that appear in molecule structures of numerous compounds. We also proved that the dominance number for a pentagonal ortho-chain of the length h is equal to the value of the expression \(⌈\frac{3h}{2}⌉\) while for a pentagonal meta-chain it is equal to h + 1.

References

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O članku

jezik rada: engleski vrsta rada: izvorni naučni članak DOI: 10.5937/vojtehg70-36576 primljen: 22.02.2022. revidiran: 22.06.2022. prihvaćen: 24.06.2022. objavljen u SCIndeksu: 28.06.2022.
metod recenzije: dvostruko anoniman