CC BY
14
Научный отдел
МАТЕМАТИКА
Известия Саратовского университета. Новая серия. Серия: Математика. Механика. Информатика. 2022. Т. 22, вып. 3. С. 278-286 Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 3, pp. 278-286 mmi.sgu.ru
https://doi.org/10.18500/1816-9791-2022-22-3-278-286 EDN: IMTPKR
Article
Forcing total outer connected monophonic number of a graph
K. Ganesamoorthy10, S. Lakshmi Priya2
1 Coimbatore Institute of Technology, Department of Mathematics, Coimbatore — 641 014, India
2CIT Sandwich Polytechnic College, Department of Mathematics, Coimbatore — 641 014, India
Kathiresan Ganesamoorthy, kvgm_2005@yahoo.co.in, https://orcid org/0000-0003-2769-1991
Shanmugam Lakshmi Priya, lakshmiuspriya@gmail.com, https:// orcid.org/0000-0001- 7367-1532
Abstract. For a connected graph G = (V., E) of order at least two, a subset T of a minimum total outer connected monophonic set S of G is a forcing total outer connected monophonic subset for S if S is the unique minimum total outer connected monophonic set containing T. A forcing total outer connected monophonic subset for S of minimum cardinality is a minimum forcing total outer connected monophonic subset of S. The forcing total outer connected monophonic number ftom (S) in G is the cardinality of a minimum forcing total outer connected monophonic subset of S. The forcing total outer connected monophonic number of G is ftom(G) = mrn{ftom(S)}, where the minimum is taken over all minimum total outer connected monophonic sets S in G. We determine bounds for it and find the forcing total outer connected monophonic number of a certain class of graphs. It is shown that for every pair a, b of positive integers with 0 < a <b and b > a + 4, there exists a connected graph G such that ftom (G) = a and cmto(G) = b, where cmto(G) is the total outer connected monophonic number of a graph. Keywords: total outer connected monophonic set, total outer connected monophonic number, forcing total outer connected monophonic subset, forcing total outer connected monophonic number
Acknowledgements: The first author's research work was supported by National Board for Higher Mathematics (NBHM), Department of Atomic Energy (DAE), Government of India (project No. NBHM/R.P.29/2015/Fresh/157).
For citation: Ganesamoorthy K., Lakshmi Priya S. Forcing total outer connected monophonic number of a graph. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 3, pp. 278-286. https://doi.org/10.18500/1816-9791-2022-22-3-278-286, EDN: IMTPKR
This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0)
Научная статья УДК 519.17
Форсирование общего внешне связного монофонического
числа графа
К. Ганееамурти10, Ш. Лакшми Прия2
1 Технологический институт Коимбатура, Индия, Коимбатур — 641 014
2Политехнический колледж Технологического института Коимбатура, Индия, Коимбатур — 641 014 Катиресан Ганееамурти, Dr., доцент математического факультета, kvgm_2005@yahoo.co.in, https: //orcid.org/0000-0003-2769-1991
Шанмугам Лакшми Прия, студентка математического факультета, lakshmiuspriya@gmail.com, https: //orcid.org/0000-0001-7367-1532
Аннотация. Для связного графа G = (V., Е) с числом вершин не менее 2 подмножество Т минимального общего внешне связного монофонического множества S графа G является сильным общим внешне связным монофоническим подмножеством для S, если S есть единственное минимальное общее внешне связное монофоническое множество, содержащее Т. Сильное общее внешне связное монофоническое подмножество для S с минимальным числом элементов есть минимальное сильное общее внешне связное монофоническое подмножество S. Сильное общее внешне связное монофоническое число ftom(S) в G есть число элементов минимального сильного общего внешне связного монофонического подмножества S. Сильное общее внешне связное монофоническое число графа G есть ftom(G) = min{ft0m(S)}, где минимум принимается над всеми минимальными общими внешне связными монофоническими множествами S в G. Мы определяем его границы и находим сильное общее внешне связное монофоническое число некоторых классов графов. Показывается, что для каждой пары а, Ь положительных целых с 0 ^ а < b и b ^ а + 4 существует связный граф G такой, что ftom(G) = а и crnto(G) = b, где cmto(G) является общим внешне связным монофоническим числом графа.
Ключевые слова: общее внешне связное монофоническое множество, общее внешне связное монофоническое число, сильное общее внешне связное монофоническое подмножество, сильное общее внешне связное монофоническое число
Благодарности: Исследовательская работа первого автора была поддержана Национальным советом по высшей математике (NBHM), Департаментом атомной энергии (DAE), Правительством Индии (проект № NBHM/R.P.29/2015/Fresh/157).
Для цитирования: Ganesamoorthy K, Lakshmi Priya S. Forcing total outer connected monophonic number of a graph [Ганесамурти К., Лакшми Прия Ш. Форсирование общего внешне связного монофонического числа графа] // Известия Саратовского университета.
Новая серия. Серия: Математика. Механика. Информатика. 2022. Т. 22, вып. 3. С. 278-286. https://doi.org/10.18500/1816-9791-2022-22-3-278-286, EDN: IMTPKR
Статья опубликована на условиях лицензии Creative Commons Attribution 4.0 International (CC-BY 4.0)
Introduction
By a graph G = (V,E) we mean a finite simple undirected connected graph. The order and size of G are denoted by p and q, respectively. For basic graph theoretic terminology we refer to Harary [1,2]. The distance d(x,y) between two vertices x and y in a connected graph G is the length of a shortest x — y path in G. An x — y path of length d(x, y) is called an x — y geodesic. A vertex v of a connected graph G is called an endvertex of G if its degree is 1. A vertex v of a connected graph G is called a support vertex of G if it is adjacent to an endvertex of G. The neighborhood of a vertex v is the set N(v) consisting of all vertices u which are adjacent with v. A vertex v is an extreme vertex if the subgraph induced by its neighbors is complete. A chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A set S of vertices of G is a monophonic set of G if each vertex v of G lies on a x — y monophonic path for some x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G and is denoted by rn(G). The monophonic number of a graph, an algorithmic aspect of monophonic concepts was introduced and studied in [3-7]. A total monophonic set of a graph G is a monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total monophonic set of G is the total monophonic number of G and is denoted by mt(G). The total monophonic number of a graph and its related concepts were studied in [8-10]. A set S of vertices in a graph G is said to be an outer connected monophonic set if S is a monophonic set of G and either S = V or the subgraph induced by V — S is connected. The minimum cardinality of an outer connected monophonic set of G is the outer connected monophonic number of G and is denoted by moc(G). The outer connected monophonic number of a graph was introduced in [11]. Very recently, outer connected monophonic concepts have been widely investigated in graph theory, such as a connected outer connected monophonic number [12], extreme outer connected monophonic graphs [13], and so on. A total outer connected monophonic set S of G is an outer connected monophonic set such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total outer connected monophonic set of G is the total outer connected monophonic number of G and is denoted by crnto(G).
The authors of this article introduced and studied the general externally total outer connected monophonic number of a graph and proved the following theorems1, which will be used further.
Theorem 1. Each extreme vertex and each support vertex of a connected graph G belong to every total outer connected monophonic set of G.
Theorem 2. For the complete graph Kp(p ^ 2), crnto(Kp) = p.
Theorem 3. For any non-trivial tree T, the set of all endvertices and support vertices of T is the unique minimum total outer connected monophonic set of G.
1Ganesamoorthy K., Lakshmi Priya S. The total outer connected monophonic number of a graph. Transactions of A. Razmadze Mathematical Institute, accepted.
Theorem 4. For any connected graph G, cmto(G) = 2 if and only if G = K2.
Throughout this paper, G denotes a connected graph with at least two vertices.
1. Main Results
Definition 1. Let S be a minimum total outer connected monophonic set of G. A subset T of S is a forcing total outer connected monophonic subset for S if S is the unique minimum total outer connected monophonic set containing T. A forcing total outer connected monophonic subset for S of minimum cardinality is a minimum forcing total outer connected monophonic subset of S. The forcing total outer connected monophonic number ftom(S) in G is the cardinality of a minimum forcing total outer connected monophonic subset of S. The forcing total outer connected monophonic number of G is ftom(G) = min{ftom(S)}, where the minimum is taken over all minimum total outer connected monophonic sets S in G.
Example 1. For the graph G in Fig. 1, it is clear that Sl = {vl,v2,v4,v5}, S2 = {vl,v4,v5,v8}, S3 = {vl,v2,v5,v6} and S4 = {vl,v5,v6,v8} are the minimum total outer connected monophonic sets of G. It is clear that no minimum total outer connected monophonic set Si(i = 1,2, 3,4) is the unique minimum total outer connected monophonic set containing any of its 1-element subsets. It is easy to see that {v2,v4} is a forcing total outer connected monophonic subset contained in Sl and ftom(Sl) = 2. Hence, we have ft0m(G) = 2. By Theorem 3, for any non-trivial tree T, the set of all endvertices and support vertices of T is the unique minimum total outer connected monophonic set of T and so ftom(T) = 0.
Theorem 5. For any connected graph G of order p, 0 ^ ftom(G) ^ cmto(G) ^ p.
Proof. By the definition of the forcing total outer connected monophonic number of a graph, it is clear that ftom(G) ^ 0. Let S be a minimum total outer connected monophonic set of G. Clearly, ftom(S) ^ |S| = cmto(G) and ftom (G) = mrn{ftom (5)}, where the minimum is taken over all minimum total outer connected monophonic sets S in G. Hence 0 < ftom(G) < cmto(G) < p. □
V2 V3 V4
V8 V7 V6
Fig. 1. Graph G with ftom(G) = 2
Remark 1. The bounds in Theorem 5 are sharp. By Theorem 3, for any non-trivial tree T,
the set of all endvertices and support vertices of T is the unique minimum total outer connected monophonic set of T and so ftom(T) = 0. By Theorem 2, for the complete graph Kp(p ^ 2), cmto(Kp) = p. Also all the inequalities in Theorem 5 can be strict. For the graph G given in Fig. 1 of order 8, it is clear that no 2-element subset or 3-element subset of V(G) is a total outer connected monophonic set of G. The minimum total outer connected monophonic sets of G are Sl = {vl,v2,v4,v5}, S2 = {vl,v4,v5,v8}, = {vl,v2,v5,v6} and S4 = {vl,v5,v6,v8} so that cmto(G) = 4. It is clear that ftom(Si) =2(i = 1,2, 3, 4) and so ftom(G) = 2. Thus 0 < ftom(G) < cmto(G) < p.
The following theorem characterizes graphs G for which the lower bound in Theorem 5 is attained and also characterizes graphs G for which ftom(G) = 1 and ftom(G) = cmto(G).
Theorem 6. Let G be a connected graph. Then
(i) ftom(G) = 0 if and only if G has the unique minimum total outer connected monophonic set;
(ii) ftom(G) = 1 if and only if G has at least two minimum total outer connected monophonic sets, one of which is the unique minimum total outer connected monophonic set containing one of its elements;
(iii) ftom(G) = cmto(G) if and only if no minimum total outer connected monophonic set of G is the unique minimum total outer connected monophonic set containing any of its proper subsets.
Proof. (i) Let ftom(G) = 0. Then, by the definition, ftom(S) = 0 for some minimum total outer connected monophonic set S of G so that the empty set p is the minimum forcing subset for S. Since the empty set p is a subset of every set, it follows that S is the unique minimum total outer connected monophonic set of G. The converse is clear.
(ii) Let ftom(G) = 1. Then by (i), G has at least two minimum total outer connected monophonic sets. Since ftom(G) = 1, there is a 1-element subset T of a minimum total outer connected monophonic set S of G such that T is not a subset of any other minimum total outer connected monophonic set of G. Thus S is the unique minimum total outer connected monophonic set containing one of its elements. The converse is clear.
(iii) Let ftom(G) = cmto(G). Then ftom(S) = cmto(G) for every minimum total outer connected monophonic set S in G. Since any total outer connected monophonic set of G needs at least two vertices, cmto(G) ^ 2 and hence ftom(G) ^ 2. Then by (i), G has at least two minimum total outer connected monophonic sets, and so the empty set is not a forcing subset for any minimum total outer connected monophonic set of G. Since ftom(G) = cmto(G), no proper subset of S is a forcing subset of S. Thus no minimum total outer connected monophonic set of G is the unique minimum total outer connected monophonic set containing any of its proper subsets.
Conversely, the data implies that G contains more than one minimum total outer connected monophonic set, and no subset of any minimum total outer connected monophonic set S other than S, is a forcing subset for S. Hence it follows that ftom (G) = cmto(G). □
Definition 2. A vertex v of G is said to be a total outer connected monophonic vertex if v belongs to every minimum total outer connected monophonic set of G.
Remark 2. If G has the unique minimum total outer connected monophonic set S, then every vertex in S is a total outer connected monophonic vertex of G. Also, if x is an extreme vertex or a support vertex of G, then x is a total outer connected monophonic vertex of G. For the graph G given in Fig. 1, vl and v5 are the total outer connected monophonic vertices of G.
The next theorem and corollary are an immediate consequence of the definitions of total outer connected monophonic vertex and a forcing total outer connected monophonic subset of G.
Theorem 7. Let G be a connected graph and let ^tom be the set of relative complements of the minimum forcing total outer connected monophonic subsets in their respective minimum total outer connected monophonic sets in G. Then nFF is the set of all total outer connected monophonic vertices of G.
Corollary 1. Let S be a minimum total outer connected monophonic set of G. Then no total outer connected monophonic vertex of G belongs to any minimum forcing total outer connected monophonic subset of S.
Theorem 8. Let M be the set of all total outer connected monophonic vertices of G. Then ftom(G) < cmto(G) - \M\.
Proof. Let S be any minimum total outer connected monophonic set of G. Then cmto(G) = |5\, M C S, and S is the unique minimum total outer connected monophonic set containing S - M. Hence ftom(G) < \S - M\ = \S\ - \M\ = cmto(G) - \M\. □
Corollary 2. If G is a connected graph with I extreme vertices and k support vertices, then ftom(G) ^ cmto(G) - (I + k).
Remark. 3. The bound in Theorem 8 is sharp. For the graph G given in Fig. 1, the minimum total outer connected monophonic sets of G are Si = {vi,v2,v4,v5}, S2 = {vi,v4,v5,v8}, S3 = {vi,v2,v5,v6} and S4 = {vi,v5,v6,v8} so that cmto(G) = 4. It is clear that ftom(Si) = 2(i = 1, 2,3,4) and so ftom(G) = 2. Also, M = {vi,v5} is the set of all total outer connected monophonic vertices of G and so ftom(G) = cmto(G) - \M\. The inequality in Theorem 8 can be strict. For the graph G given in Fig. 2, the minimum total outer connected monophonic sets of G are Mi = {vi,v2,v3,v6}, M2 = {v3,v4,v5,v6}, M3 = {v2,v3,v4,v6} and so cmto(G) = 4. It is clear that ftom(Mi) = 1 (i = 1, 2), and so ftom(G) = 1. Also, the vertices v3 and v6 are the Fig. 2. A graph G with ftom(G) < total outer connected monophonic vertices of G, < cmto(G) - \M\
we have ftom(G) < cmto(G) - \M\.
Theorem 9. If G is a connected graph with cmto(G) = 2, then ftom(G) = 0.
Proof. If cmto(G) = 2 then by Theorem 4, we have G = K2. Hence V(G) is the unique minimum total outer connected monophonic set of G. Also, by Theorem 6(z), ftom (G) =0. □
Remark 4. The converse of Theorem 9 need not be true. For the path P4 of order 4, the vertex set V(P4) is the unique minimum total outer connected monophonic set of G and so cmto(PA) = 4. By Theorem 6 (i), ftom(PA) = 0.
Theorem 10. For the complete bipartite graph G = Km^n(2 ^ m ^ n),
m + n - 1 if 2 = m ^ n,
{
ftom (G) =
4 if 3 ^ m ^ n.
Proof. Let U = {ul, u2um} and W = {wl ,w2,wn} be the partite sets of G, where m ^ n. We prove this theorem by considering two cases.
Case 1. If m = 2, then it is clear that any minimum total outer connected monophonic sets of G is of the form V(G) - {wt}(1 ^ i ^ n) or V(G) - {u3}(1 ^ j ^ m). It is easy to verify that, no minimum total outer connected monophonic set of G is the unique
minimum total outer connected monophonic set containing any of its proper subsets. Then by Theorem 6 (iii), we have ftom(G) = m + n — 1.
Case 2. If 3 ^ m ^ n, then any minimum total outer connected monophonic set of G is obtained by choosing any two elements from U as well as W, and G has at least two minimum total outer connected monophonic sets. Hence cmto(G) = 4. Clearly, no minimum total outer connected monophonic set of G is the unique minimum total outer connected monophonic set containing any of its proper subsets. Then by Theorem 6 (iii), we have ftom(G) = cmto(G) =4. □
( 0 if n = 3,
Theorem 11. For any cycle Cn(n ^ 3), ftom(Cn) = <3 if n = 4,
[2 if n ^ 5.
Proof. Let Cn : v1,v2,...,vn,v1 be a cycle of order n. We prove this theorem by considering two cases.
Case 1: n = 3. Since C3 is the complete graph of order 3, V(G3) is the unique minimum total outer connected monophonic set of C3. By Theorem 6 (i), ftom(C3) = 0.
Case 2: n ^ 4. It is clear that no 2-element subset of V(Cn) is a total outer connected monophonic set of Cn. It is easy to verify that any minimum total outer connected monophonic set of Cn consists of three consecutive vertices of Cn so that cmto(Cn) = 3. For n = 4, it is clear that no minimum total outer connected monophonic set of C4 is the unique minimum total outer connected monophonic set containing any of its proper subsets. Thus by Theorem 6 (iii), we have ftom(C4) = 3. For n ^ 5, it is clear that the set of two non-adjacent vertices of any minimum total outer connected monophonic set S of Cn is a minimum forcing total outer connected monophonic subset of S and so ftom(S) = 2. Hence ftom(Cn) =2. □
Theorem 12. For the wheel Wn = Kx + Cn-1 (n ^ 5), ftom(Wn)
{
3 if n = 5, 2 if n ^ 6.
Proof. It is clear that no 2-element subset of V(Wn) is a total outer connected monophonic set of Wn. It is easy to observe that any minimum total outer connected monophonic set of Wn consists of three consecutive vertices of Cn-l so that cmto(Wn) = 3. For n = 5, it is clear that no minimum total outer connected monophonic set of is the unique minimum total outer connected monophonic set containing any of its proper subsets. Thus by Theorem 6 (iii), we have ftom(W5) = 3. For n ^ 6, it is clear that the set of two non-adjacent vertices of any minimum total outer connected monophonic set S of Wn is a minimum forcing total outer connected monophonic subset of S and so ftom(S) = 2. Hence ftom(Wn) =2. □
Theorem 13. For any complete graph G = Kp(p ^ 2) or any non-trivial tree G = T, ftom (G) = 0.
Proof. Let G = Kp. By Theorem 2, the set of all vertices of G is the unique minimum total outer connected monophonic set of G and so by Theorem 6 (i), ftom(G) = 0. If G is a non-trivial tree, then by Theorem 3, the set of all endvertices and support vertices of G is the unique minimum total outer connected monophonic set of G and by Theorem 6
(i), ftom(G) =0. □
Theorem 14. For every pair a, b of integers such that 0 ^ a < b and b ^ a + 4, there is a connected graph G with ftom(G) = a and cmto(G) = b.
Proof. If a = 0, let G = Kb. Then by Theorem 13, ftom(G) = 0, and by Theorem 2, cmto(G) = b. Now, assume that 0 < a < b. The required graph G is obtained from the star Kl}4 having the vertex set {zl,z2,z3,z4,z5} with as the cut-vertex by adding a + b — 2 new vertices wl ,w2,..., wa, vl ,v2va, ul ,u2, Ub-a-3, x and joining each Wi(1 ^ i ^ a) to the vertices , zl and ; and joining each Vi (1 ^ i ^ a) to the vertices , and z5; and joining each ui (1 ^ i ^ b — a — 3) to the vertex ; and also joining the vertex x to the vertex zl, the vertex zl to the vertex , and the vertex to the vertex . The graph G is shown in Fig. 3. Let S = {ul,u2,..., Ub-a-3, x, zl,z5} be the set of all endvertices and support vertices of G. By Theorem 1, every total outer connected monophonic set of G contains S. It is clear that S is not a total outer connected monophonic set of G. We observe that every minimum total outer connected monophonic set of G contains exactly one vertex from the set {v^,Wj} for every (1 ^ i ^ a). Thus cmto (G) ^ b. Since Sl = S U {wl, w2wa} is a total outer connected monophonic set of G, it follows that cmto(G) = b.
Next, we show that ftom(G) = a. Since every minimum total outer connected monophonic set of G contains S, it follows from Theorem 8 that ftom (G) < cmto (G) — \S | = = b — (b — a) = a. It is clear that every minimum total outer connected monophonic set S' of G is of the form S U{xl ,X2,...,xa}, where xl £ {vl ,wl} for every i (1 ^ i ^ a). Let T be any proper subset of S' with \T\ < a. Then there is a vertex x £ S' — S such that x £ T .If x = Vi (1 < i < a), then S" = (S' — {vt})U{wt} is a minimum total outer connected monophonic set of G containing T. Similarly, if x = Wj(1 ^ j ^ a), then S"' = (S' — {wj}) U {vj} is a minimum total outer connected monophonic set of G containing T. Thus S' is not the unique minimum total outer connected monophonic set containing T and so T is not a forcing total outer connected monophonic subset of S'. This is true for all minimum total outer connected monophonic sets of G and so ft0m(G) = a. □
b-a-3
Fig. 3. A graph G with ftom(G) = a > 0 and cmto(G) = b > a
References
1. Buckley F., Harary F. Distance in Graphs. Redwood City, CA, Addison-Wesley, 1990. 335 p.
2. Harary F. Graph Theory. Addision-Wesley, 1969. 274 p.
3. Costa E. R., Dourado M. C., Sampaio R. M. Inapproximability results related to monophonic convexity. Discrete Applied Mathematics, 2015, vol. 197, pp. 70-74. https://doi.org/10. 1016/j.dam.2014.09.012
4. Dourado M. C., Protti F., Szwarcfiter J. L. Algorithmic aspects of monophonic convexity. Electronic Notes in Discrete Mathematics, 2008, vol. 30, pp. 177-182. https://doi.org/10. 1016/j.endm.2008.01.031
5. Dourado M. C., Protti F., Szwarcfiter J. L. Complexity results related to monophonic convexity. Discrete Applied Mathematics, 2010, vol. 158, pp. 1268-1274. https://doi.org/ 10.1016/j.dam.2009.11.016
6. Paluga E. M., Canoy S. R. Monophonie numbers of the join and composition of connected graphs. Discrete Mathematics, 2007, vol. 307, iss. 9-10, pp. 1146-1154. https://doi.org/ 10.1016/j.disc.2006.08.002
7. Santhakumaran A. P., Titus P., Ganesamoorthy K. On the monophonic number of a graph. Journal of Applied Mathematics & Informatics, 2014, vol. 32, iss. 1-2, pp. 255-266. https://doi.org/10.14317/JAMI.2014.255
8. Ganesamoorthy K., Murugan M., Santhakumaran A. P. Extreme-support total monophonic graphs. Bulletin of the Iranian Mathematical Society, 2021, vol. 47, pp. 159-170. https: //doi.org/10.1007/s41980-020-00485-4
9. Ganesamoorthy K., Murugan M., Santhakumaran A. P. On the connected monophonic number of a graph. International Journal of Computer Mathematics: Computer Systems Theory, 2022, vol. 7, iss. 2, pp. 139-148. https://doi.org/10.1080/23799927.2022.2071765
10. Santhakumaran A. P., Titus P., Ganesamoorthy K., Murugan M. The forcing total monophonic number of a graph. Proyecciones, 2021, vol. 40, iss. 2, pp. 561-571. https: //doi.org/10.22199/issn.0717-6279-2021-02-0031
11. Ganesamoorthy K., Lakshmi Priya S. The outer connected monophonic number of a graph. Ars Combinatoria, 2020, vol. 153, pp. 149-160.
12. Ganesamoorthy K., Lakshmi Priya S. Further results on the outer connected monophonic number of a graph. Transactions of National Academy of Sciences of Azerbaijan. Series of Physical-Technical and Mathematical Sciences, Issue Mathematics, 2021, vol. 41, iss. 4, pp. 51-59.
13. Ganesamoorthy K., Lakshmi Priya S. Extreme outer connected monophonic graphs. Communications in Combinatorics and Optimization, 2022, vol. 7, iss. 2, pp. 211-226. https://dx.doi.org/10.22049/cco.2021.27042.1184
Поступила в редакцию / Received 15.09.2021 Принята к публикации / Accepted 12.12.2021 Опубликована / Published 31.08.2022
