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Journal articles on the topic 'Vertex monophonic number'

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Published: 2 June 2025

Last updated: 31 July 2025

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1

Santhakumaran, A. P., P. Titus, and K. Ganesamoorthy. "Edge-to-vertex m-detour monophonic number of a graph." Proyecciones (Antofagasta) 37, no. 3 (2018): 415–28. https://doi.org/10.22199/issn.0717-6279-3161.

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For a connected graph G = (V, E) of order at least three, the monophonic distance dm(u, v) is the length of a longest u − v monophonic path in G. A u − v path of length dm(u, v) is called a u − v detour monophonic. For subsets A and B of V, the m-monophonic distance Dm(A, B) is defined as Dm(A, B) = max{dm(x, y) : x ∈ A, y ∈ B}. A u − v path of length Dm(A, B) is called a A − B m-detour monophonic path joining the sets A, B ⊆ V, where u ∈ A and v ∈ B. A set S ⊆ E is called an edge-to-vertex m-detour monophonic set of G if every vertex of G is incident with an edge of S or lies on a m-detour mo

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2

Titus, P., M. Subha, and S. Santha Kumari. "Monophonic graphoidal covering number of corona product graphs." Proyecciones (Antofagasta) 42, no. 2 (2023): 303–18. http://dx.doi.org/10.22199/issn.0717-6279-4781.

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In a graph G, a chordless path is called a monophonic path. A collection ψm of monophonic paths in G is called a monophonic graphoidal cover of G if every vertex of G is an internal vertex of at most one monophonic path in ψm and every edge of G is in exactly one monophonic path in ψm. The monophonic graphoidal covering number ηm(G) of G is the minimum cardinality of a monophonic graphoidal cover of G. In this paper, we find the monophonic graphoidal covering number of corona product of some standard graphs.

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3

JOHN, J., and K. UMA SAMUNDESVARI. "THE FORCING EDGE FIXING EDGE-TO-VERTEX MONOPHONIC NUMBER OF A GRAPH." Discrete Mathematics, Algorithms and Applications 05, no. 04 (2013): 1350034. http://dx.doi.org/10.1142/s1793830913500341.

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For a connected graph G = (V, E), a set Se ⊆ E(G)–{e} is called an edge fixing edge-to-vertex monophonic set of an edge e of a connected graph G if every vertex of G lies on an e – f edge-to-vertex monophonic path of G, where f ∈ Se. The edge fixing edge-to-vertex monophonic number mefev(G) of G is the minimum cardinality of its edge fixing edge-to-vertex monophonic sets of an edge e of G. A subset Me ⊆ Se in a connected graph G is called a forcing subset for Se, if Se is the unique edge fixing edge-to-vertex monophonic set of e of G containing Me. A forcing subset for Se of minimum cardinalit

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4

Gamorez, Anabel, and Sergio Canoy Jr. "Monophonic Eccentric Domination Numbers of Graphs." European Journal of Pure and Applied Mathematics 15, no. 2 (2022): 635–45. http://dx.doi.org/10.29020/nybg.ejpam.v15i2.4354.

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Let G be a (simple) undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a monophonic eccentric dominating set if every vertex in V (G) \ S has a monophonic eccentric vertex in S. The minimum size of a monophonic eccentric dominating set in G is called the monophonic eccentric domination number of G. It is shown that the absolute difference of the domination number and monophonic eccentric domination number of a graph can be made arbitrarily large. We characterize the monophonic eccentric dominating sets in graphs resulting from the join, corona, and lexi

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5

John, J., P. Arul Paul Sudhahar, and D. Stalin. "On the (M,D) number of a graph." Proyecciones (Antofagasta) 38, no. 2 (2019): 255–66. https://doi.org/10.22199/issn.0717-6279-3570.

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For a connected graph G = (V, E), a monophonic set of G is a set M ⊆ V (G) such that every vertex of G is contained in a monophonic path joining some pair of vertices in M. A subset D of vertices in G is called dominating set if every vertex not in D has at least one neighbour in D. A monophonic dominating set M is both a monophonic and a dominating set. The monophonic, dominating, monophonic domination number m(G), γ(G), γm(G) respectively are the minimum cardinality of the respective sets in G. Monophonic domination number of certain classes of graphs are determined. Connected graph of order

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6

Lourdusamy, A., S. Kither Iammal, and I. Dhivviyanandam. "Monophonic Cover Pebbling Number \((MCPN)\) of Network Graphs." Utilitas Mathematica 121, no. 1 (2024): 11–24. https://doi.org/10.61091/um121-02.

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Given a connected graph \(G\) and a configuration \(D\) of pebbles on the vertices of \(G\), a pebbling transformation involves removing two pebbles from one vertex and placing one pebble on its adjacent vertex. A monophonic path is defined as a chordless path between two non-adjacent vertices \(u\) and \(v\). The monophonic cover pebbling number, \(\gamma_{\mu}(G)\), is the minimum number of pebbles required to ensure that, after a series of pebbling transformations using monophonic paths, all vertices of \(G\) are covered with at least one pebble each. In this paper, we determine the monopho

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7

K., Ponselvi. "THE MONOPHONIC DIAMETRAL PATH FIXING MONOPHONIC NUMBER OF A GRAPH." GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES 5, no. 5 (2018): 66–69. https://doi.org/10.5281/zenodo.1251655.

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For a connected graph , let  be amonophonic diametral path of . A set  is called a monophonic set of  if every vertex of  lies on a  monophonic pathwhere and . The minimum cardinality of a  monophonic set of  is monophonic number of denoted by . A monophonic set of cardinality  is called a set of<em>G</em>. monophonic number of  certain classes of graphs are studied. Connected graphs of order  with monophonic number  and  are characterized. It is shown that for integers with , there exists a connected graph  of order , with and .

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8

Titus, P., and S. Eldin Vanaja. "Edge fixed monophonic number of a graph." Proyecciones (Antofagasta) 36, no. 3 (2017): 363–72. https://doi.org/10.22199/issn.0717-6279-2381.

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For an edge xy in a connected graph G of order p ≥ 3, a set SCV(G)is an xy-monophonic set of G if each vertex v Є V(G) lies on an x-u monophonic path or a y-u monophonic path for some element u in S. The minimum cardinality of an xy- monophonic set of G is defined as the xy-monophonic number of G, denoted by mxy (G). An xy-monophonic set of cardinality mxy (G) is called a mxy -set of G. We determine bounds for it and find the same for special classes of graphs. It is shown that for any three positive integers r, d and n ≥ 2 with 2 ≤ r ≤ d, there exists a connected graph G with monophonic radiu

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9

Santhakumaran, A. P., and P. Titus. "The vertex monophonic number of a graph." Discussiones Mathematicae Graph Theory 32, no. 2 (2012): 191. http://dx.doi.org/10.7151/dmgt.1599.

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10

Santhakumaran, A. P., and T. Venkata Raghu. "Upper double monophonic number of a graph." Proyecciones (Antofagasta) 37, no. 2 (2018): 295–304. https://doi.org/10.22199/issn.0717-6279-2929.

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A set S of a connected graph G of order n is called a double monophonic set of G if for every pair of vertices x, y in G there exist vertices u, v in S such that x, y lie on a u - v monophonic path. The double monophonic number dm(G) of G is the minimum cardinality of a double monophonic set. A double monophonic set S in a connected graph G is called a minimal double monophonic set if no proper subset of S is a double monophonic set of G. The upper double monophonic number of G is the maximum cardinality of a minimal double monophonic set of G, and is denoted by dm+(G). Some general properties

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11

Titus, P., and A. P. Santhakumaran. "Extreme Monophonic Graphs and Extreme Geodesic Graphs." Tamkang Journal of Mathematics 47, no. 4 (2016): 393–404. http://dx.doi.org/10.5556/j.tkjm.47.2016.2045.

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For a connected graph $G=(V,E)$ of order at least two, 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 monophonic set of $G$ is a set $S$ of vertices such that every vertex of $G$ lies on a monophonic path joining some pair of vertices in $S$. The monophonic number of $G$ is the minimum cardinality of its monophonic sets and is denoted by $m(G)$. A geodetic set of $G$ is a set $S$ of vertices such that every vertex of $G$ lies on a geodesic joining some pair of vertices in $S$. The geodetic number of

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12

Sethu Ramalingam, S., I. Keerthi Asir, and S. Athisayanathan. "Upper Vertex Triangle Free Detour Number of a Graph." Mapana - Journal of Sciences 16, no. 3 (2017): 27–40. http://dx.doi.org/10.12723/mjs.42.3.

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For a graph G, the x-triangle free detour set, the x-triangle free detour number, the minimal x-triangle free detour set, the upper x-triangle free detour number, are defined and studied. Certain bounds are determined and the relation with the vertex triangle free detour number of a graph is found out. Some realization problems, properties related to the upper vertex detour number, the upper vertex detour monophonic number and the upper vertex geodetic number are also studied.

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13

Varghese, Eddith Sarah, D. Antony Xavier, Ammar Alsinai, Deepa Mathew, S. Arul Amirtha Raja, and Hanan Ahmed. "Strong Total Monophonic Problems in Product Graphs, Networks, and Its Computational Complexity." Journal of Mathematics 2022 (September 8, 2022): 1–7. http://dx.doi.org/10.1155/2022/6194734.

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Let G be a graph with vertex set as V G and edge set as E G which is simple as well as connected. The problem of strong total monophonic set is to find the set of vertices T ⊆ V G , which contains no isolated vertices, and all the vertices in V G \ T lie on a fixed unique chordless path between the pair of vertices in T . The cardinality of strong total monophonic set which is minimum is defined as strong total monophonic number, denoted by s m t G . We proved the NP-completeness of strong total monophonic set for general graphs. The strong total monophonic number of certain graphs and network

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14

Titus, P., and P. Balakrishnan. "The forcing vertex detour monophonic number of a graph." AKCE International Journal of Graphs and Combinatorics 13, no. 1 (2016): 76–84. http://dx.doi.org/10.1016/j.akcej.2016.03.002.

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15

Titus, P., P. Balakrishnan, and K. Ganesamoorthy. "The connected vertex detour monophonic number of a graph." Afrika Matematika 28, no. 3-4 (2016): 311–20. http://dx.doi.org/10.1007/s13370-016-0452-x.

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16

Santhakumaran, A. P., P. Titus, and K. Ganesamoorthy. "Edge-to-vertex m-detour monophonic number of a graph." Proyecciones (Antofagasta) 37, no. 3 (2018): 415–28. http://dx.doi.org/10.4067/s0716-09172018000300415.

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17

I, Annalin Selcy, Arul Paul Sudhahar P, and Robinson Chellathurai S. "The Path Induced Edge-to-Vertex Monophonic Number of Graphs." International Journal of Mathematics Trends and Technology 66, no. 8 (2020): 82–91. http://dx.doi.org/10.14445/22315373/ijmtt-v66i8p509.

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18

Arumugam, S., P. Balakrishnan, A. P. Santhakumaran, and P. Titus. "The Upper Connected Vertex Detour Monophonic Number of a Graph." Indian Journal of Pure and Applied Mathematics 49, no. 2 (2017): 365–79. http://dx.doi.org/10.1007/s13226-018-0274-7.

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19

Mahendran, M., and R. Kavitha. "Split Detour Monophonic Sets in Graph." WSEAS TRANSACTIONS ON COMPUTERS 23 (April 9, 2024): 51–55. http://dx.doi.org/10.37394/23205.2024.23.5.

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A subset T ⊆ V is a detourmonophonic set of G if each node (vertex) x in G contained in an p-q detourmonophonic path where p, q ∈ T.. The number of points in a minimum detourmonophonic set of G is called as the detourmonophonic number of G, dm(G). A subset T ⊆ V of a connected graph G is said to be a split detourmonophonic set of G if the set T of vertices is either T = V or T is detoumonophonic set and V – T induces a subgraph in which is disconnected. The minimum split detourmonophonic set is split detourmonophonic set with minimum cardinality and it is called a split detourmonophonic number

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20

John, J., and K. Uma Samundesvari. "Total and forcing total edge-to-vertex monophonic number of a graph." Journal of Combinatorial Optimization 35, no. 1 (2017): 134–47. http://dx.doi.org/10.1007/s10878-017-0160-y.

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21

Gamorez, Anabel Enriquez, and Sergio R. Canoy Jr. "On a Topological Space Generated by Monophonic Eccentric Neighborhoods of a Graph." European Journal of Pure and Applied Mathematics 14, no. 3 (2021): 695–705. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.3990.

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In this paper, we present a way of constructing a topology on a vertex set of a graph using monophonic eccentric neighborhoods of the graph G.Â In this type of construction, we characterize those graphs that induced the indiscrete topology, the discrete topology, and a particular point topology.

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22

Moscarini, Marina, and Francesco M. Malvestuto. "Two classes of graphs in which some problems related to convexity are efficiently solvable." Discrete Mathematics, Algorithms and Applications 10, no. 03 (2018): 1850042. http://dx.doi.org/10.1142/s1793830918500428.

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Monophonic, geodesic and 2-geodesic convexities ([Formula: see text]-convexity, [Formula: see text]-convexity and [Formula: see text]-convexity, for short) on graphs are based on the families of induced paths, shortest paths and shortest paths of length [Formula: see text], respectively. We introduce a class of graphs, the class of cross-cyclicgraphs, in which every connected [Formula: see text]-convex set is also [Formula: see text]-convex and [Formula: see text]-convex. We show that this class is properly contained in the class, say [Formula: see text], of graphs in which geodesic and monoph

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23

P.Titus and K.Iyappan. "The Forcing Vertex Monophonic Number of a Graph." February 28, 2015. https://doi.org/10.5281/zenodo.22965.

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By a graph G = (V,E) we mean a finite undirected connected graph without loops or multiple edges. The order and size of G are denoted by p and q respectively. For basic graph theoretic terminology we refer to Harary.For vertices x and y in a connected graph G, the distance d(x, y) is the length of a shortest x − y path in G.

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24

Sudhahar, P. Arul Paul, and A. J. Bertilla Jaushal. "The Total Outer Independent Monophonic Dominating Parameters in Graphs." Asian Research Journal of Mathematics, June 17, 2019, 1–8. http://dx.doi.org/10.9734/arjom/2019/v14i130120.

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In this paper the concept of total outer independent monophonic domination number of a graph is introduced. A monophonic set SÍV is said to be total outer independent monophonic domination set if <S> has no isolated vertex and <V-S> is an independent set.

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25

Titus, P., and K. Iyappan. "THE UPPER VERTEX MONOPHONIC NUMBER OF A GRAPH." International Journal of Pure and Apllied Mathematics 106, no. 2 (2016). http://dx.doi.org/10.12732/ijpam.v106i2.4.

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26

Sadiquali, A., P. Arul Paul Sudhahar, and V. Lakshmana Gomathi Nayagam. "Connected monophonic domination in graphs." Discrete Mathematics, Algorithms and Applications, October 19, 2020, 2150029. http://dx.doi.org/10.1142/s1793830921500294.

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A collection of vertices in different connected graphs embraces a wholesome shift into a new collection with the properties of the couplets monophonic and dominating sets. The new collection of vertices and associated invariant with the new behavior of connected graphs are called as connected monophonic dominating set (cmd-set) and connected monophonic domination number (cmd-number), respectively. Certain initial results are studied. The cmd-number is characterized with some conditions. Some realization problems related to a connected graph by imposing conditions on the vertex count are also p

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27

Titus, P., J. Ajitha Fancy, Gyanendra Prasad Joshi, and S. Amutha. "The connected monophonic eccentric domination number of a graph." Journal of Intelligent & Fuzzy Systems, July 10, 2022, 1–10. http://dx.doi.org/10.3233/jifs-220463.

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A set S ⊆ V in a graph G is a MED-set if every vertex in V - S has a monophonic eccentric vertex in S. The MED-number γ me (G) is the cardinality of a minimum MED-set of G. A set S ⊆ V in a graph G is a CMED-set if S is a MED-set and the induced subgraph is connected. The CMED-number γ cme (G) is the cardinality of a minimum CMED-set of G. We investigate some properties of the CMED-sets. Also, we determine the bounds of the CMED-number and find the same for some standard graphs. The CMED-number has applications in security based communication networks in real life situations. This motivated us

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28

Ramos, Igor, Vinícius F. Santos, and Jayme L. Szwarcfiter. "Complexity aspects of the computation of the rank of a graph." Discrete Mathematics & Theoretical Computer Science Vol. 16 no. 2, PRIMA 2013 (2014). http://dx.doi.org/10.46298/dmtcs.2075.

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Special issue PRIMA 2013 International audience We consider the P₃-convexity on simple undirected graphs, in which a set of vertices S is convex if no vertex outside S has two or more neighbors in S. The convex hull H(S) of a set S is the smallest convex set containing S as a subset. A set S is a convexly independent set if v \not ∈ H(S\setminus \v\) for all v in S. The rank \rk(G) of a graph is the size of the largest convexly independent set. In this paper we consider the complexity of determining \rk(G). We show that the problem is NP-complete even for split or bipartite graphs with small d

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29

Sethu Ramalingam, S., and S. Athisayanathan. "Upper triangle free detour number of a graph." Discrete Mathematics, Algorithms and Applications, March 11, 2021, 2150094. http://dx.doi.org/10.1142/s1793830921500944.

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For any two vertices [Formula: see text] and [Formula: see text] in a connected graph [Formula: see text], the [Formula: see text] path [Formula: see text] is called a [Formula: see text] triangle free path if no three vertices of [Formula: see text] induce a triangle. The triangle free detour distance [Formula: see text] is the length of a longest [Formula: see text] triangle free path in [Formula: see text]. A [Formula: see text] path of length [Formula: see text] is called a [Formula: see text] triangle free detour. A set [Formula: see text] is called a triangle free detour set of [Formula:

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30

John, J. "On the vertex monophonic, Vertex Geodetic and Vertex Steiner Numbers of Graphs." Asian-European Journal of Mathematics, December 21, 2020. http://dx.doi.org/10.1142/s1793557121501710.

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