Ratio Mathematica                                                                            Volume 44, 2022 

 

 
 

 

 

 

 

The Detour Monophonic Convexity Number of 

a Graph 

 
 

M. Sivabalan*  

S. Sundar Raj†   

V. Nagarajan‡ 
 

 

 

Abstract 

A set 𝑆 is detour monophonic convexif𝐽𝑑𝑚 [𝑆} = 𝑆. The detour monophonic convexity 
number is denoted by 𝐶𝑑𝑚(𝐺), is the cardinality of a maximum proper detour 
monophonic convex subset of  𝑉.Some general properties satisfied by this concept are 
studied. The detour monophonic convexity number of certain classes of graphs are 

determined. It is shown that for every pair of integers  𝑎 and 𝑏 with 3 ≤ 𝑎 < 𝑏, there 
exists a connected graph 𝐺 such that  𝐶𝑚(𝐺) = 𝑎 and 𝐶𝑑𝑚(𝐺) = 2(𝑏 + 1), 
where𝐶𝑚(𝐺) is the monophonic convexity number of 𝐺. 
 

Keywords: convex, detour, chord, detour monophonic path, monophonic convexity 

number, detour monophonic, convexity number. 

 

AMS subject classification: 05C12,05C38§. 

 

 
 
 

 
*Register Number.12567, Research Scholar, Department of Mathematics, S.T. Hindu College, (Nagercoil 

– 629002, India); e-mail: sivabalanvkc@gmail.com        
†Department of Mathematics, Vivekananda College, (Agasteeswaram – 629701, India); 

e-mail: sundarrajvc@gmail.com 
‡Department of Mathematics, S.T. Hindu College, (Nagercoil – 629002, India); 

e-mail: sthcrajan@gmail.com Affiliated to Manonmaniam Sundaranar University, Abishekapatti,  

Tirunelveli - 627 012, Tamil Nadu, India 
§ Received on June 12 th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 

10.23755/rm.v44i0.918. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published 

under the CC-BY licence agreement. 

302

mailto:sivabalanvkc@gmail.com
mailto:sthcrajan@gmail.com


M. Sivabalan, S. Sundar Raj and V. Nagarajan, 

 

 

1. Introduction 

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 [1].  A vertex v is adjacent to another 

vertex 𝑢 if and only if there exists an edge 𝑒 =  𝑢𝑣 ∈ 𝐸(𝐺).  If  𝑢𝑣 ∈ 𝐸(𝐺), we say 
that 𝑢 is a neighbor of 𝑣 and denote by 𝑁𝐺 (𝑣), the set of neighbors of 𝑣.  A vertex 𝑣 is 
said to be universal vertex if deg𝐺 (𝑣) =  𝑝 − 1. A vertex 𝑣 is called an extreme vertex 
if the subgraph induced by 𝑣 iscomplete.  
 The length of a path is the number of its edges. Let u and v be vertices of a connected 
graph G. A shortest u-v path is also called a u-vgeodesic. The (shortest path) distance is 
defined as the length of a 𝑢-𝑣 geodesic in 𝐺 and is denoted by𝑑𝐺 (𝑢, 𝑣) or 𝑑(𝑢, 𝑣) for 
short if the graph is clear from the context. For a set 𝑆 of vertices, let 𝐼[𝑆] =
⋃ 𝐼[𝑥, 𝑦].𝑥,𝑦∈𝑆   A set 𝑆 ⊂ 𝑉 is called a convex set of 𝐺 if 𝐼[𝑆] = 𝑆. These concepts were 

studied in [1, 3] 

A chord of a path P is an edge which connects two non-adjacent vertices of P.  A u-
v path is called a monophonic path if it is a chordless path. For two vertices u and v, the 
closed interval J[u, v] consists of all the vertices lying in a u − v monophonic path 
including the vertices u and v. If u and v are adjacent, then J[u, v]  =  {u, v}. For a set M 
of vertices, let  J[M]  = ∪u,v∈M J[u, v]. Then certainly M ⊆  J[M]. A set M ⊆  V(G) is 
called a monophonic set of 𝐺 if 𝐽[𝑀]  = 𝑉. The monophonic number 𝑚(𝐺) of 𝐺 is the 
minimum order of its monophonic sets and any monophonic set of order 𝑚(𝐺) is called 
a 𝑚-set of 𝐺. A set 𝑀 ⊆ 𝑉(𝐺)  is called a monophonic convex set of G if 𝐽(𝑀) = 𝑀. 
The monophonic convexity number 𝐶𝑚(𝐺) of G is the cardinality of a maximum proper 
monophonic convex subset of V. These concepts were studied in [5-10]. 

The detour distance 𝐷(𝑢, 𝑣) between two vertices 𝑢 and 𝑣 in a connected graph 
𝐺  from 𝑢 to 𝑣 is defined as the length of a longest 𝑢-𝑣 path in 𝐺. An 𝑢-𝑣 path of length 
𝐷(𝑢, 𝑣) is called an 𝑢-𝑣 detour.  The detour monophonic distance 𝑑𝑚(u,v) between two 
vertices u and v is the length of a longest u-v monophonic path in G, Any monophonic 

path of length dm(u,v) is called  u-v detour monophonic path. For two vertices 𝑢, 𝑣 ∈
𝑉,  let 𝐽𝑑𝑚[𝑢, 𝑣] denotes  the set of all vertices that lies in 𝑢-𝑣detour monophonic path 
including 𝑢 and 𝑣, and 𝐽𝑑𝑚(𝑢, 𝑣)denotes  the set of all internal vertices that lies in 𝑢 −
𝑣detour monophonic path . For 𝑀 ⊆ 𝑉, let 𝐽𝑑𝑚 [𝑀] = ⋃ 𝐽𝑑𝑚[𝑢, 𝑣]𝑢,𝑣∈𝑀 .A set 𝑀 ⊆ 𝑉 is a 
detour monophonic set if 𝐽𝑑𝑚[𝑀] = 𝑉.The minimum cardinality of a detour 
monophonic set of G is the detour monophonic number of G and is denoted by 𝑑𝑚(𝐺). 
The detour monophonic set of cardinality 𝑑𝑚(𝐺) is called dm-set. These concepts were 
studied in [2, 4, 11]. 

 

2.The detour monophonic convexity number of a  

  Graph 

Definition 2.1. A set 𝑆 is detour monophonic convex if𝐽𝑑𝑚[𝑆} = 𝑆.  Clearly 𝑆 = {𝑣} or 
𝑆 = 𝑉 then 𝑆 is detour monophonic convex. The detour monophonic convexity number 

303



The detour monophonic convexity number of a graph 

 

 

is denoted by 𝐶𝑑𝑚(𝐺), is the cardinality of a maximum proper detour mono-phonic 
convex subset of  𝑉. 
 

Example 2.2. For the graph 𝐺in Figure 2.1, 𝑀1 = {𝑣1, 𝑣2, 𝑣3, 𝑣4, 𝑣5, 𝑣6, 𝑣7, 𝑣8}is a 𝐶𝑑𝑚-
set of 𝐺 so that 𝐶𝑑𝑚(𝐺) = 8. Also 𝑀2 = {𝑣1, 𝑣2}is a 𝐶𝑚-set of 𝐺 so that 𝐶𝑚(𝐺) = 2. 

 

Observation 2.3. Let 𝐺 be a connected graph of order 𝑝 ≥ 3. Then 2 ≤ 𝐶𝑑𝑚(𝐺) ≤ 
𝑝 − 1. 

 

Theorem 2.4. Let  𝐺 be a connected graph of order 𝑝 and 𝐺 contains an extreme vertex. 
Then 𝐶𝑑𝑚(𝐺) = 𝑝 − 1. 
Proof. Let 𝐺 contain an extreme vertex, say 𝑣. Then 𝑆 = 𝑉(𝐺) − {𝑣} is a 𝑑𝑚-convex 
set of 𝐺 so that   𝐶𝑑𝑚(𝐺) = 𝑝 − 1.  

      

Theorem 2.5. Let 𝐺 be a connected graph of order 𝑝 ≥ 3. Then2 ≤ 𝜔(𝐺) ≤ 𝐶𝑑𝑚(𝐺) ≤
𝑝 − 1,  where 𝜔(𝐺) is the clique number of 𝐺. 
Proof. Since 𝐺 is a  connected graph of order 𝑝 ≥ 3, 𝜔(𝐺) ≥ 2.  Let 𝐻 be a subgraph of 𝐺 such 
that 〈𝑉(𝐻)〉 is a maximal complete subgraph of 𝐺 so that 𝐶𝑑𝑚(𝐺) ≥ |𝑣(𝐻)| = 𝜔(𝐺). Let 𝑆 be a 
𝑑𝑚-convex set of 𝐺. Then 𝑆 is a convex set of 𝐺 so that 𝐶𝑑𝑚 (𝐺) ≤ 𝐶(𝐺). Since every convex 
set of 𝐺 is  a proper subset of  𝐺, 𝐶(𝐺) ≤ 𝑝 − 1. Therefore 2 ≤  𝜔(𝐺) ≤ 𝐶𝑑𝑚 (𝐺) ≤ 𝑝 − 1.. 
 

Corollary 2.6.  (i) For the complete graph 𝐺 = 𝐾𝑝  (𝑝 ≥ 3), 𝐶𝑑𝑚(𝐺) = 𝑝 − 1. 

(ii) For a trivial tree G of order 𝑝 ≥ 3, 𝐶𝑑𝑚(𝐺) = 𝑝 − 1. 
(iii)  For the fan graph 𝐺 = 𝐾1 + 𝑃𝑝−1  (𝑝 ≥ 4), 𝐶𝑑𝑚(𝐺) = 𝑝 − 1. 

 

Theorem 2.7.  For the cycle 𝐺 = 𝐶𝑝, (𝑝 ≥ 3), 𝐶𝑑𝑚(𝐺) = 2. 

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M. Sivabalan, S. Sundar Raj and V. Nagarajan, 

 

 

Proof. Let𝑆 = {𝑥, 𝑦}be a set of two adjacent vertices of 𝐺. Then 𝐽𝑑𝑚[𝑆] = 𝑆, it follows 
that S is a 𝑑𝑚-convex set of 𝐺 so that 𝐶𝑑𝑚 (𝐺) ≥ 2. We prove that 𝐶𝑑𝑚(𝐺) = 2. 
Suppose that 𝐶𝑑𝑚(𝐺) ≥ 3. Then there exists a dm-convex set 𝑆1 such that |𝑆1| ≥ 3. 
Hence it follows that 𝑆1 contains two independent vertices of 𝐺. Then 𝐽𝑑𝑚[𝑆1] ≠ 𝑆1. 
Therefore 𝐶𝑑𝑚(𝐺) = 2.∎ 

 

Theorem 2.8.  For the complete bipartite graph 𝐺 = 𝐾𝑚,𝑛, 𝐶𝑑𝑚(𝐺) = 2. 
Proof: Let (𝑉1, 𝑉2) be a partition of 𝐺. Since 𝜔(𝐺) = 2, 𝐶𝑑𝑚(𝐺) ≥ 2. We prove that 
𝐶𝑑𝑚(𝐺) = 2. Suppose that 𝐶𝑑𝑚(𝐺) ≥ 3. Then there exists two vertices 𝑥 and 𝑦 belong 
to the same partite 𝑉1 (or 𝑉2). Since 𝑑(𝑥, 𝑦) = 2 in 𝐺, every vertex in 𝑉1 (or 𝑉2) lie on 
𝑥 − 𝑦 detour monophonic. Hence it follows that   𝐽𝑑𝑚 [𝑆] ≠ 𝑆. Therefore 𝐶𝑑𝑚(𝐺) = 2. 

           ∎ 
Theorem 2.9.  For the wheel graph G= 𝑊𝑝 = 𝐾1 + 𝐶𝑝−1(𝑝 ≥ 4), 𝐶𝑑𝑚(𝐺) = 3. 

Proof. Let𝑉(𝐾1) = 𝑥and𝑉(𝐶𝑝−1) = {𝑣1, 𝑣2, … 𝑣𝑝−1}. Then 𝑆 = {𝑥, 𝑣1, 𝑣2} is a det- 

our monophonic convex set of 𝐺 so that 𝐶𝑑𝑚(𝐺) ≥ 3.We prove that 𝐶𝑑𝑚(𝐺) = 3. 
Suppose that 𝐶𝑑𝑚(𝐺) ≥ 4. Then there exists 𝑑𝑚-convex set 𝑆1 such that |𝑆1| ≥ 4. 
Hence it follows that 𝑆1 contains two independent vertices of 𝐺.  Then 𝐽𝑑𝑚[𝑆1] ≠ 𝑆1. 
Therefore𝐶𝑑𝑚(𝐺) = 3.             

                                                                                                  ∎ 
Theorem 2.10. For any two positive integers such that 2 ≤ 𝑎 ≤ 𝑏, there exists a 
connected graph 𝐺 such that  𝜔(𝐺) = 𝑎 and 𝐶𝑑𝑚(𝐺) = 𝑏. 
Proof.  For 𝑎 = 𝑏,let 𝐺 = 𝐾𝑎+1 − {𝑒}. Then 𝜔(𝐺) = 𝐶𝑑𝑚(𝐺) = 𝑎.For 𝑎 < 𝑏, let  𝐾𝑎 be 
the complete graph with vertices 𝑣1,𝑣2, … , 𝑣𝑎 . Let 𝑃: 𝑢1,𝑢2, … , 𝑢𝑏−𝑎 , 𝑢𝑏−𝑎+1,… , 𝑢𝑐  
where 𝑐 > 𝑏 − 𝑎 a path on 𝑐 vertices.  Let 𝐺 be the graph obtained from 𝐾𝑎 and 𝑃 by 
joining 𝑢1 with  𝑣𝑎−1and 𝑣𝑎  each𝑢𝑖 (2 ≤ 𝑖 ≤ 𝑏 − 𝑎) with 𝑣𝑎−1 and 𝑢𝑐  with 𝑣𝑎−1.  The 
graph  𝐺 is shown in Figure 2.2. 
First, we prove that 𝜔(𝐺) = 𝑎. Let 𝑆 = {𝑣1,𝑣2, … , 𝑣𝑎 }. It is clear that 𝑆 is a maximal 
complete subgraph of 𝐺 such that 𝜔(𝐺) = 𝑎. 
Next, we prove that  𝐶𝑑𝑚(𝐺) = 𝑏.  Let 𝑊 = {𝑣1,𝑣2, … , 𝑣𝑎, 𝑢1,𝑢2, … , 𝑢𝑏−𝑎}. It is clear 
that 𝑊 is a 𝑑𝑚-convex set of 𝐺 so that𝐶𝑑𝑚 (𝐺) ≥ 𝐺 . We prove that  𝐶𝑑𝑚(𝐺) = 𝑏. 
Suppose that 𝐶𝑑𝑚(𝐺) > 𝑏. Let 𝑆1 be a dm-convex set with |𝑆1| ≥ 𝑏 + 1.Then there 
exists a vertex𝑢𝑖 (𝑏 − 𝑎 + 1 ≤ 𝑖 ≤ 𝑐)such that 𝑢𝑖 ∈ 𝑆1. Then 𝐽𝑑𝑚[𝑆1] ≠
𝑆1.Therefore𝐶𝑑𝑚(𝐺) = 𝑏.                                                                                                             ∎ 

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The detour monophonic convexity number of a graph 

 

 

 
 

Theorem 2.11. For every pair of integers  𝑎 and 𝑏 with 3 ≤ 𝑎 < 𝑏, there exists a 
connected graph 𝐺 such that  𝐶𝑚(𝐺) = 𝑎 and 𝐶𝑑𝑚(𝐺) = 2(𝑏 + 1). 
Proof.  Let 𝑉(�̅�2) = {𝑥, 𝑦}.  Let 𝑃𝑖 : 𝑢𝑖,𝑣𝑖 (1 ≤ 𝑖 ≤ 𝑏)be a copy of path of order two. Let 
𝐺 be the graph obtained from �̅�2, 𝑃𝑖 (1 ≤ 𝑖 ≤ 𝑏) and 𝐾𝑎−1 by joining 𝑥 with each 
𝑢𝑖 (1 ≤ 𝑖 ≤ 𝑏) and y with each 𝑣𝑖 (1 ≤ 𝑖 ≤ 𝑏)and 𝑥 and 𝑦  with each vertex of 𝐾𝑎. The 
graph  𝐺 is shown in Figure 2.3. 
First, we prove that 𝐶𝑚(𝐺) = 𝑎. Let 𝑀 = 𝑉(𝐾𝑎) ∪ {𝑥}. Then  𝑀 is a monophonic 
convex set of 𝐺 and so 𝐶𝑚(𝐺) ≥ 𝑎.  We prove that 𝐶𝑚(𝐺) = 𝑎. Suppose that 𝐶𝑚(𝐺) ≥
𝑎 + 1. Let 𝑀1 be 𝑚-convex set with |𝑆| ≥ 𝑎 + 1. Then there exists at least one vertex, 
say 𝑥 such that 𝑥 ∈ 𝑀1 and 𝑥 ∉ 𝑀. Hence it follows that 𝑥 = 𝑢𝑖 or 𝑣𝑖  or 𝑦 for some 
𝑖 (1 ≤ 𝑖 ≤ 𝑏).  Then 𝐽𝑚[𝑀1] ≠ 𝑀1, which is a contradiction. Therefore  𝐶𝑚(𝐺) = 𝑎. 
Next we prove that 𝐶𝑑𝑚(𝐺) = 2(𝑏 + 1).  Let 𝑆 = 𝑉(𝐺) − 𝑉(𝐾𝑎).  Then 𝑆 is a detour 
monophonic convex set of 𝐺 and so   𝐶𝑑𝑚(𝐺) ≥ 2(𝑏 + 1). We prove that 𝐶𝑑𝑚(𝐺) =
2(𝑏 + 1). On the contrary  𝐶𝑑𝑚(𝐺) > 2(𝑏 + 1). Let 𝑆1 be a dm-convex set with |𝑆1| ≥
2(𝑏 + 1) + 1.  Then there exists a vertex 𝑥 ∈ 𝑆1 such that 𝑥 ∉ 𝑆. Hence it follows that 
𝑥 ∈ 𝐾𝑎. Then 𝐽𝑑𝑚[𝑆1] ≠ 𝑆1. Therefore 𝐶𝑑𝑚(𝐺) = 2(𝑏 + 1).∎ 

 
 

 

 

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M. Sivabalan, S. Sundar Raj and V. Nagarajan, 

 

 

 

 
 

3. Conclusions 
In this paper, we investigated the detour monophonic convexity number of some 

standard graphs. Also, we proved for every pair of integers  𝑎 and 𝑏 with 3 ≤ 𝑎 < 𝑏, 
there exists a connected graph 𝐺 such that  𝐶𝑚(𝐺) = 𝑎 and 𝐶𝑑𝑚(𝐺) = 2(𝑏 + 1). 

 

Acknowledgements 
The author would like to express her gratitude to the referees for their valuable 

comments and suggestions. 

 

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