Ratio Mathematica Volume 44, 2022 The Outer Connected Detour Monophonic Number of a Graph N.E. Johnwin Beaula1 S. Joseph Robin2 Abstract For a connected graph 𝐺 = (𝑉, 𝐸) of order 𝑛 β‰₯ 2, a set 𝑀 βŠ† 𝑉 is called a monophonic set of 𝐺if every vertex of 𝐺is contained in a monophonic path joining some pair of vertices in 𝑀. The monophonic number π‘š(𝐺) of 𝐺 is the minimum cardinality of its monophonic sets. If 𝑀 = 𝑉 or the subgraph 𝐺[𝑉 – 𝑀]is connected, then a detour monophonic set 𝑀 of a connected graph 𝐺 is said to be an outer connected detour monophonic setof 𝐺.The outer connecteddetourmonophonic number of 𝐺, indicated by the symbol π‘œπ‘π‘‘π‘š(𝐺), is the minimum cardinality of an outer connected detour monophonic set of 𝐺. The outer connected detour monophonic number of some standard graphs are determined. It is shown that for positive integers π‘Ÿπ‘š, π‘‘π‘šand 𝑙 β‰₯ 2 with π‘Ÿπ‘š < π‘‘π‘š ≀ 2π‘Ÿπ‘š,there exists a connected graph 𝐺withπ‘Ÿπ‘Žπ‘‘π‘šπΊ = π‘Ÿπ‘š , π‘‘π‘–π‘Žmπ‘šπΊ = π‘‘π‘šand π‘œπ‘π‘‘π‘š(𝐺)= 𝑙. Also, it is shown that for every pair of integers π‘Žand b with 2 ≀ π‘Ž ≀ 𝑏, there exists a connected graph 𝐺withπ‘‘π‘š(𝐺) = π‘Ž andπ‘œπ‘π‘‘π‘š(𝐺) = 𝑏. Keywords: chord, monophonic path, monophonic number, detour monophonic path, detour monophonic number, outer connected detour monophonic number. AMS Subject Classification: 05C383 1 Register Number.20123162092018, Research Scholar. Scott Christian College (Autonomous), Nagercoil – 629003, India. beaulajohnwin@gmail.com 2 Department of Mathematics, Scott Christian College (Autonomous), Nagercoil – 629003, India prof.robinscc@gmail.com Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627 012, Tamil Nadu, India 3 Received on June15th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.921. ISSN: 1592-7415. eISSN: 2282-8214. Β©The Authors.This paper is published under the CC-BY licence agreement 325 mailto:beaulajohnwin@gmail.com mailto:prof.robinscc@gmail.com N.E. Johnwin Beaula and S. Joseph Robin 1. Introduction A finite, undirected connected graph with no loops or many edges is referred to as a graph 𝐺 = (𝑉, 𝐸). By 𝑛 and π‘š, respectively, we indicate the order and size of 𝐺. We refer to [1] for the fundamental terms used in graph theory. If 𝑒𝑣 is an edge of G, then two vertices 𝑒 and 𝑣 are said to be adjacent. If two edges of 𝐺 share a vertex, they are said to to be adjacent. Let 𝑆 βŠ‚ 𝑉 be any subset of vertices of 𝐺. Then the graph with 𝑆 as its vertex set and all of its edges in 𝐸 having both of their end points in 𝑆 is the induced subgraph 𝐺[𝑆]. A vertex 𝑣 is an extreme vertex of a graph G if the subgraph induced by its neighbors is complete. The length of the shortest path in a connected graph G is equal to the distance d(u, v) between two vertices u and v. An u βˆ’ vgeodesic is a u βˆ’ v path with length d(u, v). An edge that connects two non-adjacent vertices of a path P is called the chord of P . A chordlessu βˆ’ v path is referred to as a monophonic path. The monophonic distance π‘‘π‘š(𝑒, 𝑣)for two vertices 𝑒 and 𝑣 in a connected graph 𝐺 is the length of a longest 𝑒 βˆ’ 𝑣 monophonic path in 𝐺. An u βˆ’ vdetour monophonic path is one that has a length of dm(u, v).The monophonic eccentricity of a vertex v, denoted by em(v) is the monophonic distance between v and a vertex farthest from 𝑣. The monophonic radius, π‘Ÿπ‘Žπ‘‘π‘š(𝐺), and the monophonic diameter, π‘‘π‘–π‘Žπ‘šπ‘š(𝐺) are the vertices respective minimum and maximum monophonic eccentricities. The closed interval π½π‘‘π‘š[𝑒, 𝑣] for two vertices 𝑒 and 𝑣 is consists of all the vertices along an 𝑒 βˆ’ 𝑣 detour monophonic path, including the vertices 𝑒 and 𝑣. If 𝑣 ∈ 𝐸 , thenπ½π‘‘π‘š[𝑒, 𝑣] = {𝑒, 𝑣}. For a set 𝑀 of vertices, let π½π‘‘π‘š[𝑀] = βˆͺ𝑒,π‘£βˆˆπ‘€ 𝐽[𝑒, 𝑣]. Then certainly 𝑀 βŠ† π½π‘‘π‘š[𝑀]. If π½π‘‘π‘š[𝑀] = 𝑉,a set 𝑀 βŠ† 𝑉(𝐺) is referred to as a detour monophonic set of 𝐺.The detour monophonic number π‘‘π‘š(𝐺) of 𝐺 is the minimum order of its detour monophonic sets.Any detour monophonic set of order π‘‘π‘š(𝐺) is referred to as an π‘‘π‘š-set of 𝐺. In [2-4], these concepts were investigated. The following theorem is used in sequel. Theorem 1.1. [4] Each extreme vertex of a connected graph 𝐺 belongs to everydetour monophonic set of 𝐺. 2. The Outer Connected Detour Monophonic Number of a Graph Definition 2.1. If 𝑀 = 𝑉 or the subgraph 𝐺[𝑉 – 𝑀]is connected, then a detour monophonic set 𝑀 of a connected graph 𝐺 is said to be an outer connected detour monophonic setof 𝐺.The outer connecteddetourmonophonic number of 𝐺, indicated by the symbol π‘œπ‘π‘‘π‘š(𝐺), is the minimum cardinality of an outer connected detour monophonic set of 𝐺. The π‘œπ‘π‘‘π‘š-set of 𝐺 is a minimum cardinality of an outer connected detour monophonic setof 𝐺. 326 The Outer Connected Detour Monophonic Number of a Graph Example 2.2. 𝑀 = {𝑣2, 𝑣4} is a π‘‘π‘š-set of the graph 𝐺 in Figure 2.1 such that π‘‘π‘š(𝐺) = 2. 𝑀 is not an outer connected detour monophonic set of 𝐺 because 𝐺[𝑉 βˆ’ 𝑀 ] is not connected, and as a result, π‘œπ‘π‘‘π‘š(𝐺) β‰₯ 3. Now since 𝑀1= {𝑣2, 𝑣3, 𝑣4}, is a π‘œπ‘π‘‘π‘š-set of 𝐺, and π‘œπ‘π‘‘π‘š(𝐺) = 3 as a result. Observation 2.3. (i) Each extreme vertex of a connected graph 𝐺 belongs to every outer connected detour monophonic set of 𝐺. (ii) No cut vertex of 𝐺 belongs to any π‘œπ‘π‘‘π‘š-set of 𝐺. (iii)For any connected graph G of order𝑛, 2 ≀ π‘‘π‘š(𝐺) ≀ π‘œπ‘π‘‘π‘š(𝐺) ≀ 𝑛. Theorem 2.4. π‘œπ‘π‘‘π‘š (𝐺) = 𝑛, for the complete graph 𝐺 = 𝐾𝑛 (n β‰₯ 2). Proof. The vertex set of 𝐾𝑛is the unique outer connected detour monophonic set of 𝐾𝑛 since every vertex of the complete graph 𝐾𝑛(𝑛 β‰₯ 2)is the extreme vertex. Therefore, π‘œπ‘π‘‘π‘š(𝐺) = 𝑛. ∎ Theorem 2.5. π‘œπ‘π‘‘π‘š(𝑇) = π‘˜, for any tree T with k end vertices. Proof. Let 𝑀 represent the collection of 𝑇′𝑠 end vertices. According to Observation 2.3(i) and (ii), π‘œπ‘π‘‘π‘š (𝑇) β‰₯ |𝑀|. 𝑀 is the unique outer connected detour monophonic set of 𝑇,since the subgraph 𝐺[𝑉 βˆ’ 𝑀] is connected. Consequently, π‘œπ‘π‘‘π‘š(𝑇) = |𝑀| = π‘˜.∎ Corollary2.6. π‘œπ‘π‘‘π‘š(𝑃𝑛) = 2 for the non-trivial path 𝑃𝑛 (𝑛 β‰₯ 3). Corollary2.7. π‘œπ‘π‘‘π‘š(𝐾1,π‘›βˆ’1) = 𝑛 βˆ’ 1 for star 𝐾1,π‘›βˆ’1(𝑛 β‰₯ 3). Theorem 2.8. π‘œπ‘π‘‘π‘š(𝐺) = 3, for the cycle 𝐺 = 𝐢𝑛(𝑛 β‰₯ 4). 𝑣1 𝐺 Figure 2.1 𝑣4 𝑣3 𝑣5 𝑣2 327 N.E. Johnwin Beaula and S. Joseph Robin Proof. Set the cycle 𝐢𝑛 to be 𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑣1. Then, 𝑀 = {𝑣1, 𝑣2, 𝑣3} is a 𝐺 ′𝑠 outer connected detour monophonic set, resulting in π‘œπ‘π‘‘π‘š(𝐺) ≀ 3. We establishπ‘œπ‘π‘‘π‘š(𝐺) = 3. Assume that π‘œπ‘π‘‘π‘š(𝐺)= 2. Then πΊβ€²π‘ π‘œπ‘π‘‘π‘š-set is 𝑀1 = {π‘₯, 𝑦}. It is obvious that π‘₯ and 𝑦 are not adjacent. A contradiction results since 𝐺[𝑉 βˆ’ 𝑀1]is not connected and 𝑀1 is not a 𝐺 ′𝑠 π‘œπ‘π‘‘π‘š βˆ’ set. ,Consequently, π‘œπ‘π‘‘π‘š(𝐺) = 3. ∎ Theorem 2.9. π‘œπ‘π‘‘π‘š (𝐺) = { 𝑠, 𝑖𝑓 π‘Ÿ = 1, 𝑠 β‰₯ 2 3, 𝑖𝑓 π‘Ÿ = 𝑠 = 2 4, 𝑖𝑓 2 < π‘Ÿ ≀ 𝑠 for the complete bipartite graph 𝐺= πΎπ‘Ÿ,𝑠 Proof. 𝐺 = πΎπ‘Ÿ,𝑠 is a tree with 𝑠 end vertices when π‘Ÿ = 1 and 𝑠 β‰₯ 2.Therefore,π‘œπ‘π‘‘π‘š (𝐾1,𝑠) = 𝑠 as per Corollary2.7.𝐺 = 𝐾2,2is the cycleC4 when π‘Ÿ = 𝑠 = 2 , Thus, according to Theorem 2.8, π‘œπ‘π‘‘π‘š(𝐾2,2) = 3. Let 2 < π‘Ÿ ≀ 𝑠.Let 𝑋 = {π‘₯1, π‘₯2, … , π‘₯π‘š},π‘Œ = {𝑦1, 𝑦2, … , 𝑦𝑛} be the bipartitions of 𝐺. Let 𝑀 = {π‘₯𝑖 , π‘₯𝑗 , π‘¦π‘˜ , 𝑦𝑙 }, where𝑖 β‰  𝑗, π‘˜ β‰  𝑙.Then M is adetour monophonic set of 𝐺. 𝑀 is anouter connected detour monophonic set of 𝐺 because the subgraph𝐺[𝑉 βˆ’ 𝑀]is connected, and as a result, π‘œπ‘π‘‘π‘š(𝐺) ≀ 4. We demonstrate that π‘œπ‘π‘‘π‘š(𝐺) = 4. Let's assume that π‘œπ‘π‘‘π‘š(𝐺) ≀ 3.Then |𝑀| ≀ 3 and there exists aπ‘œπ‘π‘‘π‘š(𝐺)-set 𝑀. If 𝑀 βŠ† 𝑋 or 𝑀 βŠ† π‘Œ then 𝐺[𝑉 βˆ’ 𝑀] is not connected. Consequently, 𝑀 βŠ‚ π‘‹π‘ˆπ‘Œ.Which suggests 𝑀 is not aouter connected detour monophonic set of G, which is in contrast with the statement made earlier. Thusπ‘œπ‘π‘‘π‘š(𝐺) = 4. ∎ Theorem 2.10 π‘œπ‘π‘‘π‘š (𝐺) = { 2 𝑖𝑓 𝑛 = 5 3 𝑖𝑓 𝑛 β‰₯ 6 for the wheel 𝐺 = 𝐾1 + πΆπ‘›βˆ’1 (𝑛 β‰₯ 5). Proof. Let's say that 𝑉(𝐾1) = π‘₯ and 𝑉(πΆπ‘›βˆ’1) = {𝑣1, 𝑣2, … … … , π‘£π‘›βˆ’1 } . 𝑀 = {𝑣1, 𝑣3,} is an π‘œπ‘π‘‘π‘š βˆ’ 𝑠𝑒𝑑 of 𝐺 for 𝑛 = 5. Therefore, π‘œπ‘π‘‘π‘š(𝐺) = 2. So, let 𝑛 β‰₯ 6. Hence it follows that π‘œπ‘π‘‘π‘š(𝐺) β‰₯ 3 Let 𝑀1 = {𝑣1, 𝑣2,𝑣3}. Then 𝑀1 is an outer connected detour monophonic set of 𝐺. Consequently, π‘œπ‘π‘‘π‘š(𝐺) = 3. ∎ Theorem 2.11 π‘œπ‘π‘‘π‘š (𝐺) = { 2 𝑖𝑓 𝑛 = 4 3 𝑖𝑓 𝑛 > 4 ,for the graph 𝐺 = 𝐾1 + π‘ƒπ‘›βˆ’1 . Proof. Let's say that 𝑉(𝐾1) = π‘₯ and 𝑉(π‘ƒπ‘›βˆ’1) = {𝑣1, 𝑣2, … … … , π‘£π‘›βˆ’1 }. 𝑀1 = {𝑣1, 𝑣3 } is aπ‘œπ‘π‘‘π‘š βˆ’ 𝑠𝑒𝑑 π‘œπ‘“ 𝐺 for n = 4 , and π‘œπ‘π‘‘π‘š(𝐺) = 2. So, let 𝑛 β‰₯ 5. Let 𝑀 = {𝑣1, π‘£π‘›βˆ’1} be the extreme vertices of 𝐺.By observation 2.3(i)𝑀 is a subset of every π‘œπ‘π‘‘π‘š βˆ’ 𝑠𝑒𝑑 π‘œπ‘“ 𝐺. Since M is not a outer connected detour monophonic set of G, π‘œπ‘π‘‘π‘š(𝐺) β‰₯ 3. Now 𝑀2 = 𝑀 βˆͺ {π‘₯} is a π‘œπ‘π‘‘π‘š βˆ’ 𝑠𝑒𝑑 π‘œπ‘“ 𝐺. π‘ π‘œ π‘‘β„Žπ‘Žπ‘‘ π‘œπ‘π‘‘π‘š(𝐺) = 3. ∎ Theorem 2.12. Consider the connected graph 𝐺, where π‘‘π‘š(𝐺) = 2. If deg (π‘₯) β‰₯ 3 for every π‘₯βˆˆπ‘‰, then π‘œπ‘π‘‘π‘š(𝐺) = 2. Proof. Let the detour monophonic set of 𝐺 be 𝑀{𝑒, 𝑣},𝑑𝑒g(π‘₯) β‰₯ 3 forπ‘₯ ∈ 𝑉, so 𝐺[𝑉 βˆ’ 𝑀]is connected. As a result, 𝑀 is an outer connected detour monophonic set of 𝐺 so that 328 The Outer Connected Detour Monophonic Number of a Graph π‘œπ‘π‘‘π‘š(𝐺) = 2. ∎ Theorem 2.13. Suppose 𝐺 is a connected graph with dπ‘š(𝐺) = 2. If 𝐺 has 2 possible outermost vertices 𝑒, 𝑣 ∈ 𝑉 that are not relatedand βˆ†[βŒ©π‘‰ βˆ’ {𝑒, 𝑣}βŒͺ] = 𝑛 βˆ’ 3.Thenπ‘œπ‘π‘‘π‘š(𝐺)= 2. Proof. Let 𝑒 and 𝑣 represent the outermost vertices of 𝐺. Let 𝑀 = {𝑒, 𝑣}. 𝑀 is therefore a 𝐺 detour monophonic set. Given that βˆ† [βŒ©π‘‰ βˆ’ {𝑒, 𝑣}βŒͺ] = 𝑛 βˆ’ 3, and that 𝑒 and 𝑣 are not adjacent outermost vertices of 𝐺, 𝐺[𝑉 – 𝑀]is connected. As a result, 𝑀 is a outer connected detour monophonic set of 𝐺, andπ‘œπ‘π‘‘π‘š(𝐺) = 2. ∎ Theorem 2.14. Let 𝐺 be a connected graph of order 𝑛 that has precisely one vertex that is not a cut vertex and has a degree of n - 1. Thenπ‘œπ‘π‘‘π‘š(𝐺) ≀ 𝑛 βˆ’ 3. Proof. Let π‘₯ represent the non-cut vertex of G at the vertex of degreen– 1.Since 𝑁(π‘₯)is not complete, there exist at least two non-adjacent vertices, say 𝑦 and 𝑧 that are both members of 𝑁(π‘₯) .There are at least two vertices say π‘₯1and π‘₯2 because 𝑁(π‘₯)is the unique vertex of degree 𝑛 βˆ’ 1, and they are both located on the 𝑦 βˆ’ 𝑧detour monophonic path such that π‘₯1 β‰  π‘₯, π‘₯2 β‰  π‘₯. 𝑀 = 𝑉(𝐺) βˆ’{π‘₯, π‘₯1,π‘₯2} is a detour monophonic set of 𝐺. 𝑀 is an outer connected detour monophonic set of 𝐺 since the subgraph G[𝑉–𝑀]is connected, which causesπ‘œπ‘π‘‘π‘š(𝐺) ≀ 𝑛 βˆ’ 3. ∎ Theorem 2.15. Let 𝐺 be an order 𝑛 β‰₯ 3. π‘œπ‘π‘‘π‘š(𝐺) ≀ 𝑛 – 1 if 𝐺 contains a cut vertex of degree n – 1. Proof. Let 𝑀 be a minimum outer connected detour monophonic set and 𝑣 be the cut vertex of degree 𝑛 βˆ’ 1 in 𝐺. Observation 2.3(ii) says that 𝑣 βˆ‰ 𝑀. It is obvious thatπ‘œπ‘π‘‘π‘š(𝐺) ≀ 𝑛 – 1, 𝑀 = 𝑉(𝐺) βˆ’ {𝑣} is a detour monophonic set of 𝐺. 𝑀 is an outer connected detour monophonic set of 𝐺 Since 𝑣 is a universal cut vertex of 𝐺, the subgraph GβŒ©π‘‰ βˆ’ 𝑀βŒͺis connected. As a result, 𝑀 is an outer connected detour monophonic set of 𝐺. which causes π‘œπ‘π‘‘π‘š(𝐺) ≀ 𝑛 βˆ’ 1. ∎ Theorem 2.16 There exists a connected graph 𝐺 with π‘Ÿπ‘Žπ‘‘π‘šπΊ= π‘Ÿπ‘š, π‘‘π‘–π‘Žπ‘šπΊ = π‘‘π‘šand π‘œπ‘π‘‘π‘š(𝐺) = 𝑙 for positive integers π‘Ÿπ‘š, π‘‘π‘šand 𝑙 β‰₯ 2 with π‘Ÿπ‘š < π‘‘π‘š ≀ 2π‘Ÿπ‘š. Proof. We make the convenient assumptions that π‘Ÿπ‘š= π‘Ÿ and π‘‘π‘š= 𝑑. Let 𝐺 = 𝐾1,𝑙 when π‘Ÿ = 1. Theorem 2.5 states that π‘œπ‘π‘‘π‘š(𝐺) = 𝑙 . Let π‘Ÿπ‘š β‰₯ 2. Let πΆπ‘Ÿ+2: 𝑣1, 𝑣2, … π‘£π‘Ÿ+2be a cycle of length π‘Ÿ + 2 and let π‘ƒπ‘‘π‘šβˆ’π‘Ÿπ‘š+1: 𝑒0, 𝑒1, 𝑒2, … , π‘’π‘‘π‘šβˆ’π‘Ÿπ‘š be that cycle. By locating 𝑣1 in πΆπ‘Ÿ+2and 𝑒0in π‘π‘‘π‘šβˆ’π‘Ÿπ‘š+1 , we may construct the graph 𝐻.The graph shown in Figure 2.2 is then created by joining each of the 𝑀𝑖vertices (1 ≀ 𝑖 ≀ 𝑙 βˆ’ 3) to the vertex π‘’π‘‘π‘šβˆ’π‘Ÿπ‘šβˆ’1 and adding new vertices𝑀1, 𝑀2, … , π‘€π‘™βˆ’3to π‘’π‘‘π‘šβˆ’π‘Ÿπ‘šβˆ’1 So, π‘Ÿπ‘Žπ‘‘π‘šπΊ = π‘Ÿπ‘š, π‘‘π‘–π‘Žπ‘šπΊ = π‘‘π‘š. The set of all 𝐺′𝑠 end vertices, π‘Š ={𝑀1, 𝑀2,…,π‘€π‘™βˆ’3,π‘’π‘‘π‘šβˆ’π‘Ÿπ‘š }shall be defined. π‘Š is then contained in every detour monophonic detour set of 𝐺 according to Observation 2.3(i). Sinceπ½π‘‘π‘š[𝑀] β‰  𝑉, π‘Š is not an outer connected detour monophonic set of 𝐺 and so π‘œπ‘π‘‘π‘š(𝐺) β‰₯ 𝑙 βˆ’ 1. π‘œπ‘π‘‘π‘š(𝐺) β‰₯ 𝑙 because it is obvious that 𝑀 is not an outer connected detour 329 N.E. Johnwin Beaula and S. Joseph Robin monophonic set of 𝐺, where𝑀 = π‘Š βˆͺ{𝑒0} and𝑒0 βˆ‰ 𝑀. π‘œπ‘π‘‘π‘š(𝐺) = 𝑙 because it is obvious that 𝑀 is an outer connected detour monophonic set of 𝐺, where 𝑀 = π‘Š βˆͺ{𝑣2, 𝑣3}. Theorem 2.17. There is a connected graph G with π‘‘π‘š(𝐺) = π‘Ž and π‘œπ‘π‘‘π‘š(𝐺) = 𝑏 for every pair of positive integers a and b such that 2 ≀ π‘Ž ≀ 𝑏. Proof. Let 𝑉(𝐾2Μ…Μ… Μ…) = {π‘₯, 𝑦}. Let a graph be created by adding additional vertices to (𝐾2Μ…Μ… Μ…) as follows: 𝑧1, 𝑧2, … , π‘§π‘Žβˆ’1, 𝑣1, 𝑣2, … , π‘£π‘βˆ’π‘Ž, and connecting each 𝑧𝑖 (1 ≀ 𝑖 ≀ π‘Ž βˆ’ 1) with π‘₯ and 𝑦. Graph 𝐺 is displayed in Figure 2.3. First, we demonstrate that π‘‘π‘š(𝐺) = π‘Ž. Assume that 𝑍 = {𝑧1, 𝑧2, … , π‘§π‘Žβˆ’1} is the collection of all end vertices of 𝐺. According to Theorem1.1, every detour monophonic set of 𝐺 has 𝑍 as a subset. Since it is obvious that 𝑍 is not a monophonic detour set of 𝐺, π‘‘π‘š(𝐺) β‰₯ π‘Ž. Now that 𝑍βˆͺ{𝑦} is a monophonic set, π‘‘π‘š(𝐺) = π‘Ž. We then demonstrate that π‘œπ‘π‘‘π‘š(𝐺) = 𝑏. 𝑍 βˆͺ {𝑦}is not an outer connected detour monophonic set of 𝐺 because 𝐺[𝑉 βˆ’ ( 𝑍 βˆͺ {𝑦})] is not connected. According to Observation 2.3(i), each outer connected detour monophonic set of 𝐺 has the vertex 𝑧𝑖 (1 ≀ 𝑖 ≀ π‘Ž βˆ’ 1). Additionally, it is simple to see that any outer connected detour monophonic set of G contains each 𝑣𝑖 (1 ≀ 𝑖 ≀ 𝑏 βˆ’ π‘Ž),which means that π‘œπ‘π‘‘π‘š(𝐺) β‰₯π‘Ž βˆ’ 1 + 𝑏 βˆ’ π‘Ž = 𝑏 – 1. Let 𝑀 = 𝑍 βˆͺ {𝑣1, 𝑣2, … , π‘£π‘βˆ’π‘Ž}. Since M is not an outer connected detour monophonic set of 𝐺, then π‘œπ‘π‘‘π‘š(𝐺) β‰₯ 𝑏. Now that 𝑀 βˆͺ {π‘₯} is an outer connected detour monophonic set of 𝐺 π‘œπ‘π‘‘π‘š(𝐺) = 𝑏. 𝐺 Figure 2.2 π‘’π‘‘π‘šβˆ’π‘Ÿπ‘š π‘’π‘‘π‘šβˆ’π‘Ÿπ‘šβˆ’2 π‘’π‘‘π‘šβˆ’π‘Ÿπ‘šβˆ’1 𝑣3 𝑣2 𝑣1 = u0 𝑒1 𝑒2 𝑀1 𝑀2 π‘€π‘™βˆ’3 π‘£π‘Ÿ+2 330 The Outer Connected Detour Monophonic Number of a Graph 3. Conclusion This article established a novel detour monophonic distance parameter called the outer connected detour monophonic number of graphs. We will develop this concept to incorporate more distance considerations in a subsequent investigation. Acknowledgements We are thankful to the referees for their constructive and detailed comments and suggestions which improved the paper overall. References [1] T. W. Haynes, S. T. Hedetniemi and P. J, Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, (1998). [2] J. John, The Forcing Monophonic and The Forcing Geodetic Numbers of a Graph, Indonesian Journal of Combinatorics 4(2), (2020), 114-125. [3] J. John and S. Panchali 2, The upper monophonic number of a graph, Int. J. Math.Combin. 4, (2010),46 – 52. [4] P. Titus, K. Ganesamoorthy and P. Balakrishnan, The Detour Monophonic Number of a Graph. J. Combin. Math. Combin. Comput., (84), (2013),179-188. 𝑣1 𝑣2 𝑣3 π‘£π‘βˆ’π‘Ž 𝑦 𝑧2 π‘§π‘Žβˆ’1 𝑧1 π‘₯ 𝐺 Figure 2.3 331