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THE UPPER CONNECTED VERTEX DETOUR MONOPHONIC NUMBER OF A GRAPH S. Arumugam1, P. Balakrishnan∗∗, A. P. Santhakumaran∗∗∗and P. Titus∗∗

National Centre for Advanced Research in Discrete Mathematics,

Kalasalingam University, Anand Nagar, Krishnankoil 626 126, Tamil Nadu, India ∗∗Department of Mathematics, University College of Engineering Nagercoil,

Anna University, Tirunelveli Region, Negercoil 629 004, India ∗∗∗Department of Mathematics, Hindustan University, Hindustan Institute of

Technology and Science, Chennai 603 103, India

e-mails: s.arumugam.klu@gmail.com; titusvino@yahoo.com, gangaibala1@yahoo.com;

apskumar1953@gmail.com

(Received 16 October 2015; after final revision 2 September 2016;

accepted 10 August 2017)

For any vertexx in a connected graphGof order n 2, a setSx V(G) is anx-detour monophonic set ofGif each vertexv∈V(G)lies on anx-ydetour monophonic path for some elementyinSx. The minimum cardinality of anx-detour monophonic set ofGis thex-detour

monophonic number ofG, denoted bydmx(G). A connectedx-detour monophonic set ofGis an

x-detour monophonic setSxsuch that the subgraph induced bySxis connected. The minimum cardinality of a connectedx-detour monophonic set ofGis the connectedx-detour monophonic

number ofG, denoted bycdmx(G). A connectedx-detour monophonic setSxofGis called a

minimal connectedx-detour monophonic set if no proper subset ofSxis a connectedx-detour

monophonic set. The upper connectedx-detour monophonic number ofG, denoted bycdm+x(G),

is defined to be the maximum cardinality of a minimal connectedx-detour monophonic set ofG.

We determine bounds and exact values of these parameters for some special classes of graphs.

We also prove that for positive integersr, dandk with2 r dandk 2, there exists a connected graphGwith monophonic radiusr, monophonic diameterdand upper connectedx

-detour monophonic numberkfor some vertexxinG. Also, it is shown that for positive integers

j, k, landnwith2 j k l n−3, there exists a connected graphGof ordernwith

dmx(G) =j, dm+

x(G) =kandcdm+x(G) =lfor some vertexxinG.

Key words : Detour monophonic path; vertex detour monophonic number; connected vertex detour monophonic number; upper connected vertex detour monophonic number.

1

Also at Department of Computer Science, Liverpool Hope University, Liverpool, UK; Department of Computer

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1. INTRODUCTION

By a graphG= (V, E)we mean a finite undirected connected graph without loops or multiple edges.

The order and size ofGare denoted bynandmrespectively. For basic graph theoretic terminology

we refer to Harary [7]. For verticesx andy in a connected graph G, the distance d(x, y) is the

length of a shortestx-y path in G. An x-y path of length d(x, y) is called an x-y geodesic. The

neighbourhood of a vertexv is the setN(v)consisting of all vertices uwhich are adjacent withv.

The closed neighbourhood of a vertexv is the setN[v] = N(v)∪ {v}. A vertex v is an extreme

vertex ofGif the induced subgraphhN[v]iis complete.

The closed interval I[x, y]consists of all vertices lying on some x-y geodesic of G, while for S V,I[S] = S

x,y∈S

I[x, y]. A setS of vertices is a geodetic set if I[S] = V,and the minimum

cardinality of a geodetic set is the geodetic numberg(G). A geodetic set of cardinalityg(G)is called

ag-set ofG. The geodetic number of a graph was introduced in [1, 8] and further studied in [2, 4].

Chartrand et al. [5] introduced the concept of geodomination number of a graph. A pair of

verticesx, yis said to geodominate a vertexvif eitherv ∈ {x, y}orvlies on somex-ygeodesic of G.A subsetSofV is called a geodominating set ofGif every vertex ofGis geodominated by some

pair of vertices inS. The cardinality of a minimum geodominating set inG is the geodomination

number ofGand its denoted byg(G).

The concept of vertex geodomination number was introduced in [9] and further studied in [10].

Letxbe a vertex of a connected graphG. A setSof vertices ofGis anx-geodominating set ofGif

each vertexvofGlies on anx-ygeodesic inGfor some elementyinS. The minimum cardinality

of anx-geodominating set ofGis defined as the x-geodomination number ofGand is denoted by gx(G).

A chord of a pathP is an edge joining any two non-adjacent vertices ofP. A pathP is called

a monophonic path if it is a chordless path. A longestx-y monophonic path P is called an x-y

detour monophonic path. For any two vertices u andv in a connected graph G, the monophonic

distancedm(u, v)fromutov is defined as the length of a longestu-vmonophonic path inG. The

monophonic eccentricityem(v)of a vertexv inGisem(v) = max{dm(v, u) : u V(G)}. The

monophonic radius,radm(G)ofGisradm(G) = min{em(v) : v V(G)}and the monophonic

diameter,diamm(G) ofGisdiamm(G) = max{em(v) : v V(G)}. The monophonic distance was introduced in [11] and further studied in [12].

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detour radius, detour diameter, detour centre and detour periphery are given in [6] and [3].

The concept of vertex detour monophonic number was introduced in [13]. Letxbe a vertex of a

connected graphG. A setS of vertices ofGis anx-detour monophonic set ofGif each vertexvof Glies on an x-y detour monophonic path inGfor some elementyinS. The minimum cardinality

of an x-detour monophonic set of G is defined as the x-detour monophonic number of G and is

denoted bydmx(G). Anx-detour monophonic set of cardinalitydmx(G)is called admx-set ofG. Several results regarding the vertex detour monophonic number and interesting applications are given

in [13]. The concept of upper vertex detour monophonic number was introduced in [14]. Anx-detour

monophonic setSx is called a minimal x-detour monophonic set if no proper subset ofSx is an x-detour monophonic set. The upperx-detour monophonic number, denoted bydm+x(G), is defined as the maximum cardinality of a minimal x-detour monophonic set of G. The concept of vertex

detour set has interesting applications in Channel Assignment Problem in radio technologies. Also the

detour matrix of a connected graph is used to discuss the applications of the detour index and

hyper-detour index of a class of graphs, which in turn, capture different aspects of certain molecular graphs

associated with molecules arising in special situations in Chemistry. Also there are applications of

vertex detour monophonic sets to security based communication network design. This motivated us

to introduce and investigate vertex detour monophonic sets in [13].

The concept of connected vertex detour monophonic number was introduced in [15]. Letxbe a

vertex of a connected graphG. A connected x-detour monophonic set ofGis anx-detour monophonic

setSxsuch that the subgraph induced bySxis connected. The minimum cardinality of a connected x-detour monophonic set ofG is defined as the connected x-detour monophonic number ofGand

is denoted bycdmx(G). A connected x-detour monophonic set of cardinalitycdmx(G) is called a cdmx-set ofG.

The following theorems will be used in the sequel.

Theorem 1.1 [13]. — Letxbe any vertex of a connected graphG.

(i) Every extreme vertex of Gother than the vertex x (whetherx is extreme or not) belongs to

everyx-detour monophonic set.

(ii) No cutvertex ofGbelongs to anydmx-set.

Theorem 1.2 [13]. — For any non-trivial tree T with kendvertices, dmx(T) = k or k−1

according asxis a cutvertex or not.

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according asxis a cutvertex or not.

Throughout this paperGdenotes a connected graph with at least two vertices.

2. MINIMAL CONNECTEDVERTEXDETOURMONOPHONIC SETS

Definition 2.1 — Letxbe any vertex of a connected graphG. A connectedx-detour monophonic set Sxis called a minimal connectedx-detour monophonic set if no proper subset ofSx is a connected x-detour monophonic set. The upper connectedx-detour monophonic number ofGis the maximum

cardinality of a minimal connectedx-detour monophonic set ofGand is denoted bycdm+ x(G).

Example 2.2 : For the graphGgiven in Figure 2.1, minimum connectedx-detour monophonic

sets, connectedx-detour monophonic number, minimal connectedx-detour monophonic sets and the

upper connectedx-detour monophonic number for all the verticesxofGare given in Table 2.1.

vertex minimal connected

x cdmx-set cdmx(G) x-detour cdm+x(G) monophonic sets

r {u, w, v} 3 {u, w, v},{u, w, r, s, t} 5

s {u, w, v} 3 {u, w, v},{u, w, r, s, t} 5

t {u, w, v} 3 {u, w, v},{u, w, r, s, t} 5

u {w, v} 2 {w, v},{w, r, s} 3 v {u, w, s, t, r}, 5 {u, w, s, t, r}, 5

{u, w, s, v, r}, {u, w, s, v, r}, {u, w, s, t, v} {u, w, s, t, v}

w {u, v} 2 {u, v},{u, s, t} 3

Table 2.1

For any vertexxin a connected graphG, every minimum connectedx-detour monophonic set is

a minimal connectedx-detour monophonic set, but the converse is not true. For the graphGgiven

G

Figure 2.1

r s

t

v u

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in Figure 2.1, {w, r, s} is a minimal connectedu-detour monophonic set but it is not a minimum

connectedu-detour monophonic set.

Theorem 2.3 — Letxbe any vertex of a connected graphG. Ify6=xis an extreme vertex ofG,

thenybelongs to every minimal connectedx-detour monophonic set ofG.

PROOF: Clearlyyis not an internal vertex of any detour monophonic path starting fromxso that ybelongs to every minimal connectedx-detour monophonic set ofG. 2

Corollary 2.4 — For any vertexxin the complete graphKnof ordern≥2,cdm+x(Kn) =n−1.

Theorem 2.5 — For any vertexxin the cycleCnof ordern≥4, we have

cdm+

x(Cn) =

 

1 if n= 4

2 if n >4.

PROOF: LetCn= (u1, u2, . . . , un, u1)be a cycle of ordern≥4and letx=u1. Ifn= 4, then Sx={u3}is the unique minimal connectedx-detour monophonic set ofCnand socdm+x(Cn) = 1. Ifn >4andnis even, thenS1={un

2+1}, S2 ={u2, u3}andS3 ={un−1, un}are the minimal

con-nectedx-detour monophonic sets ofCn. Ifnis odd, thenS1={u2, u3}, S2={un−1, un}andS3 = {un+1

2 , u

n+3

2 }are the minimal connectedx-detour monophonic sets ofCn. Hencecdm

+

x(Cn) = 2if

n >4. 2

Proposition 2.6 — IfGis any connected graph of ordernat least 4, thencdm+x(G+K1) =n, where{x}=V(K1).

PROOF: No vertex ofG+K1is an internal vertex of any detour monophonic path starting from x.HenceV(G)is the minimal connectedx-detour monophonic set ofG+xand socdm+x(G+K1) =

n. 2

Theorem 2.7 — LetWn=K1+Cn−1(n≥5)be the wheel.

(i) Ifn= 5, thencdm+

x(Wn) = 1for allxinCn−1.

(ii) Ifn >5, thencdm+x(Wn) = 3for allxinCn−1.

PROOF: LetCn−1 = (u1, u2, . . . , un−1, u1)be a cycle of order n−1 and letu be the vertex ofK1. Letxbe any vertex inCn−1, sayx =u1. Ifn= 5, thenSx ={u3}is the unique minimal connected x-detour monophonic set of G and so cdm+x(Wn) = 1. If n > 5 and n is odd, then S1={u, un−1

2 +1}, S2 ={u, u2, u3}andS3={u, un−2, un−1}are the minimal connectedx-detour

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{u, un

2, un+22 }are the minimal connectedx-detour monophonic sets ofWn. Hencecdm

+

x(Wn) = 3

ifn >5. 2

Theorem 2.8 — LetKr,s(2≤r ≤s)be the complete bipartite graph with bipartition(V1, V2).

Then

(i) cdm+

x(K2,2) = 1for any vertexx.

(ii) cdm+x(K2,s) =

 

1 ifx∈V1

s ifx∈V2and s≥3.

(iii) cdm+x(Kr,s) =

 

r ifx∈V1

s ifx∈V2 andr, s≥3.

PROOF: (i) SinceK2,2is isomorphic toC4,(i) follows from Theorem 2.5.

(ii) Letr = 2ands 3. LetV1 = {v1, v2} andV2 = {w1, w2, . . . , ws}be the bipartition of K2,s.Then any vertexwi ofV2 lies on thev1-v2 detour monophonic path(v1, wi, v2) and so{v1} is a minimal connectedx-detour monophonic set ofK2,s for x = v2.Similarly {v2}is a minimal connectedx-detour monophonic set ofK2,sforx=v1.Thuscdm+x(K2,s) = 1.

Since every vertex of V1 is adjacent to wi, no vertex of V2 is an internal vertex of any detour monophonic path starting fromwi. Thus everywi-detour monophonic set ofGcontainsS =V2 {wi}. Also, any vertexvj ofV1 lies on anwi-udetour monophonic path(wi, vj, u), whereu S. HenceS is anwi-detour monophonic set ofK2,s. Since s 3, the subgraph induced byS is not connected and hencecdm+x(K2,s)> s−1. Now,hS∪ {v1}i, is a connectedwi-detour monophonic set of maximum cardinality. Therefore,cdm+x(K2,s) =s.

(iii) The proof is similar to the second part of the proof of (ii). 2

Theorem 2.9 — LetT be any tree of ordern.

(i) Ifxis a cutvertex ofT,thencdm+x(T) =n.

(ii) Ifxis a pendant vertex ofT,then

cdm+x(T) =    

  

1 ifT is a path

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PROOF: (i) Letxbe a cutvertex ofTand letSxbe any minimal connectedx-detour monophonic set of T. By Theorem 2.3, Sx contains all extreme vertices. If Sx 6= V(T), then there exists a cutvertexvofTsuch thatv /∈Sx. Letuandwbe two endvertices belonging to different components ofT − {v}. Sincevlies on the uniqueu-wpath, it follows that the subgraph induced bySx is not connected, which is a contradiction. Hencecdm+

x(T) =n.

(ii) LetT be a tree which is not a path and letxbe an endvertex ofT. Letybe the vertex ofT

withdeg y >3such thatdm(x, y) is minimum. LetP be thex-ypath inT. ThenSx = (V(T) V(P))∪ {y}is a connectedx-detour monophonic set ofT. We claim thatSxis a minimal connected x-detour monophonic set of T. Otherwise, there is a proper subset Mx of Sx such that Mx is a connectedx-detour monophonic set ofT. By Theorem 2.3, every connectedx-detour monophonic

set ofT contains all extreme vertices except possiblyxand hence there exists a cutvertexvofT such

thatv∈Sxandv /∈Mx. LetB1, B2, . . . , Bm(m≥3)be the components ofT− {y}. Assume that xbelongs toB1.

Case 1 : v=y.

Letz B2 andw B3 be two endvertices ofT. Thenvlies on the uniquez-wdetour mono-phonic path. Sincezandwbelong toMxandv /∈Mx, the subgraph induced byMxis not connected, which is a contradiction.

Case 2 : v6=y.

Let v Bi(i 6= 1). Now, choose an endvertex u Bi such that v lies on the y-u detour monophonic path. Let a Bj(j 6= i,1) be an endvertex of T. Then y lies on the u-a detour monophonic path. Hence it follows thatv lies on theu-adetour monophonic path. Since uanda

belong toMxandv /∈Mx, the subgraph induced byMxis not connected, which is a contradiction.

Hence Sx is a minimal connectedx-detour monophonic set ofT. SinceT is a tree, Sx is the unique minimal connectedx-detour monophonic set ofT and socdm+x(T) =n−dm(x, y).

Now, letT be a path. Let xandy be the end vertices ofT.Clearly{y}is the unique minimal

connectedx-detour monophonic set ofT and socdm+x(T) = 1. 2

Corollary 2.10 — For any treeT of ordern≥3,cdm+x(T) =nif and only ifxis a cutvertex of T.

3. BOUNDS ANDREALIZATIONRESULTS FORcdm+x(G)

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These bounds are sharp. For the pathPn (n 2), cdmx(Pn) = 1for an endvertexx in Pn. For any non-trivial tree T with order n 3, cdmx+(T) = n for any cutvertex x inT. Also, for the complete graph Kn(n 2), cdmx(Kn) = cdm+x(Kn) = n−1. The inequalities can be strict. For the graph G given in Figure 2.1, cdmr(G) = 3, cdm+r(G) = 5 and n = 6. Thus

1< cdmr(G)< cdm+r(G)< n.

Theorem 3.1 — LetGbe a connected graph with at least one cutvertex v and letx V(G).

LetSxbe anx-detour monophonic set ofG.Then every component ofG−vcontains an element of Sx∪ {x}.

PROOF: Suppose there is a componentBofG−vsuch thatB contains no vertex ofSx∪ {x}. Thenx∈V−V(B). Letu∈V(B). SinceSxis anx-detour monophonic set, there exists an element y∈Sxsuch thatulies in somex-ydetour monophonic pathP = (x=u0, u1, u2, . . . , u, . . . , un = y)inG. Sincevis a cutvertex ofG, thex-usubpath ofP and theu-ysubpath ofP both containv, it

follows thatP is not a path, which is a contradiction. 2

Theorem 3.2 — Letxbe any cutvertex of a connected graphG. Then

cdm+x(G)3.

PROOF: Let Sx be any connectedx-detour monophonic set ofG. Sincexis a cutvertex of G, G−xhas at least two components. By Theorem 3.1,Sxhas vertices from each component ofG−x. Now, since the induced subgraphG[Sx]is connected,Sx must containx. ThusSx has at least three vertices and socdmx(G)3. Hencecdm+x(G)3. 2

Remark 3.3 : Ifxis not a cutvertex ofG, thenV − {x}is a connectedx-detour monophonic set

ofG. Thuscdm+x(G)≤n−1.Hence ifcdm+x(G) =n, thenxis a cutvertex ofG.

However, the converse is not true. For the graphGgiven in Figure 3.1,cdm+x(G) = 5 < nfor the cutvertexxinG.

Figure 3.1

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Remark 3.4 : Since any connected graphGcontains at least two vertices which are not cutvertices,

there is no graphGof ordernwithcdm+x(G) =nfor every vertexx.

Theorem 3.5 — For any two integerskandnwith1≤k≤nandn≥3, there exists a connected

graphGof ordernwithdm+x(G) =kfor some vertexxinG.

PROOF: We prove this theorem by considering two cases.

Case 1 : 1≤k≤n−1.

LetGbe the graph obtained from the pathPn−k = (u1, u2, . . . , un−k)of ordern−k 1and the complete graphKk of orderkwith vertex set V(Kk) = {w1, w2, . . . , wk}by joining each wi withun−k inPn−k. Letx =u1.NowS ={x, w1, w2, . . . , wk}is the set of all extreme vertices of G. Hence by Theorem 2.3,Sx =S− {x}is the unique minimal connectedx-detour monophonic set ofGand socdm+x(G) =k.

Case 2 : k=n.

LetGbe any tree of ordern. Then by Theorem 2.9(i),cdm+x(G) =nfor any cutvertexxinG.2

For every connected graphG,radm(G) ≤diamm(G). It is shown in [11] that any two positive integers a and b with a b are realizable as the monophonic radius and monophonic diameter,

respectively, of some connected graph. In the following theorem we extend this result.

Theorem 3.6 — For integersr, dandkwith2≤r≤dandk≥2, there exists a connected graph

Gwithradm(G) =r, diamm(G) =dandcdm+x(G) =kfor some vertexxinG.

PROOF: Case 1 : r=d= 2.

LetC4 = (v1, v2, v3, v4, v1)be a cycle of order4and letKk−1be a complete graph of orderk−1 with vertex setV(Kk−1) ={w1, w2, . . . , wk−1}. LetGbe the graph obtained fromC4andKk−1by joining each vertexwiwith the verticesv1, v2andv3. The graphGis shown in Figure 3.2.

It is easily verified that the monophonic eccentricity of each vertex inGis2and soradm(G) =

G

Figure 3.2

Kk−1

v1

v2

v3

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diamm(G) = 2. Letx = v4. Clearlywi(1 i k−1)is not an internal vertex of any detour monophonic path starting fromx. Therefore, eachwi(1≤i≤k−1)must belong to every minimal connectedx-detour monophonic set ofG. It is easily verified thatSx = {w1, w2, . . . , wk−1, v2}is the unique minimal connectedx-detour monophonic set ofGand socdm+x(G) =|Sx|=k.

Case 2 : 2< r=dor2≤r < d.

LetHbe the graph obtained from the cycleCr+2= (v1, v2, . . . , vr+2, v1)and the pathPd−r+1 =

[image:10.612.145.427.328.461.2]

(u0, u1, u2, . . . , ud−r) by identifying the vertex vr+1 in Cr+2 with u0 inPd−r+1 and joining each vertexui(1 i d−r)with vr+2. Now, letGbe the graph obtained from H by adding k−2 new verticesw1, w2, . . . , wk−2and joining eachwiwithv2andvr+2inH. The graphGis shown in Figure 3.3.

It is easily verified that r em(x) d for any vertex x in G. Also em(vr+2) = r and em(v1) = d. Hence radm(G) = r and diamm(G) = d. Now, let x = ud−r and let S = {v1, vr+2, w1, w2, . . . , wk−2}. Since every vertex of Glies on an x-y detour monophonic path for somey ∈Sand the induced subgraphG[S]is connected,S is a connectedx-detour monophonic set

ofG. Also anyz∈Sdoes not lie on ax-udetour monophonic path for anyu∈S−{z}.HenceSis a

minimalx-detour monophonic set ofGand socdm+x(G)≥k. Also, any minimal connectedx-detour monophonic set ofGcontains at mostkvertices and hencecdm+x(G)≤k. Hencecdm+x(G) =k.2

Since any connected x-detour monophonic set is an x-detour monophonic set, it follows that dmx(G)≤cdmx(G)≤cdm+

x(G). Now we have the following realization theorem.

Theorem 3.7 — For any three positive integersj, k andl with 2 j k l, there exists a

connected graphGwithdmx(G) =j, cdmx(G) =kandcdm+x(G) =lfor some vertexxinG.

PROOF: We consider two cases.

Case 1 : 2≤j < k≤l.

G

Figure 3.3

w1

w2

wk−2

v1

v2

vr+2

u0 =vr+1

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LetGbe the graph obtained from the pathPk−j+4 = (u1, u2, . . . , uk−j+4)by addingl−k+j−1 new verticesw1, w2, . . . , wj−2, z1, z2, . . . , zl−k+1 and joining each wi with u1, u2, u3 and u4 and joining eachziwithu1andu4inPk−j+4. The graphGis shown in Figure 3.4. Letx=u1.

We claim thatdmx(G) =j.The extreme vertexuk−j+4 belongs to everyx-detour monophonic set ofG. Also{uk−j+4}is not anx-detour monophonic set ofGand eachwi,1≤i≤j−2,is not an internal vertex of any detour monophonic path starting fromx. Thus everyx-detour monophonic

set of Gcontains S1 = {uk−j+4, w1, w2, . . . , wj−2}. It is clear that nozi lies on anyx-z detour monophonic path for any z S1. ThusS1 is not anx-detour monophonic set ofG. Now, since S2=S1∪ {u3}is anx-detour monophonic set ofG, it follows thatdmx(G) =j.

We now claim that cdmx(G) = k. Since every x-detour monophonic set of G contains S1 andS1 containsuk−j+4 and w1, every connected x-detour monophonic set of G contains the set T ={u4, u5, . . . , uk−j+3}. LetS3 =S1∪T. It is clear that nozilies on anyx-zdetour monophonic path for anyz∈S3. ThusS3is not anx-detour monophonic set ofG. FurtherS4 =S3∪ {u3}is a connectedx-detour monophonic set ofGand hencecdmx(G) =k.

Next, we show that cdm+x(G) = l. ClearlyM = S3∪ {z1, z2, . . . , zl−k+1} is a connected x-detour monophonic set ofG. We claim thatM is a minimal connectedx-detour monophonic set of G. Suppose there exists a proper subsetN ofMsuch thatN is a connectedx-detour monophonic set

ofG. Lets∈ M ands /∈N. Since every connectedx-detour monophonic set ofGcontainsS3, it

follows thats=zifor somei. Now,zidoes not lie on anyx-zdetour monophonic path for any vertex

G

Figure 3.4

u1 u2 u3 u4 u5 u6 uk

−j+3 uk−j+4

w1

w2

wj−2

z1

z2

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z N. Hence it follows thatN is not anx-detour monophonic set ofG, which is a contradiction.

ThusM is a minimal connectedx-detour monophonic set ofGand socdmx+(G)≥ |M|=l. Also, it is clear that there is no minimal connectedx-detour monophonic setM0 ofGwith|M0|> l. Hence cdm+

x(G) =l.

Case 2 : 2≤j=k≤l.

LetGbe the graph obtained from the pathP4 = (u1, u2, u3, u4) by addingl−k+j−1new

verticesw1, w2, . . . , wj−1, z1, z2, . . . , zl−k and joining eachwi with u1, u2, u3 and u4 and joining eachziwithu1andu4.The graphGis shown in Figure 3.5. Letx=u1.

By an argument similar to Case 1, the set S = {w1, w2, . . . , wj−1, u3} is both minimum x-detour monophonic set and minimum connectedx-detour monophonic set of Gand sodmx(G) = cdmx(G) =j. Also,S0={w1, w2, . . . , wj−1, z1, z2, . . . , zl−k, u4}is a minimal connectedx-detour monophonic set ofGwith maximum cardinality and socdm+x(G) =l.

In the following theorem, we construct a graph of prescribed order, monophonic diameter and

upper connected vertex detour monophonic number under suitable conditions.

Theorem 3.8 — Ifn,dandkare positive integers such that2 d n−1, 3 k nand

n−d−k+ 1 0, then there exists a connected graphGof ordernwith monophonic diameterd

andcdm+x(G) =kfor some vertexxinG.

PROOF: We prove this theorem by considering two cases.

G

Figure 3.5

u1 u2 u3 u4

w1

w2

wj−1

z1

z2

(13)

Case 1 : d= 2.

Ifk =n, then for the starG= K1,n1,we haved= 2andcdm+x(G) =nfor the cutvertexx ofG. Now, let3 k < n. LetP3 = (u1, u2, u3)be the path of order 3. Addn−3new vertices v1, v2, . . . , vn−k−1, w1, w2, . . . , wk−2 and join eachwi withu2 and join eachviwithu1, u2 andu3. Also, join eachvi,1 i ≤n−k−2,withvj, i+ 1 j n−k−1. The resulting graphGis shown in Figure 3.6. ThenGis a graph of ordernwith monophonic diameterd = 2. Letx =u1. Now every minimal connectedx-detour monophonic set ofGcontains the set of all extremal vertices S={w1, w2, . . . , wk−2}. FurtherSis not a minimal connectedx-detour monophonic set ofG. Also S∪ {y}, wherey∈V(G)−S, is not a minimal connectedx-detour monophonic set ofG. It is clear

thatS1 =S∪ {u2, u3}is a minimal connectedx-detour monophonic set ofGand socdm+x(G)≥k. Also there is no minimal connected x-detour monophonic set of G with cdm+x(G) > k. Hence cdm+x(G) =k.

Case 2 : 3≤d≤n−1.

Ifk=n, then for any treeTof ordernand diameterd, we havecdm+x(T) =nfor any cutvertex xinT. Now, let3 k < n. LetGbe the graph obtained from the pathPd = (u1, u2, . . . , ud)by addingn−dnew verticesw1, w2, . . . , wk−1, v1, v2, . . . , vn−d−k+1and joining eachwi withu1and joining eachviwithu1 andu3(see Figure 3.7). The graphGhas ordernand monophonic diameter d.

G

Figure 3.6

w

1

w

2

w

k−2

u

1

u

2

u

3

v

1

v

2

(14)

Let x = ud. Every minimal connected x-detour monophonic set of G contains the set of all extreme verticesS = {w1, w2, . . . , wk−1}.Clearly, S is not a minimal connected x-detour mono-phonic set ofGandS∪ {u1}is the unique minimal connectedx-detour monophonic set ofG.Hence cdm+

x(G) =k. 2

ACKNOWLEDGEMENT

The authors are thankful to the Department of Science and Technology for its support through the

Projects SR/S4/MS:427/07 and SR/S4/MS: 570/09. A. P. Santhakumaran is a student of S.

Aru-mugam, P. Titus is a student of A. P. Santhakumaran and P. Balakrishnan is a student of P. Titus. The

first author expresses his sincere thanks to God Almighty for His abundant blessings so that he is part

of this paper involving authors of four consecutive academic generations.

REFERENCES

1. F. Buckley and F. Harary, Distance in graphs, Addison-Wesley, Redwood City, CA, (1990).

2. F. Buckley, F. Harary and L. U. Quintas, Extremal results on the geodetic number of a graph, Scientia,

A2 (1988), 17-26.

3. G. Chartrand, H. Escuadro and P. Zhang, Detour distance in graphs, J. Combin. Math. Combin. Comput.,

53 (2005), 75-94.

4. G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks, 39(1) (2002), 1-6.

5. G. Chartrand, F. Harary, H. C. Swart and P. Zhang, Geodomination in graphs, Bull. Inst. Combin. Appl.,

31 (2001), 51-59.

6. G. Chartrand, G. L. Johns and P. Zhang, Detour number of a graph, Util. Math., 64 (2003), 97-113.

7. F. Harary, Graph theory, Addison-Wesley (1969).

u1 u2 u3 u4 ud−1 ud

v1

v2

vn−d−k+1

w1

w2

wk−1 G

(15)

8. F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modeling,

17(11) (1993), 87-95.

9. A. P. Santhakumaran and P. Titus, Vertex geodomination in graphs, Bulletin of Kerala Mathematics

Association, 2(2) (2005), 45-57.

10. A. P. Santhakumaran and P. Titus, On the vertex geodomination number of a graph, Ars Combin., 101

(2011), 137-151.

11. A. P. Santhakumaran and P. Titus, Monophonic distance in graphs, Discrete Math., Alg. and Appl., 3(2)

(2011), 159-169.

12. A. P. Santhakumaran and P. Titus, A note on monophonic distance in graphs, Discrete Math., Alg. and

Appl., 4(2) (2012), 1250018-(5 pages).

13. P. Titus and P. Balakrishnan, The vertex detour monophonic number of a graph, (Communicated).

14. P. Titus and P. Balakrishnan, The upper vertex detour monophonic number of a graph, Ars Combin., (To

appear).

15. P. Titus and P. Balakrishnan, The connected vertex detour monophonic number of a graph,

Figure

Figure 3.3.

References

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