J. Indones. Math. Soc.
Vol. 20, No. 1 (2014), pp. 1–9.
ON THE DETOUR AND VERTEX DETOUR HULL
NUMBERS OF A GRAPH
A.P. Santhakumaran
1and S.V. Ullas Chandran
21Department of Mathematics Hindustan University Hindustan Institute of Technology and Science, Padur, Chennai - 603 103, India
2Department of Mathematics, Mahatma Gandhi College, Kesavdasapuram, Pattom P.O., Thiruvanathapuram - 695 004, India
Abstract.For verticesxandyin a connected graphG, the detour distanceD(x, y)
is the length of a longestx−ypath inG. Anx−ypath of lengthD(x, y) is an x−y detour. The closed detour intervalID[x, y] consists ofx, y, and all vertices
lying on somex−y detour ofG; while forS ⊆ V(G), ID[S] =Sx,y∈SID[x, y].
A setS of vertices is a detour convex set ifID[S] =S. The detour convex hull
[S]Dis the smallest detour convex set containingS. The detour hull numberdh(G)
is the minimum cardinality among subsetsS ofV(G) with [S]D = V(G). Letx
be any vertex in a connected graph G. For a vertex y in G, denote byIG[y]x,
the set of all vertices distinct fromx that lie on somex−y detour ofG; while forS ⊆ V(G), ID[S]x =Sy∈SID[y]x. For x /∈ S,S is anx-detour set ofG if
ID[S]x=V(G)− {x}and anx-detour set of minimum cardinality is thex-detour
numberdx(G) ofG. Forx /∈S,S is anx-detour convex set ifID[S]x=S. The
x-detour convex hull ofS, [S]x
D is the smallestx-detour convex set containingS.
Thex-detour hull numberdhx(G) is the minimum cardinality among the subsets
SofV(G)− {x}with [S]x
D=V(G)− {x}. In this paper, we investigate how the
detour hull number and the vertex detour hull number of a connected graph are affected by adding a pendant edge.
Key words and Phrases:d Detour, detour number, detour hull number, x-detour number,x-detour hull number.
2000 Mathematics Subject Classification: 05C12
Abstrak. Misalkanx dan y berada di graf terhubung G, jarak detour D(x, y)
adalah panjang dari lintasanx−yyang terpanjang diG. Lintasanx−y dengan panjangD(x, y) adalah suatu detourx−y. Interval detour tertutupID[x, y] memuat
x, ydan semua titik yang berada dalam suatu detourx−ydariG; sedangkan untuk S ⊆V(G), ID[S] = Sx,y∈SID[x, y]. Himpunan titikS adalah suatu himpunan
konveks detour jikaID[S] =S. Konveks hull detour [S]Dadalah himpunan konveks
detour terkecil yang memuatS. Bilangan hull detour dh(G) adalah kardinalitas minimum diantara sub-subhimpunanSdariV(G) dengan [S]D=V(G). Misalkan
xadalah suatu titik di graf terhubungG. Untuk suatu titiky diG, dinotasikan denganIG[y]x, himpunan dari semua titik berbeda darixyang terletak pada suatu
detourx−y dariG; sedangkan untukS⊆V(G),ID[S]x=Sy∈SID[y]x. Untuk
x /∈S,Sadalah suatu himpuan detour-xdariGjikaID[S]x=V(G)−{x}dan suatu
himpuan detour-xdengan kardinalitas minimum adalah bilangan detour-x dx(G)
dariG. Untuk x /∈ S,S adalh suatu himpunan detour-x konveks jika ID[S]x =
S. Konveks hull detour-x dariS, [S]x
D adalah himpunan konveks detour-x yang
memuatS. Bilangan hull detour-x dhx(G) adalah kardinalitas minimum diantara
sub-subhimpunan S dari V(G)− {x} dengan [S]x
D = V(G)− {x}. Pada paper
ini, kami memeriksa pengaruh penambahan sisi anting dari suatu graf terhubung terhadap bilangan hull detour dan bilangan hull detour titik.
Kata kunci: Detour, bilangan detour, bilangan hull detour, bilangan detour-x, bi-langan hull detour-x.
1. Introduction
By agraph G= (V, E), we mean a finite undirected graph without loops or multiple edges. The order and size ofGare denoted bynandmrespectively. For basic definitions and terminologies, we refer to [1, 6]. For vertices x and y in a nontrivial connected graphG, thedetour distanceD(x, y) is the length of a longest x−y path inG. Anx−y path of length D(x, y) is an x−y detour. It is known that the detour distance is a metric on the vertex setV(G). Thedetour eccentricity of a vertexuis eD(u) =max{D(u, v) :v ∈V(G)}. The detour radius, radD(G) of
G is the minimum detour eccentricity among the vertices of G, while the detour diameter, diamD(G) of Gis the maximum detour eccentricity among the vertices
of G. The detour distance and the detour center of a graph were studied in [2]. The closed detour interval ID[x, y] consists of x, y, and all vertices lying on some
x−ydetour ofG; while forS⊆V(G),ID[S] =Sx,y∈SID[x, y];Sis adetour set if
ID[S] =V(G) and a detour set of minimum cardinality is thedetour numberdn(G)
3
A setS of vertices of a graph G is adetour convex set if ID[S] = S. The
detour convex hull [S]D ofS is the smallest detour convex set containing S. The
detour convex hull ofScan also be formed from the sequence{Ik
D[S], k≥0}, where
IDp[S] is thedetour convex hull [S]D and we callpas the detour iteration number
din(S) of S. A set S of vertices of G is a detour hull set if [S]D = V(G) and
a detour hull set of minimum cardinality is the detour hull number dh(G). The detour hull number of a graph was introduced and studied in [11].
For the graphGgiven in Figure 1, and S ={v1, v6},ID[S] =V − {v7}and
I2
D[S] =V. Thus S is a minimum detour hull set ofGand sodh(G) = 2. SinceS
is not a detour set and S∪ {v7} is a detour set of G, it follows from Theorem 1.2 that dn(G) = 3. Hence the detour number and detour hull number of a graph are different. Note that the setsS1 ={v1, v2} and S2 ={v2, v3, v4, v5, v7}are detour convex sets in G. Letxbe any vertex of G. For a vertex y in G, IG[y]x denotes
Figure 1. GraphGwithdh(G) = 2 and dn(G) = 3
the set of all vertices distinct fromxthat lie on somex−y detour ofG; while for S ⊆V(G),ID[S]x=Sy∈SID[y]
x. It is clear thatI
D[x]x=φ. Forx /∈S, S is an
x-detour set ifID[S]x =V(G)− {x}and an x-detour set of minimum cardinality
is thex-detour numberdx(G) ofG. Any x-detour set of cardinalitydx(G) is the
minimum x-detour setor dx-setof G. The vertex detour number of a graph was
introduced and studied in [9].
LetGbe a connected graph andxa vertex inG. LetS be a set of vertices in G such that x /∈ S. Then S is an x-detour convex set if ID[S]x = S. The
x-detour convex hull of S, [S]x
D is the smallest x-detour convex set containing S.
Thex-detour convex set can also formed from the sequence{Ik D[S]
this sequence must be constant. Letpxbe the smallest number such thatIDpx[S] x=
the x-detour iteration number dinx(S) of S. The set S is an x-detour hull set if
[S]x
D = V(G)− {x} and an x-detour hull set of minimum cardinality is the
x-detour hull numberdhx(G) ofG. Anyx-detour hull set of cardinalitydhx(G) is the
minimum x-detour hull setordx-hull set ofG.
Figure 2. G
Table 1. x-detour numbers andx-detour hull numbers ofGin Figure 2
It is clear that every minimumx-detour hull set of a connected graphGof ordern contains at least one vertex and at mostn−1 vertices. Also, since every x-detour set is ax-detour hull set, we have the following proposition. Throughout this paper G denotes a connected graph with at least two vertices. The following theorems will be used in the sequel.
Theorem 1.1. [11] Let Gbe a connected graph. Then
(i) Each detour extreme vertex of Gbelongs to every detour hull set of G. (ii) No cut vertex of Gbelongs to any minimum detour hull set ofG.
Theorem 1.2. [9]Each end vertex ofGother thanx(whether xis an end vertex or not) belongs to every minimumx-detour set ofG.
Theorem 1.3. [10]Letxbe a vertex of a connected graphG. LetSbe anyx-detour hull set ofG. Then
(i) Each x-detour extreme vertex of Gbelongs toS.
(ii) Ifvis a cut vertex ofGandC a component ofG−v such thatx /∈V(C), then S∩V(C)6=∅.
(iii) No cut-vertex of Gbelongs to any minimumx-detour hull set of G.
Theorem 1.4. [10]For any vertexxin a connected graph Gof ordern,dhx(G)≤
n−eD(x).
2. Graphs of Order nwith Vertex Detour Hull Number n−1,n−2
and n−3
5
Proof. Suppose that G =K2. Then dhx(G) = 1 = n−1. The converse follows
from Theorem 1.4.
Theorem 2.2. Let Gbe a connected graph of order n≥3. Then dhx(G) =n−2 for every vertex xinGif and only if G=K3.
Proof. Suppose that G=K3. Then it is clear thatdhx(G) = 1 =n−2 for every
vertex xin G. Conversely, suppose thatdhx(G) =n−2 for every vertex xinG.
Then by Theorem 1.4,eD(x)≤2 for every vertexxinG. It follows from Theorem
2.1 that eD(x) 6= 1 for every vertex x in G. Thus eD(x) = 2 for every vertex x
in G; or the vertex set can be partitioned intoV1 andV2 such that eD(x) = 1 for
x ∈ V1 and eD(x) = 2 for x ∈ V2. Thus either radD(G) = diamD(G) = 2; or
we have radD(G) = 1 and diamD(G) = 2. This implies that either G =K3 or
G =K1,n−1. If G =K1,n−1, then by Theorem 1.3, dhx(G) = n−1 for the cut
vertexxanddhy(G) =n−2 for any end vertexy inG, which is a contradiction to
the hypothesis. HenceG=K3.
Theorem 2.3. Let Gbe a connected graph of order n≥2. Then G=K1,n−1 if and only if the vertex setV(G)can be partitioned into two setsV1andV2 such that dhx(G) =n−1for x∈V1 anddhy(G) =n−2 for y∈V2.
Proof. Suppose thatG=K1,n−1. Thendhx(G) =n−1 for the cut vertexxin G
anddhy(G) =n−2 for any end vertexyinG. Conversely, suppose that the vertex
setV(G) can be partitioned into two setsV1 andV2 such thatdhx(G) =n−1 for
x∈V1; and we havedhy(G) =n−2 fory∈V2. Then by Theorem 1.4,eD(x) = 1 for
eachx∈V1 and eD(y) = 1 oreD(y) = 2 for each y∈V2. It follows from Theorem
2.1 thateD(y) = 2 for somey∈V2. HenceradD(G) = 1 anddiamD(G) = 2. Thus
G=K1,n−1.
Theorem 2.4. Let Gbe a connected graph of order n ≥5. Then G is a double star or G=K1,n−1+e if and only if the vertex set V(G)can be partitioned into two sets V1 andV2 such that dhx(G) =n−2 for x∈V1 anddhy(G) =n−3 for
y∈V2.
Proof. Suppose thatG is a double star orG=K1,n−1+e. Then it follows from Theorem 1.3 that dhx(G) =n−2 ordhx(G) =n−3 according to whether xis a
cut vertex of Gor not. Conversely, suppose that dhx(G) =n−2 for x∈V1 and
dhx(G) =n−3 for x∈ V2. Then by Theorem 1.4,eD(x)≤3 for everyxand so
diamD(G)≤3. It follows from Theorem 2.1 that G6=K2and so diamD(G)≥2.
IfdiamD(G) = 2, then Gis the starK1,n−1and by Theorem 2.3,dhx(G) =n−1
or dhx(G) =n−2 for every vertexx. This is a contradiction to the hypothesis.
Now, suppose that diamD(G) = 3. IfGis a tree, then Gis a double star. IfGis
is a contradiction. Thus cir(G) = 3. Also, if G contains two or more cycles, then it follows that diamD(G) ≥ 4. Hence G contains a unique triangle, say
C3 :v1, v2, v3, v1. Sincen ≥5, at least one vertex ofC3 has degree at least 3. If there are two or more vertices ofC3 having degree at least 3, thendiamD(G)≥4,
which is a contradiction. Thus exactly one vertex ofC3 has degree at least 3 and it follows thatG=K1,n−1+e. This completes the proof.
3. Detour and Vertex Detour Hull Numbers and Addition of A Pendant Edge
In this section we discuss how the detour hull number and the vertex detour hull number of a connected graph are affected by adding a pendant edge toG. Let G′ be a graph obtained from a connected graphGby adding a pendant edgeuv,
whereuis not a vertex ofGandv is a vertex ofG.
Theorem 3.1. If G′ is a graph obtained from a connected graph G by adding a
pendant edgeuv at a vertexv of G, thendh(G)≤dh(G′)≤dh(G) + 1.
minimum detour hull set of G′
. Then by Theorem 1.1, u ∈ S′
G′, then by induction hypothesis, we
have x ∈ Il
from induction hypothesis that y, z∈ Il
D[S]G. Also, since ID[y, z]G′ =ID[y, z]G,
we have x ∈ IDl+1[S]G. Let y = u or z = u, say z = u. Then y 6= u and so
by induction hypothesis,y ∈Il
D[S]G. Sincev is a cut vertex ofG′, it follows that
x∈ID[y, v]G′ =ID[y, v]G. Also, sincev∈S⊆Il
D[S]G, it follows thatx∈IDl+1[S]G.
Hence the proof of the claim is complete by induction. Now, sinceS′
is a minimum
follows from the above claim thatIr
D[S]G =V(G). ThusS is a detour hull set of
7
Remark 3.2. The bounds for dh(G′) in Theorem 3.1 are sharp. Let G′ be the
graph obtained from the graphGin Figure 3, by adding a pendant edge at one of its end vertices. Thendh(G′) =dh(G) = 2. If G′ is obtained fromG by adding a
pendant edge at one of its cut vertices, thendh(G′) =dh(G) + 1.
Figure 3. GraphGwithdh(G′) =dh(G) + 1
Theorem 3.3. Let G′ be a graph obtained from a connected graphG by adding a
pendant edge uv at a vertex v of G. Then dh(G) = dh(G′) if and only if v is a
vertex of some minimum detour hull set of G.
Proof. First, assume that there is a minimum detour hull setSofGsuch thatv∈S. LetS′ in the proof of Theorem 3.1, we can prove that S is a detour hull set of G. Since
|S|=|S′|
as in Theorem 3.1, it is straight forward to verify that In
D[S]xG ⊆IDn[S
′]x
G′ for all
Ir
be a graph obtained from a connected graphG by adding a pendant edgeuv at a vertexv of G. Then dhu(G′) =dhv(G).
Proof. LetSbe a minimumv-detour hull set ofG. Thenv /∈S. As in Theorem 3.1, it is straight forward to prove thatIn
D[S]
As in Theorem 3.1, it is straight forward to prove that In
D[T]uG′ − {v} ⊆ IDn[T]vG
Proof. The proof is similar to Theorem 3.1.
Theorem 3.7. Let G be a connected graph and x any vertex of G. Let G′
be a graph obtained from Gby adding a pendant edgeuv at a vertexv 6=xofG. Then dhx(G) =dhx(G′) if and only ifv belongs to some minimum x-detour hull set of
G.
9
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