Applied Mathematics, 2011, 2, 705-710
doi:10.4236/am.2011.26093 Published Online June 2011 (http://www.SciRP.org/journal/am)
On Eccentric Digraphs of Graphs
Medha Itagi Huilgol*, Syed Asif Ulla S., Sunilchandra A. R.
Department of Mat hematics, Ba ngal ore University, Bangalore, India
Received March 11, 2011; revised April 9, 2011; accepted April 12, 2011
Abstract
The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G. A vertex v is an
eccentric vertex of vertex u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a
graph (digraph) G is the digraph that has the same vertex as G and an arc from u to v exists in ED(G) if and
only if v is an eccentric vertex of u in G. In this paper, we have considered an open problem. Partly we have
characterized graphs with specified maximum degree such that ED(G) = G.
Keywords: Eccentric Vertex, Eccentric Degree, Eccentric Digraph, Degree Sequence,Eccentric Degree
Sequence
1. Introduction
A directed graph or digraph G consists of a finite
nonempty set called vertex set with vertices and
edge set of ordered pairs of vertices called arcs;
that is represents a binary relation on

VG


EG

EG


VG

.
Throughout this paper, a graph is a symmetric digraph;
that is, a digraph G such that implies
If

is an arc, it is said that u is
adjacent to v and also that v is adjacent from u. The set of
vertices which are from (to) a given vertex v is denoted
 
,uv EG

vu 

E G

,.




,uv
by and its cardinality is the out-degree
 
NuNu



of v [in-degree of v]. A walk of length k from a vertex u
to a vertex v in G is a sequence of vertices
012 1
=,,,,,=
kk
uuuuu uv



,uu
such that each pair
1ii is an arc of G. A digraph G is strongly
connected if there is a u to v walk for any pair of vertices
u and v of G. The distance d(u,v) from u to v is the length
of a shortest u to v walk. The eccentricity e(v) of v is the
distance to a farthest vertex from v. If

,=distu veuvudi
GGGG 
we say that v is an eccentric
vertex of u. We define whenever there is
no path joining the vertices u and v. The radius rad(G)
and diameter diam(G) are minimum and maximum ec-
centricities, respectively. As in [2], the sequential join
123 k of graphs 12 is the
graph formed by taking one copy of each of the graphs
12 and adding in additional edges from each
vertex of i to each vertex in 1i, for 11

,=stu v
,,,
k
GG G
,,,
k
GG G
GG ik

.
Throughout this paper, means G and H are ='G H
isomorphic. The reader is referred to Buckley and Harary
[2] and Chartrand and Lesniak [3] for additional, un-
defined terms.
Buckley [4] defines the eccentric digraph


ED G of
a graph G as having the same vertex set as G and there is
an arc from u to v if v is an eccentric vertex of u. The
paper [4] presents the eccentric digraphs of many classes
of graphs including complete graphs, complete bipartite
graphs, antipodal graphs and cycles and gives various
interesting general structural properties of eccentric
digraphs of graphs. The antipodal digraph of a digraph G
denoted by


A
G, has the vertex set as G with an arc
from vertex v in


A
G if and only if v is an antipodal
vertex of u in G; that is



.m G,=u v

dist dia

This
notion of antipodal digraph of a digraph was introduced
by Johns and Sleno [5] as an extension of the definition
of the antipodal graph of a graph given by Aravamudhan
and Rajendran [6]. It is clear that
A
G is a subgraph
of


ED G, and


=

A
GEDG
if and only if G is
self centered.
In [7] Akiyama et al. have defined eccentric graph of a
graph G, denoted by e, has the same set of vertices as
G with two vertices u and v being adjacent in e if and
only if either v is an eccentric vertex of u in G or u is an
eccentric vertex of v in G, that is
GG






,=min ,
GG
distu veev
*Part of this paper [1] was presented at the International Conference on
Emerging Trends in Mathematics and Computer Applications, MEPCO
Schlenk Engineering College, Sivakasi, India (Dec. 2010) and had
a
p
p
eared in the Proceedin
g
s of the same.
G
u. Note that is the
underlying graph of
e
G


GED .
In [8] Boland and Miller introduced the concept of the

M. I. HUILGOL ET AL. 706
if a
eccentric digraph of a digraph. In [9] Gimbert et al. have
proved that=
e
Gnd only if G is self-centered.
In the same paper, the authors have characterized eccen-
tric digraphs in terms of complement of the reduction of
G, denoted by

EDG
G Given a digraph G of order n, a
reduction of G, doted by G, is derived from G by en
removing all its arcs incident from vertices with out-
degree 1n. Note that )(GED is a subgraph of .

G
In [9] , Gmbert et al. died on the behaviofihave stuour
se
. The iterated se-
G and t
. Basic Results
this section we list some results which are quite
ph has at least
on
2. There exists no directed cycle in an eccen-
tr
a directed cycle with edge being
di
dges cther
ed
quences of iterated eccentric digraphs. Given a positive
integer 2k , the th
k iterated eccentric digraph of G
is writte
 

1kk
DGED EDG
, where

0=ED GG
entric with the smallest
integer numbers 0>p
 and 0t such that

=
tp
EDGED  We cathe period of
antities are denoted p(G) and t(G)
respectively. In [8,10] Boland et al. have discussed many
interesting results about eccentric digraphs. Also they
have listed open problems about these graphs. One of
these open problems is being discussed mainly in this
paper. We have characterized graphs with specified
maximum degree such that

=EDGG .
n asE
of ecc
il of G; t
=
1
ED
digra

t
G.
qu
and
 
=GED G
phs concerns

ll p
hese
quence
the ta
2
In
evident for eccentric digraphs of graphs.
Remark 1. Since every vertex in a gra
e vertex at eccentric distance it follows that every
vertex in an eccentric digraph will have out degree at
least one.
Remark
ic digraph.
Let C be uv
e 1 rectedom uv as shown below in Figur .
The other ean be bidirectional. If all the o
fr
ges except uv
 are bidirectional then a symmetric
edge 1 vy indicates the equality of eccentric values of v
and ikewise
 
1
y. L

 
==
n
eccyecc x
12
==ecceccyy. Also edge
x
usymmHence,
 
=ecc ueccx

eccecc u. This co
arc u

to tha
is
existence of
etric.
1
the directed

. So also,
 
===veccyecc xtradicts the
v as u being tail
has less eccentricity as comparedt of .
The same argument can be extended to a directed
n
v
cy
r a graph
to
symmetric cycle having a
pe
cle with more than one directed arc, as the eccen-
tricities go on increasing in the same direction.
The above two conditions are not sufficient fo
be an eccentric digraph.
For example consider a
ndant vertex adjacent to one of the vertices on the
cycle. The pendant vertex having in-degree zero and
out-degree one as in Figure 2.
Vertex i
x
is at eccentric distance from. Let
be
u
p
v
adjaceno u and lying on the eccentric ath con-
necting u and i
t t
x
. All the vertices in the graph except
i
xare adistanc atmost 1n from u, where n is
eccentricity of u. This implies v bng adjacet to
u will have eccentricity n. Buv lying on the
metric circle can have eccentricity 1n. There-
fore the above graph cannot be an eccentriaph.
Also we give a counter example for a problem gi
t e
the
sym
ei n
t
c digr
ven
in
2.2 (p. 41) [2]: If G is self-centered
w
[2] , as follows:
Problem 3, Ex.
ith radius 2 , then G is self-centered with radius 2.
Counter Exampleonsider 7
C, join the vertice a
: Cst
distance 2 in .
7
C Let G be tresulting graph with he


=2rad G. Considering G, we observe that 7 GC

;
that is G is self-centered of radius 3.
. Grahs with Isomorphic Eccen3ptric
case of undirected graphs, Buckley [4] proved that the
Digraph
In
Figure 1. Directed cycle C.
Figure 2. Directed cycle with unidirectional edge u-xi.

M. I. HUILGOL ET AL.707
ecc - entric digraph of a graph G is equal to its com
plement, ()= ,ED GGif and ly if G is either a
self-centeradius two or G is the union of
2k complete graphs. In [9], Gimbert et al. have
that the eccentric digraph

ED G is symmetric
if and only if G is self centered.
Here we are looking at gr
on
red graph of
aphs which have their
ec
or which
Odd cycles are graphs with minimum
nu
radius
3,
proved
centric digraphs isomorphic to themselves. So by
Gimbert’s result these graphs are self-centered graphs. In
this section we consider self-centered, undirected graphs.
The following observations are easily justified.
Remark 3. Odd cycles is a class of graphs f

=.GG
.
ED
Remark 4
mber of edges and maximum eccentricity on given
number of vertices such that

=.EDGG
Remark 5. For a self-centewithred graph G 
the complement G is self-centered with radius equ
to two. Hence al GG, and GG, and\


ED G is
isomorphic to araph o subgf G. F
th urther,ing Buckley’s result [4], we can say at by us

==EDGGG .
That is if


=,ED GED G then G

=EDG.
k 6. Complete graphs is another class of
grRemar
aphs for which


=EDGG .
Remark 7. It is easy to see that for graphs upto order
7, the only graphs for which


=EDGG , are
23455677
,,, ,,,,
K
KKKCKKC
ic g
.
hraphs havRemark 8. Two isomorpe their eccen-
tr
hown inn Figure 3, we give a pair
of
raph with ra-
di
ic digraphs isomorphic, but the converse need not be
true always.
As an example, as s
non-isomorphic, self-centered graphs with same
eccentricity having one eccentric digraph.
Lemma 9. Let G be a self-centered g
us 2, then


=EDG if and only if G is self-
complementary.
Proof. Given self-cente
G
red graph be self-com-
pl
G
ementary with radius 2. Then by Buckley’s cha-
racterization theorem [4],


==DGGG . Conversley,
consider a self-centered gra with

=EDGG . Then,

E
ph of radius 2,
=,EDGG that is

==GEDG
Lemma 10.
G
. Het.  nce tresul with eccen-
tr
he
All self-centered graphs G
icity greater than or equal to 3 with Ghaving
=1, =1,periodtail satisfies th condition e


=ED G
a self-centered graph
G.
with eccen- Proof. Let G be
tricity 3.Then G is self-centered graph with eccen-
tricity al to 2. nce, equHe


==;EDGGG that is


2
EDGED G. If Gand tail has ,
that is
=1period 1=
 
ED then
2
ED GG



2
ED GGGED . 
But


2
EDGED G
Figure 3. ED(G) = ED(H).





2=ED GEDGEDGG

. Hence the result.
For connected graph to be isomorphic to

G


ED G
ld not be
rathy
ce
mber of
the necessary c shou
nique eccentric node as defined by Parthasa
and Nandakumar [11], for
ssary condition is that thfor ev
v of a gr
fr o
asing order.
ondition is that the graph
ugraph
. Also
e
th

GEDG, the ne-
egree say ery vertex of d
k, there must exist anoer vertex with k nu
eccentric vertices. This can be defined as eccentric
degree of a vertex.
Definition 11. For a vertex aph G is
defined to be the number of vertices at eccentric distance
om v. Also the eccentric degree sequencef a graph is
defined as a listing of eccentric degrees of vertices
written in non-incre
So for


=ED GG, the eccentric degree sequen of
G should be equal to the degree sequence of G. But this
condition is not sufficient as seen in the example below,
depicted as Figure 4. Here both G and
ce


ED G have
their degree seqence and eccentric degree sequences as


49
3,2 , but


ED GG.
Next, we consider self-centered graphs with given
maximum degree


G. By [2],

22,Gpr
for a self-centered graph G with radius
r
. Our next
result shows that there is no possibility of having a graph
with


=GEDG, with


=22prG

.
Proposition 12. There does not exist a graph G with


=22Gpr

, hat

such t

=G. ED G
Proof. Let G be a self-centered graph with


=22Gpr

 Let

uVG such that
deg u implies =2pr2

. Partitntoion the vertex set i sets lying

M. I. HUILGOL ET AL. 708
then 1
r
x
will have u as the oy eccentric veex, a
ntradiction.
imilar contradiction is arrived if 1
r
nl rt
co
S
x
is adjacent to
both 2
r
x
and 2
1r
x

Same argum can be appliedn
case of
ent i
Figure 4. G and ED(G) having same eccentric degree se-
quence and degre e se quence, but ED(G) ≇ G.
at distance fromand name them as i u , 1
i
A
ir

.
Since 2, =2deg pur 2. As G is 2=
1 rpA
self-centered, it cannot contain a cut-vertex and hence
2, 2
i
A
ir. Hence, 22r

vertices are needed to
onsideration,
tices. Hence
satisfyh under c
with
the cnditionsf t
t possible to constr
o ohe grap
t a grap
but we have

22=23ppr r ver,
it is nouch


=EDGG ,
with

=22Gpr.
connectecentered graph G with

=21Gpr is isomorphic to its eccentric digraph
if and only i is of the form

22
21,2
p
pr   with structure
Theorem 13. Ad self-
f its degree sequence



121222
2
,
pr
K
KFKKHKH
KHr times


 
where
F
is the gjoining one vertex of raph obtained by
21
to
pr
K one vertex of 2
K
and remaining 2pr

vertices to one vertex of 2
K
, and
H
is the 1
f
actor

removed from successive 22
K
K.
Proo Let G be a self-centered graph with

=21Gpr. Let u be a vertex of G with
21p r. As seen in the above Lemma
f.
each =deg u,
i
A
should contain at least two vertices each, for G to
satisfy


=GEDG, wicenth
s
self-terednnot
ining
ess and
ma be
22r vertir
at least two ie
ing unique entric node graph. Since the re
e to be ditributed into 1r sets with
ach set, it follows that each i
ecc
ces a
n
A
has
vertices.
Let ththe set k
exactly two
e vertices in
A
be labelled 1
k
x
and 2
k
x
for all k, 2kr. Since G has no cutvertex, 1
1r
x

be adjacent to at least one vertexr should of
A
, let us
say, to 1
r
x
and similarly let 2
1r
x
 be adjacent to 2 r
x
.
Degree of 1
r
x
is at le
22
ast tw be adj to
of
o, so it canacent any
1
,
rr
x
x or both.
2
r
x
. Therefore, 1
r
x
and 2
r
x
are mutually
adjahave degree at least two. There are two ps
1
P an2
ij
P where
ii
cath
t be equal to j.
r cases are proved as follows:
Cla 1: The vertices
ent t
d
o
Othe
im
o
1111 1jx 1 1
11122331 1
11 11111
11 211
=, , , , , , ,,,,,,
,,,,,,,;,=1
kk kk
kk rrrrr
Puexexexeex
ex xexexij

 


and
22 2
2112
,, , ,
iji ij
Puexe
233
222222
11 11
22222
211
=, , ,,
,,,,,,,
,,,,;2,21,
kkkkkk
rrrrr
xex
exexex
xexexijpr
 
 


i need n
1
1k
x
 are adjacent to either 1 k
x
2
k
x
Suppose, 1
r
x
is adjacent to 2
1r
x

,
or , bu
Proof o
x bel
t not both; for.
f the claimvertex eac
verteonging to
all
: Since
, 21kkr
G has no cut h
k
A
, 21kr, is adjacent to at
least one vertex in 1k
A

and 1k
A
. Without loss gene-
rality let each vertex 1
1k
x

be adjacent to 1
k
x
, for all
, 21krk

. Let us cder k onsi
A
, with vertices 1
k
x
and 2
k
x
. Let 1
1k
x

be adjacent to 2
k
x
aloithng w 1
k
x
.
Then eccentricity of 1
1k
x

can be at most 1r

as any
vertex lying on the p 1an be at most at a
distance 1rath P or 2
ij
P c

from 1k
1
x

. And any vertex on the path
ij
P2 can be at most at a distance of 2r fr 2
k
om
x
. In
any case if 1
1k
x

is adjacent to both2
k
x
and 1
k
x
then
ceases to be seentered, a contradiction and hence
proof of the claim.
Claim 2: Each , 2
i
G
thelf-c
1
A
kr

 is independent, that
is,
,
2
=
i
A
K
Phe re
.
roof of the claim: First we prove tsult for 1r
A

.
If the verti1
1r
ces
x

and 2
1r
x
 beloing to1r
ng
A

are
adjacent then 1
1r
x

will have u as the only eccentric
vertex, a contradicti proat on ves th12r
A
=K

For any other
.
k
A
, 2kr2

, if 1
k
x
and 2
k
x
are
adjacent then, eccentricity of 1
k
x
and 2
k
x
will t
most r − 2 as vertices on P or ij P2 will be at distance
at most r − 2 m 1
k
be a
1
fro
x
or 2
k
x
and hence for all k,
2i
Claim 3: 1
21, =krAK.
2
x
and 2
2
x
do not haneve comon ors
in
m ighb
1
A
.
Proof of the claim: As in case of the veces of 1r rti
A

,
the vertices 1
2
x
and 2
2
x
of 2
A
will have their
eccentricity equal to 1r

ifhey have a common t
neighbor in 1
A
, h
1
ence . th claime
Claim 4: 2
x
is adjacent to 1
1
x
and 2
2
x
is adjacent to
all er p-othtices of
Proof of taim : If
2r ver
he clA.
1
1
2
x
is adjacent to

M. I. HUILGOL ET AL.709

=
1=1 , 1<<2
ki
k
k
x
ip r, then two vertices 2
r
x
and 1r
2
x

have eccentric degree k + 1, but only one vertex 1
2
x
has
del dicgree equa 1k, a contration.
If 1
2
to
x
and 2
2
x
have dege k each ,that is if
2k1
re
=2 1pr, then r
x
, r
2
x
, 1r
1
x
 and 2 1r
x

have
eccentric degree equal to+ 1, but only two k vertices1
2
x
and 2
2
x
ha
Hence,
ve de
1
gree k + 1.
2
x
is adjacent to 1 x1not a and 1 x2
1, 1<<2
i djacent to
x
ip r.
Finding the etric degree of all ccenrtices of G we
see that, ecc. deg
ve

1=2, 11
k
x
kr and ecc.deg

1=2
k
y, except for 1
1r
x
, where

11
kk
yNx.
Silarly, ecc. deg

2=
mi2,
k 11
x
kr and ecc.deg

2
y2 
gr

2

=2
k, where. Hence, the eccentric de-

kk
yNx
ence of G
ee sequence of G is

22
21,2
p
pr   , which is same
as that of degree sequ
Claim 5:
.
121
=
p
r
.
Proof Wet for any twvertices
AK
of the claim: show thao
1
t
x
and 1
s
x
where 1,ts  are not , 1, 2tsp r
eed to consadjacentn. Here we ider two possibilities:
Case 1):

1
12
t
x
Nx

and 2

2
1
s
x
Nx
Case 2):

2
12
t
x
Nx and 1
s

2
2
x
Nx
In case 1), 1
t
x
will have only one eccentric vertex, a
contradiction.
In case 2), deg


1
t
x
,deg 13then only the ver-
tic 1

s
x
es r
x
or 2
r
x
of r
A
and 1
1r
x
 or 2 1r
x
 of 1r
A

the cartof along
pa r , si
n have ve as ecices
2
ij
PA1 nc
centric vertices,
ths 1 P oe 1
2
x
and 2
2
x
t sh
co
do noare a
mmon neighr in 1 bo
A
, as claimed in 3.
By Claim 4at 1 , we see th2
x
is adjacent to 1
1
x
and
2
x
2th vertis adjacent to all oer pices of 1
− 2r
A
, implies
t when degrees t tha of 1
x
and 1
s
x
c
a
emcedegree other
i ve
he
fof
Cgiv
m
emark 1tices
re changed by mak-
ing th adjant, thccentriny o
vertex of does not change, hence we will not be able to
get

=E G a contradion pros the claim.
Referring all the claims we conclude tat
e ef a
G

DG, ct
is of th
rm defined in the statement othe theorem.
onverse is easy to observe that if G is as en in the
statement of the theore, then

=GG . 
In the next result we consider a particular case of
graphs with

=EDGG , that is, odd cycles.
R a labelled 21 ,1
n
Cn
, two ver
ED
4. In
,
ij
v are at eccentric distance in

21n
ED C, if and
on
v
ly if

,=
2
Gij
dv v or
n1n
2

.
Remark 15. For unlabelled itti odd cycles,eraons of
can be packed into

21n
ED C n
K
,

13
1=
22
ns 
,
nED G

, whereas,

2
rad Cn
1= n.
In case of labelled odd cyclesnce of ED(G)'s
e packed into K,pif the permutatumber of
the seque
can bion on p n
defined by vertices





1=1, 2=2, =1,2,3,,;
1=2 2,=1,2,,
ffirii r
fir iir



 23,
since there are
is a product of three cyclicns of length
respectively.
permutatio,r1, r,
Proposition 16. There exists a self-centered graph G,
such that


=EDGG , containing an odd cycle.
Proof. Let be
p
C be a cycle, whose vertices are
labeled as . Let . Define a
function onices
1, 2, 3,
the setvert
,p
of

=1,2,3, ,Sp
of
p
C as










  
21 =1, 112.fipiip
1=1, 2=232,112,ffipi ip 
Now, we partition of set of vertices of
p
C into


123
,, m S, , SSS ere, wh






12 1
= 2,, 2, SSf S
where n is the least positteg
2
1,=2, 2,, 2
n
f f
ive iner such that


2=
12 n
f whereas


2
n
f is obtained by applyi ng
f
on n2, times. Similarly,

 


2
=, , ,, ,
m
m
Slflfl fl
S S


1
123 1
,
m
lS S


 
where is the least positive integer such that m


=
1m
f
ll
. It is clear that 123 =
m
SSSS S
and =, 1,ij j,i
jm i
SS


 . Now, for each vertex
i
S we fine sets of vertices not in of de
p
C by
 


2
=, ,,
i
ii ii
jj jj
Sflflfln for each , and
whose adjacencies are to the ertices of
i
=1,2,3,j v
p
C defined by: If


g
f
l, when is adjacent 1lp
to


h
f
l, then,


g
ij
f
l is adjacent to




glNf ,


h
ij
f
l and


h
ij
f
l is adjac to




,
hg
i
lf lent j;
othese,
Nf


g
ij
f
lis adjacent to


g
Nfl and


h
rwi ij
f
l
is acent to dja




h
Nfl . So we get a self-ce
graph with
ntered
radius


12p,isfying

=ED GG
sa
vertices
t, as
l on
p
C and respective

, 1
ij
m
f
llp

,
e same distance from ther vertices in the graph. By
Remark 15, we get
hav o


G. 
Fng is an ee of a self-cente
ED G
ollow
radius own iing
i xampl
4, as shn Figure 5, satisfy
red graph with


ED GG

,
with 9 C, asSo base.


=1,2,S ,9 and




45
,,SS
12
=,,
p
VC SS, where
3
S






12
=1,=2,54, =7.SS S
345
,8, =, =3,6,9S S

M. I. HUILGOL ET AL. 710
Figure 5. ED(G) ≅ G.
For we have , 14,
i
Si








  
    
1112132111
1 231
2222233331
231
323341424
23123
=1, =1, =1; =2,5,8,
=2,5,8 , =2,5,8; =4,
=4, =4; =7, =7, =7.
SSSS
SSS
SSSSS
3
4. References
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e Institute of Com-
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[1] M. A. S. Ulla an
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