Table Of ContentDirected Metric Dimension of
Oriented Graphs with Cyclic Covering
Sigit Pancahayani and Rinovia Simanjuntak
4
1 Combinatorial Mathematics Research Group
0 Faculty of Mathematics and Natural Sciences
2 Institut Teknologi Bandung, Bandung 40132, Indonesia
r e-mail: [email protected], [email protected]
p
A
1
Abstract
2
Let D be a strongly connected oriented graph with vertex-set V
]
O and arc-set A. The distance from a vertex u to another vertex v,
d(u,v) is the minimum length of oriented paths from u to v. Sup-
C
pose B = {b ,b ,b ,...b } is a nonempty ordered subset of V. The
1 2 3 k
.
h representationofavertexvwithrespecttoB,r(v|B),isdefinedasa
t vector (d(v,b ),d(v,b ),...,d(v,b )). If any two distinct vertices u,v
a 1 2 k
satisfyr(u|B)(cid:54)=r(v|B),thenB issaidtobearesolvingsetofD. If
m
the cardinality of B is minimum then B is said to be a basis of D
[ and the cardinality of B is called the directed metric dimension of
D.
2
v Let G be the underlying graph of D admitting a Cn-covering. A
9 Cn-simple orientation is an orientation on G such that every Cn in
2 Disstronglyconnected. Thispaperdealswithmetricdimensionsof
9 orientedwheels,orientedfans,andamalgamationoforientedcycles,
0 all of which admitting C -simple orientations.
n
.
1
0 Keywords: directed metric dimension, oriented graphs, oriented wheels,
4 oriented fans, amalgamation of oriented cycles, simple-C orientation.
n
1
:
v
i
X 1 Introduction
r
a
Let D = (V,A) be a strongly connected oriented graph with vertex-set V
andarc-setA. Foru,v ∈V(D),thedistancefromutov,d(u,v),isthemin-
imum length of oriented paths from u to v. Suppose B = {b ,b ,b ,...b }
1 2 3 k
is a nonempty ordered subset of V. The representation of a vertex v with
1
respect to B, r(v|B), is defined as a vector (d(v,b ),d(v,b ),...,d(v,b )). If
1 2 k
any two distinct vertices u,v satisfy r(u|B)(cid:54)=r(v|B), then B is said to be
a resolving set of D. If the cardinality of B is minimum, then B is said to
be a basis of D and the cardinality of B is called the directed metric di-
mension of D, dim(D). This notion was introduced by Chartrand, Raines
and Zhang [5] as an analog to the undirected version of metric dimension
introduced by Slater [13] and Harary and Melter [8]. It is obvious that
not every oriented graph has directed dimension, however necessary and
sufficient conditions for the dimension of an oriented graph to be defined
are still unknown.
Unlike the undirected version, not many results have been known on di-
rected metric dimension. Characterization of graphs with particular di-
rectedmetric dimensionisonlyknown forone-dimensionalorientedgraph.
Theorem 1.1. [5] Let D be a nontrivial oriented graph of order n. Then,
dim(D)=1 if and only if there exists a vertex v in D such that
(i) D admits an oriented Hamiltonian path P with a terminal vertex v
such that id(v)=1 and
(ii) if P in (i) is in the form v ,v ,...,v ,v, then for each pair i,j of
n−1 n−2 1
integer with 1≤i<j ≤n−1, the oriented graph D−E(P) does not
admit the arc in the form (v ,v ).
j i
Other results include directed metric dimension of oriented trees [5], tour-
naments [11], and some Cayley digraphs [3].
In [6], Chartrand, Raines and Zhang defined a parameter called the upper
orientable dimension ofagraphG, ORD(G), whichisthemaximumvalue
of dim(D) among the orientations D of G for which dim(D) is defined.
In this paper we deal with the strongly connected oriented graphs admit-
ting a cycle covering. The graphs under consideration are oriented wheels,
oriented fans, oriented friendship graphs, and amalgamation of oriented
cycles.
A wheel, W , is defined as K +C and a fan, F , is K +P . We shall
n 1 n m,n m n
recallthe(undirected)metricdimensionsofthewheelW andthefanF ,
n 1,n
which are essentially the same on general cases, studied by Buczkowski et
al [1] and Caceres et al [2], respectively.
Theorem 1.2. [1]
(cid:26)
3, n=3,6;
dim(W )=
n (cid:98)2n+2(cid:99), otherwise.
5
2
Theorem 1.3. [2]
1, for n=1;
2, for n=2,3;
dim(F )=
1,n 3, for n=6;
(cid:98)2n+2(cid:99), otherwise.
5
For n≥2, let C ={C |i=1,2,...,n} be a collection of n cycles. The ver-
ti
tex amalgamation of cycles in C, Amal{C }n , is the graph constructed
ti i=1
by joining the cycles in C on a common vertex called the terminal vertex.
The edge amalgamation of cycles in C, Edge−Amal{C }n , is the graph
ti i=1
constructedbyjoiningncyclesonacommonedgecalledtheterminal edge.
Iswadi et al [9] determined the dimension of Amal{C }n and Simanjun-
ti i=1
taket al [12]determinedthedimensionofEdge−Amal{C }t asstated
ni i=1
bellow.
Theorem 1.4. [9] Let Amal{C }n be a vertex amalgamation of n cycles
ti i=1
that consists of n number of odd cycles and n number of even cycles.
1 2
Then,
(cid:26)
n , for n =0;
dim(Amal{C }n )= 1 2
ti i=1 n +2n −1, for n >0.
1 2 2
Theorem 1.5. [12]
t−2≤dim(Edge−Amal{C }t )≤t.
ni i=1
In this paper, we will also consider a generalization of vertex and edge
amalgamations, which is the path amalgamation.
2 Main Result
We start by introducing the notion of simple orientation.
Definition 2.1. Let G be a graph with cyclic covering. An orientation on
G is called C -simple if all directed C s in the resulting oriented graph
n n
are strong.
3
2.1 Directed Metric Dimension of Oriented Wheels
For n ≥ 3, a wheel, W , is defined as K + C , where the vertex c in
n 1 n
K is called the center and the vertices v ,v ,...,v in C is called the
1 1 2 n n
outer vertices. First we shall characterize oriented wheels which admitting
C -simple orientations.
3
Lemma 2.2. There exists a C -simple orientation on W if and only if n
3 n
is even.
Proof. Suppose n is odd. Considering all possible orientations for the edge
cv ,(c,v )or(v ,c)willresultinthelastC inW beingnotstrong. Forthe
1 1 1 3 n
sufficiency, we define the arc set A = {(c,v ),(v ,v ),(v ,v )|i odd}∪
i i i+1 i i−1
{(v ,c)|ieven}∪{(v ,v )}whichyieldsaC -simpleorientationonW .
i 1 n 3 n
Theorem 2.3. If W is a wheel with even n admitting a C -simple orien-
n 3
tation then
(cid:26)
2, for n=4;
dim(W )=
n n −1, for n≥6.
2
Proof. There are only two possibilities of C -simple orientation on W :
3 n
(A) whereA={(c,v ),(v ,v ),(v ,v )|iodd}∪{(v ,c)|ieven}∪{(v ,v )},
i i i+1 i i−1 i 1 n
or
(B) whereA={(c,v ),(v ,v ),(v ,v )|ieven}∪{(v ,c)|iodd}∪{(v ,v )}.
i i i+1 i i−1 i n 1
ThusW doesnotcontainanorientedHamiltonianpathandsobyTheorem
n
1.1, dim(W )>1.
n
For n = 4, let B = {v ,v }, then under orientation A or B, r(c|B) =
1 2
(1,2) or (2,1), r(v |B) = (3,1) or (3,2) and r(v |B) = (2,3) or (1,3),
3 4
respectively. Therefore, dim(W )=2.
4
For n ≥ 6, based on a C -simple orientation on W , we define a partition
3 n
of V(W ): V = {c}, V = {v |d(c,v ) = 1}, and V = {v |d(c,v ) = 2}.
n 0 1 i i 2 i i
Since d(x,y) = 2 for x ∈ V , y ∈ V and d(x,y) = 3 for x,y ∈ V then we
2 1 2
could have at most one vertex in V omitted from a resolving set. Thus
2
dim(W ) ≥ n − 1. For the upper bound, let B be a subset of V of
n 2 2
cardinality n −1 and b ∈ B, then d(v,b) = 1 or 4, for v ∈ V , d(c,b) = 2,
2 1
and d(v,b) = 3, for v ∈ V \B. Thus in r(v|B), v ∈ V , the coordinates
2 1
related to the neighbors of v are 1 and those related to the non-neighbors
of v are 4. Additionally, r(c|B) = (2,2,...,2) and r(v|B) = (3,3,...,3)
4
for v ∈ V \B. Therefore B resolves W and dim(W ) ≤ n −1, and we
2 n n 2
obtain dim(W )= n −1.
n 2
ByLemma2.2, awheelW withnoddisnotC -simple, howeverwecould
n 3
consider a subgraph of W admitting a C -simple; the subgraph is a fan
n 3
F =K +P , where V(P )={v ,v ,...,v }.
1,n 1 n n 1 2 n
Theorem 2.4. If W is a wheel with odd n which contains a fan F
n 1,n
admitting a C -simple orientation then
3
1, for n=5;
n−3, for n≥7, od(c)>id(c) or
dim(W )= 2
n od(c)<id(c) and (v ,v )∈E(W );
n−1, otherwise. n 1 n
2
Proof. For n = 3, it is easy to check that for all 4 possible C -simple
3
orientations on W , we could apply Lemma 1.1 to obtain dim(W )=1.
3 3
For n=5, it is obvious that there is no oriented Hamiltonian path in all 4
possible C -simple orientations on W , and by Lemma 1.1, dim(W ) > 1.
3 5 5
Based on a C -simple orientation on W , we define a partition of V(W ):
3 n n
V = {c}, V = {v |d(c,v ) = 1}, and V = {v |d(c,v ) = 2}. Let B be a
0 1 i i 2 i i
2-subsetofV andb∈B,theninr(v|B),v ∈V ,thecoordinatesrelatedto
2 1
the neighbors of v are 1 and those related to the non-neighbors of v are 4.
Additionally, r(c|B)=(2,2) and r(v|B)=(3,3) for v ∈V \B. Therefore
2
dim(W )≤2.
n
For n ≥ 7, as in n = 5 before, we define a partition V ,V ,V of V(W ).
0 1 2 n
We shall consider two cases separately. (i) If od(c)>id(c) or od(c)<id(c)
and (v ,v ) ∈ E(W ) then d(x,y) = 2 for x ∈ V , y ∈ V and d(x,y) = 3
1 n n 2 1
for x,y ∈ V . Thus we could have at most one vertex in V omitted from
2 2
a resolving set. Thus dim(W ) ≥ |V | − 1. (ii) If od(c) < id(c) and
n 2
(v ,v )∈E(W ) then the distances will be the same as in case (i) except
n 1 n
ford(v ,v ),whichobviouslyis1. ThusatmostonevertexinV alongwith
n 1 2
v couldbeomittedfromaresolvingsetandsodim(W )≥|V |−2. Forthe
n n 2
upper bound, we shall the two afore-mentioned cases. (i) If od(c) > id(c)
or od(c) < id(c) and (v ,v ) ∈ E(W ) then let B be a subset of V of
1 n n 2
cardinality|V |−1andb∈B. Thusd(v,b)=1or4,forv ∈V ,d(c,b)=2,
2 1
andd(v,b)=3,forv ∈V \B. Thereforeinr(v|B),v ∈V ,thecoordinates
2 1
related to the neighbors of v are 1 and those related to the non-neighbors
of v are 4. Additionally, r(c|B) = (2,2,...,2) and r(v|B) = (3,3,...,3)
for v ∈ V \B. In other words, B resolves W and dim(W ) ≤ |V |−1.
2 n n 2
(ii) If od(c) < id(c) and (v ,v ) ∈ E(W ) then let B be a subset of V
n 1 n 2
5
of cardinality |V |−2, where v ∈/ B, and b ∈ B. We then obtain the
2 n
samerepresentationsasincase(i)exceptforr(v |B)wherethecoordinate
n
related to v is 1 and the coordinates related to the other vertices are 3.
1
ThusBresolvesW anddim(W )≤|V |−2. Thiscompletestheproof.
n n 2
2.2 Directed Metric Dimension of Oriented Fans
Afan,F ,isdefinedasK +P ,wheretheverticesc ,c ,...,c inK
m,n m n 1 2 m m
is called the centers.
Theorem 2.5. If F is a fan admitting C −simple orientation then
m,n 3
1 for m=1 and n=2,3,4;
mm,−1, ffoorr mm≥≥22 aanndd nn==23,;4;
dim(F )= m+1, for m≥2 and n=5;
m,n
nnn2−−2+11 m++mm−2−−,21,, fffooorrr nnn eoovddeddn,,,nnn≥≥≥776,,;aanndd oodd((cci))><iidd((cci)),,∀∀ii.;
2 i i
Proof. For m = 1 and n = 2,3,4, all F s have a directed Hamiltonian
m,n
path which satisfies the premises of Theorem 1.1. (See Figure 1).
Figure1: AllpossibleC -simpleorientationsofthefanF ,n=2,3,4. (A
3 1,n
Hamiltonian path is depicted in dotted line.)
For m≥2 and n=2, we have d(c ,v)=d(c ,v) for v ∈P and d(c ,c )=
i j 2 i j
3∀i(cid:54)=j. Thusatmostonecentervertexcouldbeomittedfromaresolving
set and so dim(F )≥m−1. With B ={c |1≤i≤m−1}, it is easy to
m,n i
see that each vertex has distinct representation with respect to B, and so
dim(F )≤m−1.
m,2
6
For the rest of the cases, based on a C -simple orientation on F , we
3 m,n
defineapartitionofV(F ): V ={c ,c ,...,c }, V ={v |d(c,v )=1},
m,n 0 1 2 m 1 i i
and V = {v |d(c,v ) = 2}. For i = 1,2,...,m, d(c ,v) = d(c ,v) for
2 i i i j
v ∈ V ∪ V and d(c ,c ) = 3 ∀i (cid:54)= j; thus at most one center vertex
1 2 i j
could be omitted from a resolving set. Moreover, we have d(x,y) = 2 for
x ∈ V , y ∈ V and d(x,y) = 3 for x,y ∈ V , and so we could have at
2 1 2
most one vertex in V omitted from a resolving set. Thus dim(F ) ≥
2 m,n
(m−1)+(|V |−1) = m+|V |−2. For the upper bound, let B be a
2 2 1
subset of V of cardinality m−1 and B be a subset of V of cardinality
0 2 2
|V |−1. For b ∈ B , we have d(c ,b ) = 3, for c ∈ V \B , d(v,b ) = 2,
2 1 1 i 1 i 0 1 1
for v ∈ V , and d(v,b ) = 1, v ∈ V . For b ∈ B , we have d(c ,b ) = 2,
1 1 2 2 2 i 2
d(v,b ) = 1 or 4, for v ∈ V , and d(v,b ) = 3, for v ∈ V \ B . Let
2 1 2 2 2
B = B ∪ B , then in r(v|B), v ∈ V , the coordinates related to the
1 2 1
centers are 2, to the neighbors of v are 1, and to the non-neighbors of v
are 4. Additionally, r(c |B) = (3,3,...,3,2,2,...,2) for c ∈ V \B and
i i 0 1
r(v|B)=(1,1,...,1,3,3,...,3) for v ∈V \B. Therefore B resolves F
2 m,n
and dim(F ) ≤ m+|V |−2, and we obtain dim(F ) = m+|V |−2.
m,n 2 m,n 2
Depends on parity of n and degrees of the centers, |V | is either n, or
2 2
n−1, or n−1 +1. Substituting |V | with appropriate number proves the
2 2 2
theorem.
2.3 2-Dimensional Oriented Wheels and Fans
Here we shall construct an orientation on W and F in such a way that
n 1,n
the directed metric dimension is 2.
Theorem 2.6. For n ≥ 3, there exists an orientation on the wheel W
n
such that dim(W )=2.
n
Proof. For 3 ≤ n ≤ 7 we have a C -simple orientation for W such that
3 n
dim(W ) = 2 as in Theorem 2.3 and 2.4. For n ≥ 8, consider a subgraph
n
F withvertex-set{c,v ,v ,...,v }. Applyasimple−C orientationwith
1,7 1 2 7 3
od(c) > id(c) on the F . We then apply an orientation such that there
1,7
exists an oriented path from v to v , and lastly, for v ∈V(W )\V(F ),
1 7 n 1,7
set an arc from c to v.
The next step is to determine the directed dimension of W under such
n
an orientation. Choose B = {v ,v }. Then the representations of all
2 4
vertices are r(c|B) = (2,2),r(v |B) = (1,4),r(v |B) = (0,3),r(v |B) =
1 2 3
(1,1),r(v |B) = (3,0),r(v |B) = (4,1),r(v |B) = (3,3),r(v |B) = (4,4),
4 5 6 7
andr(v |B)=(i−7+4,i−7+4), for8≤i≤n. SinceW doesnotadmit
i n
an oriented Hamiltonian path then dim(W )=2.
n
7
Theorem 2.7. Forn≥3, thereexistsanorientationonthefanF such
m,n
that dim(F )=2.
m,n
Proof. For n = 3, let the arc set of F be A = {(v ,v ),(v ,v ),(v ,c),
1,3 1 2 3 2 1
(v ,c),(c,v )}. It is clear that B = {v ,v } is a resolving set. Since
2 3 2 3
such orientation does not admit an oriented Hamiltonian path, we have
dim(F ) = 2. For n = 4, let the arc set of F be A = {(v ,v ),(v ,v ),
1,3 1,4 1 2 3 2
(v ,v ),(v ,c),(v ,c),(c,v ),(v ,c)}. Thus we have B = {v ,v } as a re-
3 4 1 2 3 4 2 3
solving set and, since F does not admit an oriented Hamiltonian path,
1,4
we have dim(F )=2.
1,4
for n≥5, consider a subgraph F with vertex-set {c,v ,v ,v ,v }. Give
1,4 1 2 3 4
F an orientation as in the previous case. Then apply an orientation
1,4
such that there exists an oriented path from v to v and, lastly, for v ∈
n 4
V(W )\V(F ), set an arc from u to v. Now, let B = {v ,v }. The
n 1,7 2 3
representationofallverticesarer(c|B)=(2,1),r(v |B)=(1,2),r(v |B)=
1 2
(0,2),r(v |B)=(0,1),r(v |B)=(3,2), and r(v |B)=(i−4+3,i−4+2),
3 4 i
for 5 ≤ i ≤ n. Since F does not admit the directed Hamiltonian path,
1,n
then dim(F )=2.
1,n
Figure 2: The 2-dimensional oriented W and F .
n 1,n
So far we have obtained directed dimension for wheels and fans admit-
ting C -simple orientations and construct 2-dimensional oriented wheels
3
and fans. A question arisen after this study is whether the C -simple ori-
3
entation gives the maximum possible dimension, i.e. the ORD(W ) and
n
ORD(F ),oristhereanorientationwhichprovideslargerdimension? An-
n
other interesting question is whether all values up to the ORD(W ) or
n
ORD(F ) are achievable.
n
Problem 2.8. For n≥3, determine ORD(W ) and ORD(F ).
n m,n
8
Problem 2.9. Let k be an integer in [2,ORD(W )] and k be an inte-
W n F
ger in [2,ORD(F )]. Does there exist an orientation on W such that
m,n n
dim(W ) = k ? Similarly, does there exist an orientation on F such
n W m,n
that dim(F )=k ?
m,n F
2.4 Directed Metric Dimension of Amalgamation of
Oriented Cycles
Let C ={C |i=1,2,...,t} be a collection of t strongly oriented cycles. A
ni
path amalgamation of cycles in C, denoted by P −Amal{C }t , where
x ni i=1
2≤x≤min{|V(C )|}−1, is the graph constructed by joining the cycles
ni
in C on a common path or order x called the terminal path. We shall
denote the vertices is P −Amal{C }n as follow: P = v v ...v v
x ti i=1 x 1 2 x−1 x
and C = v v ...v vi vi ...vi vi v . We shall use the notation
ni 1 2 x x+1 x+2 ni−1 ni 1
P for the path vi vi ...vi v .
ni x+1 x+2 ni−1 ni
Figure 3: A P −Amal{C }t of strongly oriented cycles.
x ni i=1
Theorem 2.10. Let P − Amal{C }t be a path amalgamation of t
x ni i=1
strongly oriented cycles then dim(P −Amal{C }t )=t−1.
x ni i=1
Proof. IfanarbitrarysetS doesnotcontainavertexinP andP where
ni nj
i(cid:54)=j, then r(vi |S)=r(vj |S) and so S is not a resolving set. Thus each
ni nj
resolvingsetneedstocontainavertexfromeachpairofP andP . This
ni nj
results in dim(P −Amal{C }t )≥t−1.
x ni i=1
Next, consider B = {v1 ,v2 ,...,vt−1 }. For a vertex u ∈ P , we have
n1 n2 nt−1 ni
d(u,vi ) ≤ n −1, while for a vertex u ∈ P ,j (cid:54)= i, d(u,vi ) = d(u,v )+
ni i nj ni 1
(x−1)+(n −x)=d(u,v )+n −1 and for a vertex u∈P , d(u,vi )=
i 1 i x ni
9
d(u,v )+n −x. This guarantees that each vertex will have distinct rep-
x i
resentation with respect to B. Thus B is a resolving set and dim(P −
x
Amal{C }t )≤t−1. This completes the proof.
ni i=1
IfthepathP isoforder1thenwehaveavertexamalgamationofstrongly
x
oriented cycles and if it is of order 2 then we have an edge amalgamation
of strongly oriented cycles. Thus the following corollaries hold.
Corollary2.11. LetAmal{C }t beavertexamalgamationoftstrongly
ni i=1
oriented cycles then dim(Amal{C }t )=t−1.
ni i=1
Corollary 2.12. Let Edge−Amal{C }t be an edge amalgamation of
ni i=1
t strongly oriented cycles then dim(Edge−Amal{C }t )=t−1.
ni i=1
References
[1] P.S.Buczkowski,G.Chartrand,C.Poisson,P.Zhang,Onk-dimensional
graphs and their bases, Periodica Math. Hungarica 46 (2003) 9-15.
[2] J. Caceres, C. Hernando, M. Mora, I. M. Pelayo, M. L. Puertas, C.
Seara, On the metric dimension of some families of graphs, Elec. Notes
Disc. Math. 22 (2005) 129-133.
[3] Melodie Fehr, Shonda Gosselin, Ortrud R. Oellermann, The metric di-
mension of Cayley digraphs, Disc. Math. 306 (2006) 31-41.
[4] G. Chartrand, L. Eroh, M. Johnson, O.R. Oellermann, Resolvability in
Graphs and the Metric Dimension of a Graph, Disc. Appl. Math. 105
(2000) 99-113.
[5] G. Chartrand, M. Raines, P. Zhang, The Directed Distance Dimension
of Oriented Graphs, Math. Bohemica 125 (2000) 155-168.
[6] G. Chartrand, M. Raines, P. Zhang, On the dimension of oriented
graphs, Util. Math. 60 (2001) 139-151.
[7] R. Diestel, Graph Theory, Third Edition, Springer-Verlag, New York
(2005).
[8] F. Harary and R.A. Melter, On The Metric Dimension of a Graph, Ars
Combin., 2 (1976) 191-195.
[9] H. Iswadi, E.T. Baskoro, A.N.M. Salman, R. Simanjuntak, The Metric
DimensionofAmalgamationofCycles,FarEastJ.Math.Sci.41(2010)
19-31.
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