Artikel Meta Ilafiani
ON THE PARTITION DIMENSION OF CAVEMAN GRAPH,
GENERALIZED PETERSEN GRAPH, AND Km ∗2 Kn GRAPH
Meta Ilafiani, Tri Atmojo Kusmayadi
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Sebelas Maret University
Abstract. Let G be a connected graph with vertex set V (G) = {v1 , v2 , . . . , vn } and
edge set E(G) = {e1 , e2 , . . . , en }. Those vertices are divided into k-partition, denoted by
S1 , S2 , . . . , Sk . Let Π = {S1 , S2 , . . . , Sk } be an ordered k-partition. The representation
for every vertex V (G) of Π is a minimum distance of a vertex to other vertices, denoted
by r(v|Π) = (d(v, S1 ), d(v, S2 ), . . . , d(v, Sk )). If every vertex has distinct representation,
Π is called a resolving k-partition. Minimum cardinality of k-partition of V (G) is called
by partition dimension of G, denoted by pd(G). In this research, we determine partition
dimension of caveman graph C(n, k), generalized Petersen graph P (n, k), and Km ∗2 Kn
graph.
Keywords : Partition dimension, caveman graph, generalized Petersen graph, Km ∗2 Kn
graph.
1. Introduction
There are many concepts in graph discussed by researchers. One of those
concepts is partition dimension. This concept is introduced by Chartrand et al. [4]
in 1998. Partition dimension of G or pd(G) is minimum cardinality of a resolving
k-partition of V (G).
The concept of partition dimension has been applied frequently in some graph
classes. In 1998, Chartrand et al. [4] studied about partition dimension of path
graph, complete graph, and cycle graph. Then, Asmiati [2] found partition dimension
of amalgamation of stars graph in 2012. Thereafter, in 2016, Apriliani [1] determined
partition dimension of antiprism graph, mongolian tent graph, and stacked book
graph. Also in 2016, Dewi [5] determined partition dimension of some graph classes,
such as lollipop graph, generalized Jahangir graph, and Cn ∗2 Km graph. In this
research, we determine partition dimension of caveman graph C(n, k) for n, k ≥ 3,
generalized Petersen graph P (n, k) for n ≥ 5, k = 2 and n ≥ 8, k = 3, and Km ∗2 Kn
graph for m, n ≥ 3.
2. Partition Dimension
The following definition and lemma are about partition dimension given by
Chartrand et al. [4].
Definition 2.1. Let G be a connected graph. For a subset S of V (G) and a vertex
v of G, the distance d(v, S) between v and S is defined as d(v, S) = min{d(v, x)|x ∈
S}. For an ordered k-partition Π = {S1 , S2 , . . . , Sk } of V (G) and a vertex v of
G, the representation of v with respect to Π is defined as the k-vector r(v|Π) =
(d(v, S1 ), d(v, S2 ), . . . , d(v, Sk )). The partition Π is called a resolving partition if the
1
ON THE PARTITION DIMENSION . . .
M. Ilafiani, T. A. Kusmayadi
k-vectors r(v|Π), v ∈ V (G), are distinct. The minimum k for which there is a
resolving k-partition of V (G) is the partition dimension pd(G) of G.
Lemma 2.1. Let G be a connected graph, then
(1) pd(G) = 2 if only if G = Pn for n ≥ 2 and
(2) pd(G) = n if only if G = Kn .
3. Main Results
3.1. Partition Dimension of Caveman Graph.
Watts [6] defined the caveman graph denoted by C(n, k) as a graph formed
by modifying set of k-cliques or caves with removing one edge from each clique and
using it to connect to a neighboring clique such that all n cliques form a single,
1
unbroken loop. The vertex set of C(n, k) is V (Cn,k ) = {v11 , v21 , . . . , vk−1
, u11 , v12 ,
2
n
v22 , . . . , vk−1
, u21 , . . . , vk−1
, un1 } with k ≥ 3 and n ≥ 3. The values of n are divided
into three types, they are n = 3m, n = 3m + 1, and n = 3m + 2 for m ∈ N.
Lemma 3.1. Let G be C(n, 4) graph and a subgraph of C(n, k) for n ≥ 3, k ≥ 5.
Let Π = {S1 , S2 , . . . , Sk−1 } be a resolving partition of C(n, k). If d(a, w) = d(vbc , w)
with a ∈ V (G), vbc ∈ V (C(n, k)) − V (G), w ∈ V (C(n, k)) − {a, vbc }, 2 ≤ b ≤ k − 3,
0 < c ≤ n, then a and vbc belong to distinct elements of Π.
Proof. Let Π = {S1 , S2 , . . . , Sk−1 } be a resolving partition of C(n, k) and let d(a, w) =
d(vbc , w). Suppose a and vbc are in same elements of Π, say Si . Then, d(a, Si ) =
d(vbc , Si ) for 0 < i ≤ k − 1 and so r(a|Π) = r(vbc |Π). Consequently, Π is not
a resolving partition for C(n, k). Hence, a and vbc belong to distinct elements of
Π.
Theorem 3.1. Let C(n, k) be caveman graph for n ≥ 3 and k ≥ 3, then
{
3,
for k = 3, 4;
pd(C(n, k)) =
k − 1,
for k ≥ 5.
Proof. The proof is divided into four cases by the values of k and n.
(1) For k = 3 and n = 3m with m ∈ N.
Let Π = {S1 , S2 , S3 } be a resolving partition of C(n, k) with S1 = {vsj , up1 }
for 1 ≤ s ≤ k − 1, 0 < j ≤ ⌈ n3 ⌉, 0 < p ≤ n3 , S2 = {vsl , uq1 } for 1 ≤ s ≤ k − 1,
⌈ n3 ⌉ < l ≤ ⌈ n3 ⌉ + m, n3 < q ≤ n3 + 1, S3 = {vsh , ur1 } for 1 ≤ s ≤ k − 1,
⌈ n3 ⌉ + m < h ≤ n, n3 + 1 < r ≤ n. Then, we obtain the representations of
every vertex in V (C(n, k)) with respect to Π are
{
{
(2j, 0, 2m − 2j + 3),
a = 1;
(0, 2m − 2j + 3, 2j),
a = 1;
l
j
r(va |Π) =
r(va |Π) =
(0, 2m − 2j + 2, 2j − 1),
(2j − 1, 0, 2m − 2j + 2), a = 2.
a = 2.
2
2017
ON THE PARTITION DIMENSION . . .
r(val |Π) =
{
(2m − 2j + 3, 2j, 0),
(2m − 2j + 2, 2j − 1, 0),
M. Ilafiani, T. A. Kusmayadi
(0, 2m − 2p + 1, 2j), b = p;
a = 1;
(2j, 0, 2m − 2p + 1),
r(ub1 |Π) =
a = 2.
(2j, 0, 2m − 2p + 1),
b = q;
b = r.
Since the representations of every vertex in V (C(n, k)) for k = 3 and
n = 3m with respect to Π are distinct, Π is a resolving partition with three
elements.
(2) For k = 3 with n = 3m + 1 and n = 3m + 2, m ∈ N.
Let Π = {S1 , S2 , S3 } be a resolving partition of C(n, k) with S1 = {vsj , up1 } for
1 ≤ s ≤ k − 1, 0 < j ≤ ⌈ n3 ⌉, 0 < p ≤ ⌈ n3 ⌉ − 1, S2 = {vsl , uq1 } for 1 ≤ s ≤ k − 1,
⌈ n3 ⌉ < l ≤ ⌈ n3 ⌉ + m, ⌈ n3 ⌉ − 1 < q ≤ ⌈ n3 ⌉ + m, S3 = {vsh , ur1 } 1 ≤ s ≤ k − 1,
⌈ n3 ⌉ + m < h ≤ n, ⌈ n3 ⌉ + m < r ≤ n. We divide the representations of every
vertex in V (C(n, k)) with respect to Π into two subcases according to the
values of n.
(a) For n = 3m + 1.
The representations of every vertex in V (C(n, k)) with respect to Π are
{
{
(2l − 2m − 1, 0, 4m − 2l + 5), a = 1;
(0, 2m − 2j + 4, 2j),
a = 1;
l
j
r(va |Π) =
r(va |Π) =
(0, 2m − 2j + 3, 2j − 1),
r(vah |Π)
=
{
(2l − 2 − 2m, 0, 4m − 2l + 6),
a = 2.
(6m − 2h + 5, 2h − 4m − 2, 0),
a = 1;
(6m − 2h + 4, 2h − 4m − 3, 0),
a = 2.
r(ub1 |Π)
=
a = 2.
(0, 2m − 2p + 2, 2p),
b = p;
(2q − 2m − 1, 0, 4m − 2q + 3), b = q;
(6m − 2r + 3, 2r − 4m − 2, 0),
b = r.
(b) For n = 3m + 2.
The representations of every vertex in V (C(n, k)) to Π are
{
{
(2l − 2m − 1, 0, 4m − 2l + 5), a = 1;
(0, 2m − 2j + 4, 2j),
a = 1;
l
j
r(va |Π) =
r(va |Π) =
(0, 2m − 2j + 3, 2j − 1),
r(vah |Π)
=
{
a = 2.
(2l − 2 − 2m, 0, 4m − 2l + 6),
a = 2.
(0, 2m − 2p + 2, 2p),
b = p;
a = 1;
b
(2q − 2m − 1, 0, 4m − 2q + 3), b = q;
r(u1 |Π) =
a = 2.
(6m − 2r + 5, 2r − 4m − 2, 0), b = r.
(6m − 2h + 7, 2h − 4m − 2, 0),
(6m − 2h + 6, 2h − 4m − 3, 0),
It can be checked that the representations of every vertex in V (C(n, k))
with respect to Π are distinct. Therefore, Π is a resolving partition with three
elements.
(3) For k = 4 and n ≥ 3.
Let Π = {S1 , S2 , S3 } be a resolving partition of C(n, k) with S1 = {vsj , up1 }
for 1 ≤ s ≤ k − 1, 0 < j ≤ ⌈ n3 ⌉, 0 < p ≤ ⌈ n3 ⌉ − 1, S2 = {vsl , uq1 } for
1 ≤ s ≤ k − 1, ⌈ n3 ⌉ < l ≤ ⌈ n3 ⌉ + m,⌈ n3 ⌉ − 1 < q ≤ ⌈ n3 ⌉ + m, S3 = {vsh , ur1 }
for 1 ≤ s ≤ k − 1, ⌈ n3 ⌉ + m < h ≤ n, ⌈ n3 ⌉ + m < r ≤ n. We divide the
representations of every vertex in V (C(n, k)) with respect to Π into three
subcases based on the values of n.
3
2017
ON THE PARTITION DIMENSION . . .
M. Ilafiani, T. A. Kusmayadi
(a) For n = 3m with m ∈ N.
The representations of every vertex in V (C(n, k)) with respect to Π are
a = 1;
a = 1;
(2j, 0, 2m − 2j + 3),
(0, 2m − 2j + 3, 2j),
l
j
(2j − 1, 0, 2m − 2j + 2), a = k − 2;
(0, 2m − 2j + 2, 2j − 1), a = k − 2; r(va |Π) =
r(va |Π) =
(0, 2m − 2j + 2, 2),
r(vah |Π) =
(2m − 2j + 3, 2j, 0),
(2m − 2j + 2, 2j − 1, 0),
(6m − 2q + 2, 2, 0),
(2, 0, 2m − 2l + 2),
a = 2.
a = 2.
a = 1;
(0, 2m − 2p + 1, 2j), b = p;
b
(2p, 0, 2m − 2p + 1), b = q;
a = k − 2; r(u1 |Π) =
(2p, 0, 2m − 2p + 1), b = r.
a = 2.
(b) For n = 3m + 1 with m ∈ N.
The representations of every vertex in V (C(n, k)) with respect to Π are
a = 1;
a = 1;
(0, 2m − 2j + 4, 2j),
(2l − 3, 0, 4m − 2l + 5),
j
l
r(va |Π) =
(0, 2m − 2j + 3, 2j − 1), a = k − 2; r(va |Π) =
(2l − 2m − 2, 0, 4m − 2l + 6), a = k − 2;
(0, 2m − 2j + 3, 2j),
r(vah |Π)
=
a = 2.
(2l − 2m − 1, 0, 2l − 2m − 4),
(4m − 2h + 4, 2h − 4m − 2, 0), a = 1;
(4m − 2h + 5, 2h − 2m − 2, 0), a = k − 2;
(6m − 2h + 4, 2h − 4m − 2, 0), a = 2.
r(ub1 |Π)
=
(0, 2m − 2p + 2, 2p),
a = 2.
b = p;
(2q − 2m − 1, 0, 4m − 2q + 3), b = q;
(6m − 2r + 3, 4m − 2r − 2, 0), b = r.
(c) For n = 3m + 2 with m ∈ N.
The representations of every vertex in V (C(n, k)) with respect to Π are
a = 1;
a = 1;
(0, 2m − 2j + 4, 2j),
(2l − 3, 0, 4m − 2l + 5),
j
l
r(va |Π) =
(0, 2m − 2j + 3, 2j − 1), a = k − 2; r(va |Π) =
(2l − 2m − 2, 0, 4m − 2l + 6), a = k − 2;
(0, 2m − 2j + 3, 2j),
r(vah |Π)
=
a = 2.
(2l − 2m − 1, 0, 2l − 2m − 4),
(6m − 2h + 7, 2h − 4m − 2, 0), a = 1;
(6m − 2h + 2, 2h − 4m − 3, 0), a = k − 2;
r(ub1 |Π)
=
(0, 2m − 2p + 2, 2p),
b = p;
(2q − 2m − 1, 0, 4m − 2q + 3), b = q;
(6m − 2h + 2, 2h − 4m − 2, 0), a = 2.
(6m − 2r + 5, 6m − 2r + 4, 0),
We obtain that the representations of every vertex in V (C(n, k)) for
k = 4 and n ≥ 3 with respect to Π are distinct. Then, Π is a resolving
partition with three elements.
(4) For k ≥ 5 and n ≥ 3.
Let Π = {S1 , S2 , . . . , Sk−1 } be a resolving partition of C(n, k). By using
c
} for 0 < i ≤ k − 4, 1 ≤
Lemma 3.1, we obtain the partition as Si = {vi+1
p
j
c ≤ n; Sk−3 = {vs , u1 } − Si for 1 ≤ s ≤ k − 1, 0 < j ≤ ⌈ n3 ⌉, 0 < p ≤ 1,
Sk−2 = {vsl , uq1 } − Si for 1 ≤ s ≤ k − 1, ⌈ n3 ⌉ < l ≤ ⌈ n3 ⌉ + m, 1 < q ≤ 2,
Sk−1 = {vsh , ur1 } − Si for 1 ≤ s ≤ k − 1, ⌈ n3 ⌉ + m < h ≤ n, 2 < r ≤ 3. We
divide the representations of every vertex in V (C(n, k)) with respect to Π
into two subcases under the values of n.
4
a = 2.
2017
b = r.
ON THE PARTITION DIMENSION . . .
M. Ilafiani, T. A. Kusmayadi
(a) For n = 3m with m ∈ N.
The representations of every vertex in V (C(n, k)) with respect to Π
j
c
l
c
are r(vi+1
|Π) = (d(vi+1
, Si ), 1, 2m − 2j + 2, 2j), r(vi+1
|Π) = (d(vi+1
, Si ), 2j, 1, 2m −
h
c
2j + 2), r(vi+1
|Π) = (d(vi+1
, Si ), 2m − 2j + 2, 2j, 1), r(v1j |Π) = (1, . . . , 0, 2m − 2j +
j
j
3, 2j), r(vk−2
|Π) = (1, . . . , 0, 2m − 2j + 2, 2j − 1), r(vk−3
|Π) = (1, . . . , 0, 2m − 2j +
l
2, 2), r(v1l |Π) = (1, . . . , 2j, 0, 2m − 2j + 3), r(vk−2
|Π) = (1, . . . , 2j − 1, 0, 2m − 2j +
l
2), r(vk−3
|Π) = (1, . . . , 2, 0, 2m − 2l + 2), r(v1h |Π) = (1, . . . , 2m − 2j + 3, 2j, 0),
h
h
r(vk−2
|Π) = (1, . . . , 2m − 2j + 2, 2j − 1, 0), r(vk−3
|Π) = (1, . . . , 6m − 2j + 2, 2, 0),
r(up1 |Π) = (1, . . . , 0, 2m − 2p + 1, 2p), r(uq1 |Π) = (1, . . . , 2p, 0, 2m − 2p + 1), r(ur1 |Π) =
c
c
(1, . . . , 2p, 0, 2m − 2p + 1). We obtain d(vi+1
, Si ) = 0 for vi+1
∈ Si and
c
c
d(vi+1
, Si ) = 1 for vi+1
∈
/ Si .
(b) For n = 3m + 1 with m ∈ N.
The representations of every vertex in V (C(n, k)) with respect to Π
j
c
l
are footnotesize r(vi+1
|Π) = (d(vi+1
, Si ), 1, 2m − 2j + 3, 2j),r(vi+1
|Π) =
c
h
c
, Si ), 6m −
|Π) = (d(vi+1
, Si ), 2l − 2m − 1, 1, 4m − 2l + 4), r(vi+1
(d(vi+1
j
j
2h + 5, 2h − 4m − 2, 1),r(v1 |Π) = (1, . . . , 0, 2m − 2j + 4, 2j),r(vk−2
|Π) =
j
(1, . . . , 0, 2m−2j+3, 2j−1), r(vk−3 |Π) = (1, . . . , 0, 2m−2j+3, 2j),r(v1l |Π) =
l
|Π) = (1, . . . , 2j, 0, 4m − 2l + 6),
(1, . . . , 2l − 3, 0, 4m − 2l + 5), r(vk−2
l
r(vk−3 |Π) = (1, . . . , 2l − 2m − 1, 0, 2l − 2m − 4),r(v1h |Π) = (1, . . . , 4m −
h
2h + 5, 2h − 2m − 2, 0),r(vk−2
|Π) = (1, . . . , 4m − 2h + 5, 2h − 2m − 2, 0),
j
r(v1 |Π) = (1, . . . , 6m − 2h + 4, 2h − 4m − 2, 0),r(up1 |Π) = (1, . . . , 0, 2m −
2p + 2, 2p),r(uq1 |Π) = (1, . . . , 2q − 2m − 1, 0, 4m − 2q + 3), r(ur1 |Π) =
c
c
(1, . . . , 6m−2r+3, 4m−2r−2, 0). We obtain d(vi+1
, Si ) = 0 for vi+1
∈ Si
c
c
and d(vi+1 , Si ) = 1 for vi+1 ∈
/ Si .
(c) For n = 3m + 2 with m ∈ N.
The representations of every vertex in V (C(n, k)) with respect to Π are
j
c
l
c
r(vi+1
|Π) = (d(vi+1
, Si ), 1, 2m − 2j + 3, 2j),r(vi+1
|Π) = (d(vi+1
, Si ), 2j + 1, 1, 2m −
h
c
|Π) = (d(vi+1
, Si ), 2m − 2j + 3, 2j, 1). r(v1j |Π) = (1, . . . , 0, 2m − 2j +
2j + 2),r(vi+1
j
j
4, 2j),r(vk−2
|Π) = (1, . . . , 0, 2m−2j+3, 2j−1),r(vk−3
|Π) = (1, . . . , 0, 2m−2j+3, 2j).
l
|Π) = (1, . . . , 2l − 2m − 2, 0, 4m − 2l +
r(v1l |Π) = (1, . . . , 2l − 3, 0, 4m − 2l + 5),r(vk−2
l
6),r(vk−3
|Π) = (1, . . . , 2l − 2m − 1, 0, 2l − 2m − 4). r(v1h |Π) = (1, . . . , 6m − 2h +
h
h
7, 2h − 4m − 2, 0),r(vk−2
|Π) = (1, . . . , 6m − 2h + 2, 2h − 4m − 3, 0),r(vk−3
|Π) =
(1, . . . , 6m − 2h + 2, 2h − 4m − 2, 0). r(ub1 |Π) = (1, . . . , 0, 2m − 2p + 2, 2p), r(ub1 |Π) =
(1, . . . , 2q − 2m − 1, 0, 4m − 2q + 3), r(ub1 |Π) = (1, . . . , 6m − 2r + 5, 6m − 2r + 4, 0).
c
c
c
c
We obtain d(vi+1
, Si ) = 0 for vi+1
∈ Si and d(vi+1
, Si ) = 1 for vi+1
∈
/ Si .
5
2017
ON THE PARTITION DIMENSION . . .
M. Ilafiani, T. A. Kusmayadi
Since the representations of every vertex in V (C(n, k)) for k ≥ 5 and
n ≥ 3 with respect to Π are distinct, Π is a resolving partition with (k − 1)elements.
This completes the proof of the theorem.
3.2. Partition Dimension of Generalized Petersen.
Biggs [3] defined generalized Petersen graph denoted by P (n, k) is a 3-regular
graph with 2n vertices x0 , x1 , . . . , xn−1 , y0 , y1 , . . . , yn−1 and edges {xi , yi }, {xi , xi+1 },
{yi , yi+k }, for all i ∈ {0, 1, . . . , n − 1}, where the subscripts are reduced modulo n.
Theorem 3.2. Let P (n, k) be a generalized Petersen graph for (n ≥ 5, k = 2) and
(n ≥ 8, k = 3), then pd(P (n, k)) = 4.
Proof. Let Π = {S1 , S2 , S3 , S4 } be a resolving partition with S1 = {y0 , x0 }, S2 =
{y1 , x1 }, S3 = {ya , xa } for 2 ≤ a ≤ ⌈ n2 ⌉ − 1, and S4 = {yb , xb } for ⌈ n2 ⌉ ≤ b ≤ n − 1.
The representations of every vertex in V (P (n, k)) for (n ≥ 5, k = 2) and
(n ≥ 8, k = 3) with respect to Π are r(y0 |Π) = (0, 2, 1, a), r(y1 |Π) = (2, 0, 1, 1), r(yi |Π) =
(d(yi , S1 ), d(yi , S2 ), 0, d(yi , S4 )), r(yk |Π) = (d(yj , S1 ), d(yj , S2 ), d(yj , S3 ), 0), r(x0 |Π) = (0, 1, 2, 1),
r(x0 |Π) = (1, 0, 1, 2), r(xj |Π) = (d(xj , S1 ), d(xj , S2 ), 0, d(xj , S4 )), r(xl |Π) = (d(xl , S1 ), d(xl , S2 ),
d(xl , S3 ), 0) with i = 2, . . . , ⌈ n2 ⌉ − 1, k = ⌈ n2 ⌉, . . . , n − 1, j = 2, . . . , ⌈ n2 ⌉ − 1, j =
⌈ n2 ⌉, . . . , n − 1. We obtain a = 2 for n = 5, k = 2 and a = 1 for n ≥ 6, k = 2, 3 where
d(yi , S1 ), d(yi , S2 ), d(yi , S3 ), d(yi , S4 ), d(xi , S1 ), d(xi , S2 ), d(xi , S3 ), d(xi , S4 ) ≤ ⌊ n2 ⌋ so
that r(y0 |P i) ̸= r(y1 |P i) ̸= . . . ̸= r(yn−1 |P i) ̸= r(x0 |P i) ̸= r(x1 |P i) ̸= . . . ̸=
r(xn−1 |P i). Hence, Π is a resolving partition of P (n, k) graph for (n ≥ 5, k = 2)
and (n ≥ 8, k = 3) with four elements and pd(P (n, k)) = 4.
3.3. Partition Dimension of Km ∗2 Kn Graph.
Km ∗2 Kn graph with m ≥ 3 and n ≥ 3 is built from edge-amalgamation or
amalgamating an edge belongs to Km and other belongs to Kn . Let vertex set
V (Km ) = {v1 , v2 , . . . , vm } and V (Kn ) = {u1 , u2 , . . . , un }. An amalgamated edge
consists of two vertices, denoted by x and y. Vertex x is defined by amalgamating
vertex u1 and v1 . Vertex y is defined by amalgamating vertex u2 and v2 . So, we
obtain the vertex set V (Km ∗2 Kn ) = {x, y, v3 , v4 , . . . , vm , u3 , u4 , . . . , un }.
Theorem 3.3. Let Km ∗2 Kn graph be an edge-amalgamation graph between Km
and Kn for m, n ≥ 3, then
m,
for m, n = 3 and m, n = 4;
n + m − 4, for m = 3, n ≥ 4 and m ≥ 4, n = 3;
pd(Km ∗2 Kn ) =
n + m − 5, for m = 4, n ≥ 5 and m ≥ 5, n = 4;
n + m − 6,
for m ≥ 5, n ≥ 5.
6
2017
ON THE PARTITION DIMENSION . . .
M. Ilafiani, T. A. Kusmayadi
Proof. The proof is divided into four cases.
(1) Case m, n = 3 and m, n = 4.
The proof is divided into 2 subcases based on the values of m and n.
(a) For m, n = 3.
Let Π = {S1 , S2 , S3 } be a resolving set with S1 = {x, v3 }, S2 = {y},
and S3 = {u3 }. The representations of every vertex are r(x|Π) =
(0, 1, 1), r(y|Π) = (1, 0, 1),r(v3 |Π) = (0, 1, 2), r(u3 |Π) = (1, 1, 0). Since
the representations are distinct so Π is a resolving set of K3 ∗2 K3 so
pd(K3 ∗2 K3 ) = m.
(b) For m, n = 4.
Let Π = {S1 , S2 , S3 , S4 } be a resolving set with S1 = {x, v3 }, S2 =
{y, v4 }, S3 = {u3 }, and S4 = {u4 }. The representations of every
vertex are r(x|Π) = (0, 1, 1, 1), r(y|Π) = (1, 0, 1, 1), r(v3 |Π) = (0, 1, 2, 2),
r(u3 |Π) = (1, 1, 0, 1), r(v4 |Π) = (1, 0, 2, 2), r(u4 |Π) = (1, 1, 1, 0). Since
the representations are distinct so pd(K4 ∗2 K4 ) = m.
(2) Case m = 3, n ≥ 4 and m ≥ 4, n = 3.
Let Π = {S1 , S2 , . . . , Sm+n−4 } be a resolving set of Km ∗2 Kn for m = 3
and n ≥ 4 with S1 = {x, u3 }, S2 = {y, u4 }, S3 = {v3 }, Si = {ui+1 } for
3 < i ≤ n−1. Si is accomplished when n ≥ 5. When n = 4, the resolving set
consists of only S1 , S2 , S3 . The representations of every vertex are r(x|Π) =
(0, 1, 1, 1, . . .), r(y|Π) = (1, 0, 1, 1, . . .), r(u3 |Π) = (0, 1, 2, d(u3 , Si )), r(u4 |Π) =
(1, 0, 2, d(u4 , Si )), r(v3 |Π) = (1, 1, 0, d(v3 , Si )), r(un |Π) = (1, 1, 2, d(un , Si ))
with d(u3 , Si ) = d(u4 , Si ) = d(un , Si ) = 1 for un ̸∈ Si and d(v3 , Si ) = 2.
Since the representations are distinct so Π is a resolving set of Km ∗2 Kn for
m = 3 and n ≥ 4 so pd(Km ∗2 Kn ) = m + n − 4. Then, we show partition
dimension of Km ∗2 Kn with m ≥ 4 and n = 3. Since Km ∗2 Kn for m ≥ 4
and n = 3 is isomorphic to Km ∗2 Kn for m = 3 and n ≥ 4, we obtain the
partition dimension are same, pd(Km ∗2 Kn ) = m + n − 4.
(3) Case m = 4, n ≥ 5 and m ≥ 5, n = 4.
Let Π = {S1 , S2 , . . . , Sm+n−5 } be a resolving set of Km ∗2 Kn for m = 4
and n ≥ 5 with S1 = {x, v3 , u3 }, S2 = {y, u4 }, S3 = {v4 }, Si = {ui+1 }
when 4 < i ≤ n − 1. The representations of every vertex are r(x|Π) =
(0, 1, 1, 1, . . .), r(y|Π) = (1, 0, 1, 1, . . .), r(u3 |Π) = (0, 1, 2, d(u3 , Si )), r(u4 |Π) =
(1, 0, 2, d(u4 , Si )), r(v3 |Π) = (0, 1, 1, d(v3 , Si )), r(v4 |Π) = (1, 1, 0, d(v4 , Si )),
r(un |Π) = (1, 1, 2, d(un , Si )) with d(u3 , Si ) = d(u4 , Si ) = d(un , Si ) = 1
for un ̸∈ Si and d(v3 , Si ) = d(v4 , Si ) = 2. Since the representations are
distinct so Π is a resolving set of Km ∗2 Kn for m = 4 and n ≥ 5 so
7
2017
ON THE PARTITION DIMENSION . . .
M. Ilafiani, T. A. Kusmayadi
pd(Km ∗2 Kn ) = m + n − 5. Then, we show partition dimension of Km ∗2 Kn
with m ≥ 5 and n = 4. Since Km ∗2 Kn for m ≥ 5 and n = 4 is isomorphic
to Km ∗2 Kn for m = 4 and n ≥ 5, we obtain the partition dimension are
same, pd(Km ∗2 Kn ) = m + n − 5.
(4) Case m ≥ 5 and n ≥ 5.
Let Π = {S1 , S2 , . . . , Sn+m−6 } be a resolving set of Km ∗2 Kn for m ≥ 5
and n ≥ 5 with S1 = {x, v3 , u3 }, Si = {vi+2 }, S2 = {y, v4 , u4 }, Sj = {uj+1 }
when 2 < i ≤ m − 2 dan m − 2 < j ≤ n + m − 6. The representations of
every vertex are r(x|Π) = (0, 1, 1, 1, . . .), r(y|Π) = (1, 0, 1, 1, . . .), r(v3 |Π) =
(0, 1, d(v3 , Si ), d(v3 , Sj )), r(v4 |Π) = (1, 0, d(v4 , Si ), d(v4 , Sj )), r(v5 |Π) = (1, 1,
d(v5 , Si ), d(v5 , Sj )), r(vm |Π) = (1, 1, d(vm , Si ), d(vm , Sj )), r(u3 |Π) = (0, 1, d(u3 , Si ),
d(u3 , Sj )), r(u4 |Π) = (1, 0, d(u4 , Si ), d(u3 , Sj )), r(u5 |Π) = (1, 1, d(u5 , Si ), d(u3 , Sj ),
r(un |Π) = (1, 1, d(un , Si ), d(u3 , Sj )) with d(v3 , Si ) = d(v4 , Si ) = d(v5 , Si ) =
d(vm , Si ) = 1, d(v3 , Sj ) = d(v4 , Sj ) = d(v5 , Sj ) = d(vm , Sj ) = 2, d(u3 , Si ) =
d(u4 , Si ) = d(u5 , Si ) = d(un , Si ) = 2, and d(u3 , Sj ) = d(u4 , Sj ) = d(u5 , Sj ) =
d(un , Sj ) = 1. Since the representations are distinct so Π is a resolving set
of Km ∗2 Kn for m ≥ 5 and n ≥ 5 so pd(Km ∗2 Kn ) = m + n − 6.
4. Conclusion
Based on the discussion above, we conclude that partition dimension of caveman
graph, generalized Petersen graph, and Km ∗2 Kn graph are each described in
Theorem 3.1, Theorem 3.2, and Theorem 3.3, respectively.
References
[1] Apriliani, T., Dimensi Partisi dari Graf Antiprisma, Graf Mongolian Tent, dan Graf Stacked
Book, Tugas Akhir, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Sebelas
Maret, Surakarta, 2016.
[2] Asmiati, Partition Dimension of Amalgamation of Stars, Bulletin of Mathematics 04 (2012),
No. 02, 161-167.
[3] Biggs, N. L., Algebraic Graph Theory, Second Edition, Cambridge University Press, England,
1993.
[4] Chartrand, G., E. Salehi, and P. Zhang, The Partition Dimension of a Graph, Aequationes
Math. 59 (2000), 45-54.
[5] Dewi, M.P.K., Dimensi Partisi dari Graf Lollipop, Graf Generalized Jahangir, dan Graf Cn ∗2
Km , Tugas Akhir, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Sebelas
Maret, Surakarta, 2016.
[6] Watts, D. J., Networks, Dynamics, and the Small-World Phenomenon, Amer. J. Soc. 105
(1999), 493-527.
8
2017
GENERALIZED PETERSEN GRAPH, AND Km ∗2 Kn GRAPH
Meta Ilafiani, Tri Atmojo Kusmayadi
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Sebelas Maret University
Abstract. Let G be a connected graph with vertex set V (G) = {v1 , v2 , . . . , vn } and
edge set E(G) = {e1 , e2 , . . . , en }. Those vertices are divided into k-partition, denoted by
S1 , S2 , . . . , Sk . Let Π = {S1 , S2 , . . . , Sk } be an ordered k-partition. The representation
for every vertex V (G) of Π is a minimum distance of a vertex to other vertices, denoted
by r(v|Π) = (d(v, S1 ), d(v, S2 ), . . . , d(v, Sk )). If every vertex has distinct representation,
Π is called a resolving k-partition. Minimum cardinality of k-partition of V (G) is called
by partition dimension of G, denoted by pd(G). In this research, we determine partition
dimension of caveman graph C(n, k), generalized Petersen graph P (n, k), and Km ∗2 Kn
graph.
Keywords : Partition dimension, caveman graph, generalized Petersen graph, Km ∗2 Kn
graph.
1. Introduction
There are many concepts in graph discussed by researchers. One of those
concepts is partition dimension. This concept is introduced by Chartrand et al. [4]
in 1998. Partition dimension of G or pd(G) is minimum cardinality of a resolving
k-partition of V (G).
The concept of partition dimension has been applied frequently in some graph
classes. In 1998, Chartrand et al. [4] studied about partition dimension of path
graph, complete graph, and cycle graph. Then, Asmiati [2] found partition dimension
of amalgamation of stars graph in 2012. Thereafter, in 2016, Apriliani [1] determined
partition dimension of antiprism graph, mongolian tent graph, and stacked book
graph. Also in 2016, Dewi [5] determined partition dimension of some graph classes,
such as lollipop graph, generalized Jahangir graph, and Cn ∗2 Km graph. In this
research, we determine partition dimension of caveman graph C(n, k) for n, k ≥ 3,
generalized Petersen graph P (n, k) for n ≥ 5, k = 2 and n ≥ 8, k = 3, and Km ∗2 Kn
graph for m, n ≥ 3.
2. Partition Dimension
The following definition and lemma are about partition dimension given by
Chartrand et al. [4].
Definition 2.1. Let G be a connected graph. For a subset S of V (G) and a vertex
v of G, the distance d(v, S) between v and S is defined as d(v, S) = min{d(v, x)|x ∈
S}. For an ordered k-partition Π = {S1 , S2 , . . . , Sk } of V (G) and a vertex v of
G, the representation of v with respect to Π is defined as the k-vector r(v|Π) =
(d(v, S1 ), d(v, S2 ), . . . , d(v, Sk )). The partition Π is called a resolving partition if the
1
ON THE PARTITION DIMENSION . . .
M. Ilafiani, T. A. Kusmayadi
k-vectors r(v|Π), v ∈ V (G), are distinct. The minimum k for which there is a
resolving k-partition of V (G) is the partition dimension pd(G) of G.
Lemma 2.1. Let G be a connected graph, then
(1) pd(G) = 2 if only if G = Pn for n ≥ 2 and
(2) pd(G) = n if only if G = Kn .
3. Main Results
3.1. Partition Dimension of Caveman Graph.
Watts [6] defined the caveman graph denoted by C(n, k) as a graph formed
by modifying set of k-cliques or caves with removing one edge from each clique and
using it to connect to a neighboring clique such that all n cliques form a single,
1
unbroken loop. The vertex set of C(n, k) is V (Cn,k ) = {v11 , v21 , . . . , vk−1
, u11 , v12 ,
2
n
v22 , . . . , vk−1
, u21 , . . . , vk−1
, un1 } with k ≥ 3 and n ≥ 3. The values of n are divided
into three types, they are n = 3m, n = 3m + 1, and n = 3m + 2 for m ∈ N.
Lemma 3.1. Let G be C(n, 4) graph and a subgraph of C(n, k) for n ≥ 3, k ≥ 5.
Let Π = {S1 , S2 , . . . , Sk−1 } be a resolving partition of C(n, k). If d(a, w) = d(vbc , w)
with a ∈ V (G), vbc ∈ V (C(n, k)) − V (G), w ∈ V (C(n, k)) − {a, vbc }, 2 ≤ b ≤ k − 3,
0 < c ≤ n, then a and vbc belong to distinct elements of Π.
Proof. Let Π = {S1 , S2 , . . . , Sk−1 } be a resolving partition of C(n, k) and let d(a, w) =
d(vbc , w). Suppose a and vbc are in same elements of Π, say Si . Then, d(a, Si ) =
d(vbc , Si ) for 0 < i ≤ k − 1 and so r(a|Π) = r(vbc |Π). Consequently, Π is not
a resolving partition for C(n, k). Hence, a and vbc belong to distinct elements of
Π.
Theorem 3.1. Let C(n, k) be caveman graph for n ≥ 3 and k ≥ 3, then
{
3,
for k = 3, 4;
pd(C(n, k)) =
k − 1,
for k ≥ 5.
Proof. The proof is divided into four cases by the values of k and n.
(1) For k = 3 and n = 3m with m ∈ N.
Let Π = {S1 , S2 , S3 } be a resolving partition of C(n, k) with S1 = {vsj , up1 }
for 1 ≤ s ≤ k − 1, 0 < j ≤ ⌈ n3 ⌉, 0 < p ≤ n3 , S2 = {vsl , uq1 } for 1 ≤ s ≤ k − 1,
⌈ n3 ⌉ < l ≤ ⌈ n3 ⌉ + m, n3 < q ≤ n3 + 1, S3 = {vsh , ur1 } for 1 ≤ s ≤ k − 1,
⌈ n3 ⌉ + m < h ≤ n, n3 + 1 < r ≤ n. Then, we obtain the representations of
every vertex in V (C(n, k)) with respect to Π are
{
{
(2j, 0, 2m − 2j + 3),
a = 1;
(0, 2m − 2j + 3, 2j),
a = 1;
l
j
r(va |Π) =
r(va |Π) =
(0, 2m − 2j + 2, 2j − 1),
(2j − 1, 0, 2m − 2j + 2), a = 2.
a = 2.
2
2017
ON THE PARTITION DIMENSION . . .
r(val |Π) =
{
(2m − 2j + 3, 2j, 0),
(2m − 2j + 2, 2j − 1, 0),
M. Ilafiani, T. A. Kusmayadi
(0, 2m − 2p + 1, 2j), b = p;
a = 1;
(2j, 0, 2m − 2p + 1),
r(ub1 |Π) =
a = 2.
(2j, 0, 2m − 2p + 1),
b = q;
b = r.
Since the representations of every vertex in V (C(n, k)) for k = 3 and
n = 3m with respect to Π are distinct, Π is a resolving partition with three
elements.
(2) For k = 3 with n = 3m + 1 and n = 3m + 2, m ∈ N.
Let Π = {S1 , S2 , S3 } be a resolving partition of C(n, k) with S1 = {vsj , up1 } for
1 ≤ s ≤ k − 1, 0 < j ≤ ⌈ n3 ⌉, 0 < p ≤ ⌈ n3 ⌉ − 1, S2 = {vsl , uq1 } for 1 ≤ s ≤ k − 1,
⌈ n3 ⌉ < l ≤ ⌈ n3 ⌉ + m, ⌈ n3 ⌉ − 1 < q ≤ ⌈ n3 ⌉ + m, S3 = {vsh , ur1 } 1 ≤ s ≤ k − 1,
⌈ n3 ⌉ + m < h ≤ n, ⌈ n3 ⌉ + m < r ≤ n. We divide the representations of every
vertex in V (C(n, k)) with respect to Π into two subcases according to the
values of n.
(a) For n = 3m + 1.
The representations of every vertex in V (C(n, k)) with respect to Π are
{
{
(2l − 2m − 1, 0, 4m − 2l + 5), a = 1;
(0, 2m − 2j + 4, 2j),
a = 1;
l
j
r(va |Π) =
r(va |Π) =
(0, 2m − 2j + 3, 2j − 1),
r(vah |Π)
=
{
(2l − 2 − 2m, 0, 4m − 2l + 6),
a = 2.
(6m − 2h + 5, 2h − 4m − 2, 0),
a = 1;
(6m − 2h + 4, 2h − 4m − 3, 0),
a = 2.
r(ub1 |Π)
=
a = 2.
(0, 2m − 2p + 2, 2p),
b = p;
(2q − 2m − 1, 0, 4m − 2q + 3), b = q;
(6m − 2r + 3, 2r − 4m − 2, 0),
b = r.
(b) For n = 3m + 2.
The representations of every vertex in V (C(n, k)) to Π are
{
{
(2l − 2m − 1, 0, 4m − 2l + 5), a = 1;
(0, 2m − 2j + 4, 2j),
a = 1;
l
j
r(va |Π) =
r(va |Π) =
(0, 2m − 2j + 3, 2j − 1),
r(vah |Π)
=
{
a = 2.
(2l − 2 − 2m, 0, 4m − 2l + 6),
a = 2.
(0, 2m − 2p + 2, 2p),
b = p;
a = 1;
b
(2q − 2m − 1, 0, 4m − 2q + 3), b = q;
r(u1 |Π) =
a = 2.
(6m − 2r + 5, 2r − 4m − 2, 0), b = r.
(6m − 2h + 7, 2h − 4m − 2, 0),
(6m − 2h + 6, 2h − 4m − 3, 0),
It can be checked that the representations of every vertex in V (C(n, k))
with respect to Π are distinct. Therefore, Π is a resolving partition with three
elements.
(3) For k = 4 and n ≥ 3.
Let Π = {S1 , S2 , S3 } be a resolving partition of C(n, k) with S1 = {vsj , up1 }
for 1 ≤ s ≤ k − 1, 0 < j ≤ ⌈ n3 ⌉, 0 < p ≤ ⌈ n3 ⌉ − 1, S2 = {vsl , uq1 } for
1 ≤ s ≤ k − 1, ⌈ n3 ⌉ < l ≤ ⌈ n3 ⌉ + m,⌈ n3 ⌉ − 1 < q ≤ ⌈ n3 ⌉ + m, S3 = {vsh , ur1 }
for 1 ≤ s ≤ k − 1, ⌈ n3 ⌉ + m < h ≤ n, ⌈ n3 ⌉ + m < r ≤ n. We divide the
representations of every vertex in V (C(n, k)) with respect to Π into three
subcases based on the values of n.
3
2017
ON THE PARTITION DIMENSION . . .
M. Ilafiani, T. A. Kusmayadi
(a) For n = 3m with m ∈ N.
The representations of every vertex in V (C(n, k)) with respect to Π are
a = 1;
a = 1;
(2j, 0, 2m − 2j + 3),
(0, 2m − 2j + 3, 2j),
l
j
(2j − 1, 0, 2m − 2j + 2), a = k − 2;
(0, 2m − 2j + 2, 2j − 1), a = k − 2; r(va |Π) =
r(va |Π) =
(0, 2m − 2j + 2, 2),
r(vah |Π) =
(2m − 2j + 3, 2j, 0),
(2m − 2j + 2, 2j − 1, 0),
(6m − 2q + 2, 2, 0),
(2, 0, 2m − 2l + 2),
a = 2.
a = 2.
a = 1;
(0, 2m − 2p + 1, 2j), b = p;
b
(2p, 0, 2m − 2p + 1), b = q;
a = k − 2; r(u1 |Π) =
(2p, 0, 2m − 2p + 1), b = r.
a = 2.
(b) For n = 3m + 1 with m ∈ N.
The representations of every vertex in V (C(n, k)) with respect to Π are
a = 1;
a = 1;
(0, 2m − 2j + 4, 2j),
(2l − 3, 0, 4m − 2l + 5),
j
l
r(va |Π) =
(0, 2m − 2j + 3, 2j − 1), a = k − 2; r(va |Π) =
(2l − 2m − 2, 0, 4m − 2l + 6), a = k − 2;
(0, 2m − 2j + 3, 2j),
r(vah |Π)
=
a = 2.
(2l − 2m − 1, 0, 2l − 2m − 4),
(4m − 2h + 4, 2h − 4m − 2, 0), a = 1;
(4m − 2h + 5, 2h − 2m − 2, 0), a = k − 2;
(6m − 2h + 4, 2h − 4m − 2, 0), a = 2.
r(ub1 |Π)
=
(0, 2m − 2p + 2, 2p),
a = 2.
b = p;
(2q − 2m − 1, 0, 4m − 2q + 3), b = q;
(6m − 2r + 3, 4m − 2r − 2, 0), b = r.
(c) For n = 3m + 2 with m ∈ N.
The representations of every vertex in V (C(n, k)) with respect to Π are
a = 1;
a = 1;
(0, 2m − 2j + 4, 2j),
(2l − 3, 0, 4m − 2l + 5),
j
l
r(va |Π) =
(0, 2m − 2j + 3, 2j − 1), a = k − 2; r(va |Π) =
(2l − 2m − 2, 0, 4m − 2l + 6), a = k − 2;
(0, 2m − 2j + 3, 2j),
r(vah |Π)
=
a = 2.
(2l − 2m − 1, 0, 2l − 2m − 4),
(6m − 2h + 7, 2h − 4m − 2, 0), a = 1;
(6m − 2h + 2, 2h − 4m − 3, 0), a = k − 2;
r(ub1 |Π)
=
(0, 2m − 2p + 2, 2p),
b = p;
(2q − 2m − 1, 0, 4m − 2q + 3), b = q;
(6m − 2h + 2, 2h − 4m − 2, 0), a = 2.
(6m − 2r + 5, 6m − 2r + 4, 0),
We obtain that the representations of every vertex in V (C(n, k)) for
k = 4 and n ≥ 3 with respect to Π are distinct. Then, Π is a resolving
partition with three elements.
(4) For k ≥ 5 and n ≥ 3.
Let Π = {S1 , S2 , . . . , Sk−1 } be a resolving partition of C(n, k). By using
c
} for 0 < i ≤ k − 4, 1 ≤
Lemma 3.1, we obtain the partition as Si = {vi+1
p
j
c ≤ n; Sk−3 = {vs , u1 } − Si for 1 ≤ s ≤ k − 1, 0 < j ≤ ⌈ n3 ⌉, 0 < p ≤ 1,
Sk−2 = {vsl , uq1 } − Si for 1 ≤ s ≤ k − 1, ⌈ n3 ⌉ < l ≤ ⌈ n3 ⌉ + m, 1 < q ≤ 2,
Sk−1 = {vsh , ur1 } − Si for 1 ≤ s ≤ k − 1, ⌈ n3 ⌉ + m < h ≤ n, 2 < r ≤ 3. We
divide the representations of every vertex in V (C(n, k)) with respect to Π
into two subcases under the values of n.
4
a = 2.
2017
b = r.
ON THE PARTITION DIMENSION . . .
M. Ilafiani, T. A. Kusmayadi
(a) For n = 3m with m ∈ N.
The representations of every vertex in V (C(n, k)) with respect to Π
j
c
l
c
are r(vi+1
|Π) = (d(vi+1
, Si ), 1, 2m − 2j + 2, 2j), r(vi+1
|Π) = (d(vi+1
, Si ), 2j, 1, 2m −
h
c
2j + 2), r(vi+1
|Π) = (d(vi+1
, Si ), 2m − 2j + 2, 2j, 1), r(v1j |Π) = (1, . . . , 0, 2m − 2j +
j
j
3, 2j), r(vk−2
|Π) = (1, . . . , 0, 2m − 2j + 2, 2j − 1), r(vk−3
|Π) = (1, . . . , 0, 2m − 2j +
l
2, 2), r(v1l |Π) = (1, . . . , 2j, 0, 2m − 2j + 3), r(vk−2
|Π) = (1, . . . , 2j − 1, 0, 2m − 2j +
l
2), r(vk−3
|Π) = (1, . . . , 2, 0, 2m − 2l + 2), r(v1h |Π) = (1, . . . , 2m − 2j + 3, 2j, 0),
h
h
r(vk−2
|Π) = (1, . . . , 2m − 2j + 2, 2j − 1, 0), r(vk−3
|Π) = (1, . . . , 6m − 2j + 2, 2, 0),
r(up1 |Π) = (1, . . . , 0, 2m − 2p + 1, 2p), r(uq1 |Π) = (1, . . . , 2p, 0, 2m − 2p + 1), r(ur1 |Π) =
c
c
(1, . . . , 2p, 0, 2m − 2p + 1). We obtain d(vi+1
, Si ) = 0 for vi+1
∈ Si and
c
c
d(vi+1
, Si ) = 1 for vi+1
∈
/ Si .
(b) For n = 3m + 1 with m ∈ N.
The representations of every vertex in V (C(n, k)) with respect to Π
j
c
l
are footnotesize r(vi+1
|Π) = (d(vi+1
, Si ), 1, 2m − 2j + 3, 2j),r(vi+1
|Π) =
c
h
c
, Si ), 6m −
|Π) = (d(vi+1
, Si ), 2l − 2m − 1, 1, 4m − 2l + 4), r(vi+1
(d(vi+1
j
j
2h + 5, 2h − 4m − 2, 1),r(v1 |Π) = (1, . . . , 0, 2m − 2j + 4, 2j),r(vk−2
|Π) =
j
(1, . . . , 0, 2m−2j+3, 2j−1), r(vk−3 |Π) = (1, . . . , 0, 2m−2j+3, 2j),r(v1l |Π) =
l
|Π) = (1, . . . , 2j, 0, 4m − 2l + 6),
(1, . . . , 2l − 3, 0, 4m − 2l + 5), r(vk−2
l
r(vk−3 |Π) = (1, . . . , 2l − 2m − 1, 0, 2l − 2m − 4),r(v1h |Π) = (1, . . . , 4m −
h
2h + 5, 2h − 2m − 2, 0),r(vk−2
|Π) = (1, . . . , 4m − 2h + 5, 2h − 2m − 2, 0),
j
r(v1 |Π) = (1, . . . , 6m − 2h + 4, 2h − 4m − 2, 0),r(up1 |Π) = (1, . . . , 0, 2m −
2p + 2, 2p),r(uq1 |Π) = (1, . . . , 2q − 2m − 1, 0, 4m − 2q + 3), r(ur1 |Π) =
c
c
(1, . . . , 6m−2r+3, 4m−2r−2, 0). We obtain d(vi+1
, Si ) = 0 for vi+1
∈ Si
c
c
and d(vi+1 , Si ) = 1 for vi+1 ∈
/ Si .
(c) For n = 3m + 2 with m ∈ N.
The representations of every vertex in V (C(n, k)) with respect to Π are
j
c
l
c
r(vi+1
|Π) = (d(vi+1
, Si ), 1, 2m − 2j + 3, 2j),r(vi+1
|Π) = (d(vi+1
, Si ), 2j + 1, 1, 2m −
h
c
|Π) = (d(vi+1
, Si ), 2m − 2j + 3, 2j, 1). r(v1j |Π) = (1, . . . , 0, 2m − 2j +
2j + 2),r(vi+1
j
j
4, 2j),r(vk−2
|Π) = (1, . . . , 0, 2m−2j+3, 2j−1),r(vk−3
|Π) = (1, . . . , 0, 2m−2j+3, 2j).
l
|Π) = (1, . . . , 2l − 2m − 2, 0, 4m − 2l +
r(v1l |Π) = (1, . . . , 2l − 3, 0, 4m − 2l + 5),r(vk−2
l
6),r(vk−3
|Π) = (1, . . . , 2l − 2m − 1, 0, 2l − 2m − 4). r(v1h |Π) = (1, . . . , 6m − 2h +
h
h
7, 2h − 4m − 2, 0),r(vk−2
|Π) = (1, . . . , 6m − 2h + 2, 2h − 4m − 3, 0),r(vk−3
|Π) =
(1, . . . , 6m − 2h + 2, 2h − 4m − 2, 0). r(ub1 |Π) = (1, . . . , 0, 2m − 2p + 2, 2p), r(ub1 |Π) =
(1, . . . , 2q − 2m − 1, 0, 4m − 2q + 3), r(ub1 |Π) = (1, . . . , 6m − 2r + 5, 6m − 2r + 4, 0).
c
c
c
c
We obtain d(vi+1
, Si ) = 0 for vi+1
∈ Si and d(vi+1
, Si ) = 1 for vi+1
∈
/ Si .
5
2017
ON THE PARTITION DIMENSION . . .
M. Ilafiani, T. A. Kusmayadi
Since the representations of every vertex in V (C(n, k)) for k ≥ 5 and
n ≥ 3 with respect to Π are distinct, Π is a resolving partition with (k − 1)elements.
This completes the proof of the theorem.
3.2. Partition Dimension of Generalized Petersen.
Biggs [3] defined generalized Petersen graph denoted by P (n, k) is a 3-regular
graph with 2n vertices x0 , x1 , . . . , xn−1 , y0 , y1 , . . . , yn−1 and edges {xi , yi }, {xi , xi+1 },
{yi , yi+k }, for all i ∈ {0, 1, . . . , n − 1}, where the subscripts are reduced modulo n.
Theorem 3.2. Let P (n, k) be a generalized Petersen graph for (n ≥ 5, k = 2) and
(n ≥ 8, k = 3), then pd(P (n, k)) = 4.
Proof. Let Π = {S1 , S2 , S3 , S4 } be a resolving partition with S1 = {y0 , x0 }, S2 =
{y1 , x1 }, S3 = {ya , xa } for 2 ≤ a ≤ ⌈ n2 ⌉ − 1, and S4 = {yb , xb } for ⌈ n2 ⌉ ≤ b ≤ n − 1.
The representations of every vertex in V (P (n, k)) for (n ≥ 5, k = 2) and
(n ≥ 8, k = 3) with respect to Π are r(y0 |Π) = (0, 2, 1, a), r(y1 |Π) = (2, 0, 1, 1), r(yi |Π) =
(d(yi , S1 ), d(yi , S2 ), 0, d(yi , S4 )), r(yk |Π) = (d(yj , S1 ), d(yj , S2 ), d(yj , S3 ), 0), r(x0 |Π) = (0, 1, 2, 1),
r(x0 |Π) = (1, 0, 1, 2), r(xj |Π) = (d(xj , S1 ), d(xj , S2 ), 0, d(xj , S4 )), r(xl |Π) = (d(xl , S1 ), d(xl , S2 ),
d(xl , S3 ), 0) with i = 2, . . . , ⌈ n2 ⌉ − 1, k = ⌈ n2 ⌉, . . . , n − 1, j = 2, . . . , ⌈ n2 ⌉ − 1, j =
⌈ n2 ⌉, . . . , n − 1. We obtain a = 2 for n = 5, k = 2 and a = 1 for n ≥ 6, k = 2, 3 where
d(yi , S1 ), d(yi , S2 ), d(yi , S3 ), d(yi , S4 ), d(xi , S1 ), d(xi , S2 ), d(xi , S3 ), d(xi , S4 ) ≤ ⌊ n2 ⌋ so
that r(y0 |P i) ̸= r(y1 |P i) ̸= . . . ̸= r(yn−1 |P i) ̸= r(x0 |P i) ̸= r(x1 |P i) ̸= . . . ̸=
r(xn−1 |P i). Hence, Π is a resolving partition of P (n, k) graph for (n ≥ 5, k = 2)
and (n ≥ 8, k = 3) with four elements and pd(P (n, k)) = 4.
3.3. Partition Dimension of Km ∗2 Kn Graph.
Km ∗2 Kn graph with m ≥ 3 and n ≥ 3 is built from edge-amalgamation or
amalgamating an edge belongs to Km and other belongs to Kn . Let vertex set
V (Km ) = {v1 , v2 , . . . , vm } and V (Kn ) = {u1 , u2 , . . . , un }. An amalgamated edge
consists of two vertices, denoted by x and y. Vertex x is defined by amalgamating
vertex u1 and v1 . Vertex y is defined by amalgamating vertex u2 and v2 . So, we
obtain the vertex set V (Km ∗2 Kn ) = {x, y, v3 , v4 , . . . , vm , u3 , u4 , . . . , un }.
Theorem 3.3. Let Km ∗2 Kn graph be an edge-amalgamation graph between Km
and Kn for m, n ≥ 3, then
m,
for m, n = 3 and m, n = 4;
n + m − 4, for m = 3, n ≥ 4 and m ≥ 4, n = 3;
pd(Km ∗2 Kn ) =
n + m − 5, for m = 4, n ≥ 5 and m ≥ 5, n = 4;
n + m − 6,
for m ≥ 5, n ≥ 5.
6
2017
ON THE PARTITION DIMENSION . . .
M. Ilafiani, T. A. Kusmayadi
Proof. The proof is divided into four cases.
(1) Case m, n = 3 and m, n = 4.
The proof is divided into 2 subcases based on the values of m and n.
(a) For m, n = 3.
Let Π = {S1 , S2 , S3 } be a resolving set with S1 = {x, v3 }, S2 = {y},
and S3 = {u3 }. The representations of every vertex are r(x|Π) =
(0, 1, 1), r(y|Π) = (1, 0, 1),r(v3 |Π) = (0, 1, 2), r(u3 |Π) = (1, 1, 0). Since
the representations are distinct so Π is a resolving set of K3 ∗2 K3 so
pd(K3 ∗2 K3 ) = m.
(b) For m, n = 4.
Let Π = {S1 , S2 , S3 , S4 } be a resolving set with S1 = {x, v3 }, S2 =
{y, v4 }, S3 = {u3 }, and S4 = {u4 }. The representations of every
vertex are r(x|Π) = (0, 1, 1, 1), r(y|Π) = (1, 0, 1, 1), r(v3 |Π) = (0, 1, 2, 2),
r(u3 |Π) = (1, 1, 0, 1), r(v4 |Π) = (1, 0, 2, 2), r(u4 |Π) = (1, 1, 1, 0). Since
the representations are distinct so pd(K4 ∗2 K4 ) = m.
(2) Case m = 3, n ≥ 4 and m ≥ 4, n = 3.
Let Π = {S1 , S2 , . . . , Sm+n−4 } be a resolving set of Km ∗2 Kn for m = 3
and n ≥ 4 with S1 = {x, u3 }, S2 = {y, u4 }, S3 = {v3 }, Si = {ui+1 } for
3 < i ≤ n−1. Si is accomplished when n ≥ 5. When n = 4, the resolving set
consists of only S1 , S2 , S3 . The representations of every vertex are r(x|Π) =
(0, 1, 1, 1, . . .), r(y|Π) = (1, 0, 1, 1, . . .), r(u3 |Π) = (0, 1, 2, d(u3 , Si )), r(u4 |Π) =
(1, 0, 2, d(u4 , Si )), r(v3 |Π) = (1, 1, 0, d(v3 , Si )), r(un |Π) = (1, 1, 2, d(un , Si ))
with d(u3 , Si ) = d(u4 , Si ) = d(un , Si ) = 1 for un ̸∈ Si and d(v3 , Si ) = 2.
Since the representations are distinct so Π is a resolving set of Km ∗2 Kn for
m = 3 and n ≥ 4 so pd(Km ∗2 Kn ) = m + n − 4. Then, we show partition
dimension of Km ∗2 Kn with m ≥ 4 and n = 3. Since Km ∗2 Kn for m ≥ 4
and n = 3 is isomorphic to Km ∗2 Kn for m = 3 and n ≥ 4, we obtain the
partition dimension are same, pd(Km ∗2 Kn ) = m + n − 4.
(3) Case m = 4, n ≥ 5 and m ≥ 5, n = 4.
Let Π = {S1 , S2 , . . . , Sm+n−5 } be a resolving set of Km ∗2 Kn for m = 4
and n ≥ 5 with S1 = {x, v3 , u3 }, S2 = {y, u4 }, S3 = {v4 }, Si = {ui+1 }
when 4 < i ≤ n − 1. The representations of every vertex are r(x|Π) =
(0, 1, 1, 1, . . .), r(y|Π) = (1, 0, 1, 1, . . .), r(u3 |Π) = (0, 1, 2, d(u3 , Si )), r(u4 |Π) =
(1, 0, 2, d(u4 , Si )), r(v3 |Π) = (0, 1, 1, d(v3 , Si )), r(v4 |Π) = (1, 1, 0, d(v4 , Si )),
r(un |Π) = (1, 1, 2, d(un , Si )) with d(u3 , Si ) = d(u4 , Si ) = d(un , Si ) = 1
for un ̸∈ Si and d(v3 , Si ) = d(v4 , Si ) = 2. Since the representations are
distinct so Π is a resolving set of Km ∗2 Kn for m = 4 and n ≥ 5 so
7
2017
ON THE PARTITION DIMENSION . . .
M. Ilafiani, T. A. Kusmayadi
pd(Km ∗2 Kn ) = m + n − 5. Then, we show partition dimension of Km ∗2 Kn
with m ≥ 5 and n = 4. Since Km ∗2 Kn for m ≥ 5 and n = 4 is isomorphic
to Km ∗2 Kn for m = 4 and n ≥ 5, we obtain the partition dimension are
same, pd(Km ∗2 Kn ) = m + n − 5.
(4) Case m ≥ 5 and n ≥ 5.
Let Π = {S1 , S2 , . . . , Sn+m−6 } be a resolving set of Km ∗2 Kn for m ≥ 5
and n ≥ 5 with S1 = {x, v3 , u3 }, Si = {vi+2 }, S2 = {y, v4 , u4 }, Sj = {uj+1 }
when 2 < i ≤ m − 2 dan m − 2 < j ≤ n + m − 6. The representations of
every vertex are r(x|Π) = (0, 1, 1, 1, . . .), r(y|Π) = (1, 0, 1, 1, . . .), r(v3 |Π) =
(0, 1, d(v3 , Si ), d(v3 , Sj )), r(v4 |Π) = (1, 0, d(v4 , Si ), d(v4 , Sj )), r(v5 |Π) = (1, 1,
d(v5 , Si ), d(v5 , Sj )), r(vm |Π) = (1, 1, d(vm , Si ), d(vm , Sj )), r(u3 |Π) = (0, 1, d(u3 , Si ),
d(u3 , Sj )), r(u4 |Π) = (1, 0, d(u4 , Si ), d(u3 , Sj )), r(u5 |Π) = (1, 1, d(u5 , Si ), d(u3 , Sj ),
r(un |Π) = (1, 1, d(un , Si ), d(u3 , Sj )) with d(v3 , Si ) = d(v4 , Si ) = d(v5 , Si ) =
d(vm , Si ) = 1, d(v3 , Sj ) = d(v4 , Sj ) = d(v5 , Sj ) = d(vm , Sj ) = 2, d(u3 , Si ) =
d(u4 , Si ) = d(u5 , Si ) = d(un , Si ) = 2, and d(u3 , Sj ) = d(u4 , Sj ) = d(u5 , Sj ) =
d(un , Sj ) = 1. Since the representations are distinct so Π is a resolving set
of Km ∗2 Kn for m ≥ 5 and n ≥ 5 so pd(Km ∗2 Kn ) = m + n − 6.
4. Conclusion
Based on the discussion above, we conclude that partition dimension of caveman
graph, generalized Petersen graph, and Km ∗2 Kn graph are each described in
Theorem 3.1, Theorem 3.2, and Theorem 3.3, respectively.
References
[1] Apriliani, T., Dimensi Partisi dari Graf Antiprisma, Graf Mongolian Tent, dan Graf Stacked
Book, Tugas Akhir, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Sebelas
Maret, Surakarta, 2016.
[2] Asmiati, Partition Dimension of Amalgamation of Stars, Bulletin of Mathematics 04 (2012),
No. 02, 161-167.
[3] Biggs, N. L., Algebraic Graph Theory, Second Edition, Cambridge University Press, England,
1993.
[4] Chartrand, G., E. Salehi, and P. Zhang, The Partition Dimension of a Graph, Aequationes
Math. 59 (2000), 45-54.
[5] Dewi, M.P.K., Dimensi Partisi dari Graf Lollipop, Graf Generalized Jahangir, dan Graf Cn ∗2
Km , Tugas Akhir, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Sebelas
Maret, Surakarta, 2016.
[6] Watts, D. J., Networks, Dynamics, and the Small-World Phenomenon, Amer. J. Soc. 105
(1999), 493-527.
8
2017