Artikel Meta Ilafiani

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GENERALIZED PETERSEN GRAPH, AND Km∗2Kn 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, . . . , v}_n_} _and

edge setE(G) ={e1, e2, . . . , en}. Those vertices are divided intok-partition, denoted by

S1, S2, . . . , Sk. Let Π = {S1, S2, . . . , Sk} be an ordered k-partition. The representation for every vertexV(G) of Π is a minimum distance of a vertex to other vertices, denoted byr(v|Π) = (d(v, S1), d(v, S2), . . . , d(v, Sk)). If every vertex has distinct representation, Π is called a resolvingk-partition. Minimum cardinality ofk-partition of V(G) is called by partition dimension ofG, denoted bypd(G). In this research, we determine partition dimension of caveman graphC(n, k), generalized Petersen graphP(n, k), andKm∗2Kn graph.

Keywords: Partition dimension, caveman graph, generalized Petersen graph,Km∗2Kn 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 andn≥8, k = 3, andKm∗2Kn 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 distanced(v, S)between v andS is defined asd(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

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k-vectors r(v|Π), v ∈ V(G), are distinct. The minimum k for which there is a resolving k-partition ofV(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 ofk-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, unbroken loop. The vertex set of C(n, k) is V(Cn,k) = {v1

1, v21, . . . , vk1−1, u 1 1, v12,

v2

2, . . . , vk2−1, u 2

1, . . . , vnk−1, u

n

1} 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(v

c b, w)

with a ∈V(G), vc

b ∈V(C(n, k))−V(G), w∈ V(C(n, k))− {a, vcb}, 2≤b ≤ k−3, 0< c≤n, then a and vc

b belong to distinct elements of Π.

Proof. Let Π ={S1, S2, . . . , Sk−1}be a resolving partition ofC(n, k) and letd(a, w) =

d(vc

b, w). Suppose a and v c

b are in same elements of Π, say Si. Then, d(a, Si) =

d(vc

b, Si) for 0 < i ≤ k − 1 and so r(a|Π) = r(vcb|Π). Consequently, Π is not a resolving partition for C(n, k). Hence, a and vc

b belong to distinct elements of

Π.

Theorem 3.1. Let C(n, k) be caveman graph for n≥3 and k ≥3, then

pd(C(n, k)) =

{

3, fork= 3,4;

k−1, fork≥5.

Proof. The proof is divided into four cases by the values ofk and n.

(1) For k = 3 andn = 3m with m ∈N_.

Let Π = {S1, S2, S3} be a resolving partition of C(n, k) with S1 = {vjs, u p

1}

for 1≤s≤ k−1, 0 < j ≤ ⌈n

3⌉, 0< p≤

n

3, S2 ={v

l s, u

q

1} for 1≤ s≤k−1,

⌈n

3⌉ < l ≤ ⌈

n

3⌉+m,

n

3 < q ≤

n

3 + 1, S3 = {v

h

s, ur1} for 1 ≤ s ≤ k −1,

⌈n

3⌉+m < h ≤ n,

n

3 + 1 < r ≤ n. Then, we obtain the representations of

every vertex in V(C(n, k)) with respect to Π are

r(vj a|Π) =

{

(0,2m−2j+ 3,2j), a= 1; (0,2m−2j+ 2,2j−1), a= 2. r(v

l a|Π) =

{

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r(vl

n= 3m with respect to Π are distinct, Π is a resolving partition with three elements.

It can be checked that the representations of every vertex inV(C(n, k)) with respect to Π are distinct. Therefore, Π is a resolving partition with three elements.

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(a) For n= 3m with m∈N_.

We obtain that the representations of every vertex in V(C(n, k)) for

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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 byP(n, k) is a 3-regular graph with 2n verticesx0, 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 modulon.

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, andS4 ={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 =

⌈n

2⌉, . . . , n−1. We obtain a= 2 forn= 5, k= 2 anda= 1 forn≥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

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 andpd(P(n, k)) = 4.

3.3. Partition Dimension ofKm∗2KnGraph.

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 setV(Km∗2Kn) ={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

pd(Km∗2Kn)=

   

  

m, form, n= 3andm, n= 4;

n+m−4, form= 3, n≥4 andm≥4, n= 3;

n+m−5, form= 4, n≥5 andm≥5, n= 4;

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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 ofm 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∗2K3) = 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∗2K4) = m.

(2) Case m= 3, n ≥4 andm ≥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 whenn ≥5. Whenn= 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 ofKm∗2Kn for

m = 3 and n ≥ 4 so pd(Km ∗2Kn) = m+n−4. Then, we show partition

dimension of Km∗2Kn with m ≥ 4 and n = 3. Since Km∗2 Kn for m ≥ 4 and n = 3 is isomorphic to Km ∗2Kn for m = 3 and n ≥ 4, we obtain the partition dimension are same, pd(Km∗2Kn) = m+n−4.

(3) Case m= 4, n ≥5 andm ≥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

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pd(Km∗2Kn) =m+n−5. Then, we show partition dimension ofKm∗2Kn with m ≥5 and n = 4. Since Km∗2Kn 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∗2Kn) =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∗2Kn for m≥5 and n≥5 so pd(Km∗2Kn) = 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 Mathematics04_(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 GrafCn∗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