On Minimum Metric Dimension ofCirculant Networks

On Minimum Metric Dimension of Circulant Networks

1

BHARATI RAJAN,

INDRA RAJASINGH,

CHRIS MONICA. M*

and

PAUL MANUEL

1

Department of Mathematics, Loyola College, Chennai 600 034 (India)

2

Department of Information Science, Kuwait University, Kuwait 13060 Email: [email protected]

A B S TR ACT

Let M = { v ₁ , v ₂ ,..., v _n } be an ordered set of vertices in a graph G. Then ( d ( u , v ₁ ), d ( u , v ₂ ),..., d ( u , v _n )) is called the M-coordinates of a vertex u of G. The set M is called a metric basis if the vertices of G have distinct M-coordinates. A minimum metric basis is a set M with minimum cardinality. The cardinality of a minimum metric basis of G is called minimum metric dimension and is denoted by  (G). This concept has wide applications in motion planning and in the field of robotics. In this paper we determine the minimum metric dimension of certain classes of circulant networks. We prove that for circulant graphs G(n;  ^{{1, 2}),}  ^{(G (n;} _ {1, 2}) = 3, when n = 4 l, 4 l + 2, 4 l + 3, l 1 and 2 < (G (n; ±{1, 2})  4, for 4 l+ 1, l 1. We have similar results for circulant digraphs G(n;  {1, 2, 3}) and certain subclasses of circulant graphs.

Key words: metric basis, metric dimension, circulant graphs, robotics, Cayley graph.

1. INTRODUCTION

Major issues involved in the design of interconnection networks are quick

communication among vertices, high robustness, rich structure in the sense of embeddable properties, fault tolerance and VLSI. The circulant graphs are an important class of topological structures of interconnection networks. They are symmetric with simple structures and easy extendability. Circulant graphs have

*This research is supported by The

Major Project-No.F.8-5-2004(SR) of University

Grants Commission, India.

been used for decades in the design of computer and telecommunication networks due to their optimal fault- tolerance and routing capabilities

. The term circulant comes from the nature of its adjacency matrix; a matrix is circulant if all its rows are periodic rotations of the first one. Circulant matrices have also been employed for designing binary codes. Circulant graphs also constitute the basis for designing certain data alignment networks for complex memory systems

. Most of the earlier research concentrated on using the circulant graphs to build inter- connection networks for distributed and parallel systems

. For example, undirected circulant networks arise in the context of Mesh Connected Com- puter suited for parallel processing of data, such as the well-known ILLIAC type computers

. Generally, the ILLIAC network with n

processors can be represented as a circulant graph G(n

;

±{1, n}).

By using circulant graph we can adapt the performance of the network to user needs. For a given number of processors and a given crossbar switch technology, we can choose the perfor- mance of the network. If subsequently, t he us e r ne e ds t o inc re a se t his performance we can increase the degree of the circulant graph without changing the number of processors.

The opposite modification is also possible, we can increase the number of processors without changing the degree of the circulant graph. This

flexibility which is not possible with other topologies, allows us to optimize the ratio price/performance according the user. The family of circulant graphs includes the complete graph and the cycle among its members.

Circulant graphs are intensively researched in computer science, graph theory and discrete mathematics.

2. An Overview of the Paper

Let M = { v ₁ , v ₂ ,..., v _n } ^{be an} ordered set of vertices in a graph G.

T he n ( d ( u , v ₁ ), d ( u , v ₂ ),..., d ( u , v _n )) ^is called the M-coordinates of a vertex u of G. The set M is called a metric basis if the vertices of G have distinct M- coordinates. A minimum metric basis is a set M with minimum cardinality.

If M is a metric basis then it is clear that for each pair of vertices u and v of V \ M, there is a vertex w  M such that d(u, w)  d(v, w). The cardinality of a minimum metric basis of G is called minimum metric dimension and is denoted by  (G); the members of a metric basis are called landmarks. A minimum metric dimension (MMD) problem is to find a minimum metric basis.

This problem has application in the field of robotics. A robot is a mechanical device which is made to move in space with obstructions around.

It has neither the concept of direction

nor that of visibility. But it is assumed

that it can sense the distances to a set

of landmarks. Evidently, if the robot knows its distances to a sufficiently large set of landmarks its position in space is uniquely determined.

The concept of metric basis and minimum metric basis has appeared in the literature under a different name as early as 1975. Slater in

and later in

had called these sets as locating sets and reference sets respectively. Slater called the cardinality of a reference set as the location number of G. He described the usefulness of these ideas when working with sonar and loran stations. Independently, Harary and Melter

discovered these concepts as well but used the terms metric basis and metric dimension, rather than reference set and location number. Chartrand et al.

have called a metric basis and a minimum metric basis as a resolving set and minimum resolving set. We adopt the terminology of Harary and Melter.

If G has p vertices then it is clear that 1  (G)  p – 1. Also for the complete graph K

, the cycle C

and the complete bipartite graph K

, the minimum metric dimensions are  (K

)

= p – 1,  ^(C

) = 2 and  ^(K

) = m + n – 2

. This problem has been studied for grids

, trees, multi-dimen- sional grids

, Petersen graphs

, Torus Net works

, B e ne s a nd B utt e rf ly networks

and Honeycomb networks

.

Garey and Johnson

noted that this problem is NP-complete for general graphs by a reduction from 3-dimen- sional matching. Recently Manuel et a l.

have prove d t ha t t he M M D problem is NP-complete for bipartite graphs by a reduction from 3-SAT narrowing down the gap between the polynomial classes and NP-complete classes for the MMD problem.

In this paper, we derive a minimum metric basis for certain classes of circulant networks.

3. Topological Properties of Circulant Networks

T he c irc ulant ne t work is a natural generalization of double loop network, which was first considered by Wong and Coppersmith

. A circulant undirected graph, denoted by G(n;±

{1, 2… j}), 1<j  n/2, n  3 is defined as an undirected graph consisting of the vertex set V = {0, 1 … n – 1} and the edge set E = {(i, j): j  i  s (mod n ) , s{1, 2 … j}}.

The graph in Figure 1 is the circulant graph G(8; ±{1, 2}). It is also clear that G(n;±1) is an undirected cycle C _n and G(n; ±{1, 2 … n/2}) is a complete graph K _n . We observe e that C _n = G(n, ±1) is a subgraph of G(n; ±{1, 2 … j}) for every j, 1 < j 

n/2.

Figure 1. Circulant undirected graph G (8; {1,2})

A circulant digraph was proposed by Elspas and Turner

. A circulant digraph, denoted by G(n; {1, 2 … j}), 1<j  n–1, n2 is defined as a digraph consisting of the vertex set V = {0, 1

… n – 1} and the edge set E = {(i, j):

) (mod n s

i

j   , s  {1, 2 … j}}. See Figure 2. It is clear that G(n; 1) is a directed cycle C _n and G(n; {1,2…n–1 }) is a complete digraph K _n ^.

Theoretical properties of circulant graphs have been studied extensively and are surveyed in

. A circulant graph

Figure 2. Circulant digraph G (8; {1, 2})

is hamiltonian, and is a Cayley graph.

In this paper, we make use of an interesting property of circulant graphs, that it is diametrically uniform

.

For each vertex u of a graph G, the maximum distance d(u, v) to any ot he r ve rt ex v of G is c a lle d its eccentricity and is denoted by ecc(u).

In a graph G, the maximum value of eccentricity of vertices of G is called the diameter of G and is denoted by

 . Let G be a graph with diameter  . A vertex v of G is said to be diametrically opposite to a vertex u of G, if d

(u, v)=  . A graph G is said to be a diametrically uniform graph if every vertex of G has at least one diametrically opposite vertex.

The set of diamet rically opposite vertices of a vertex x in G is denoted by D(x).

4. Minimum Metric Dimension of Circulant Networks

Theorem 1

: Let G be a graph with minimum metric dimension 2 and let {u,v}V be a metric basis in G.

Then the following are true:

a) There is a unique shortest path between u and v.

b) The degree of each u and v is at most 3.

Since G(n;±{1, 2 … j}), 1<j 

n/2  is 2 j-re gula r, we have the following Lemma as an application of Theorem 1.

0 7

6

5

4

3

2

Lemma 1:  (G(n; ±{1, 2 … j}))

> 2, 1 < j  n/2.

Proposition 1: The undirected c ircula nt gra ph G (n;± {1 , 2 }) is diametrically uniform. Moreover, for any vertex x in G(n; ±{1, 2}), the set D(x) of diametrically opposite vertices of x satisfies

1. D ( x )  1 if n = 4l + 2 2. D ( x )  2 if n = 4l + 3 3. D ( x )  3 if n = 4l 4. D ( x )  4 if n = 4l + 1.

T he proof of Propos it ion 1 follows by considering the breadth first search tree rooted at the vertex x of the graph.

Definition 1: For any vertex x of G(n; {1, 2}), the geometric diameter of C _n through x is called a mirror.

For any x  V ^{, if} D (x ) ^{is odd,,} then the mirror through x pas ses through one of the members of D(x).

This vertex is denoted by x*.

Proposition 2: Let t =  ^{(  n 1 ) / 2}  ^.

Then







 



















 ^{ }

3 4 }

, {

2 4

*}

{

1 4 }

, , , {

4 }

,

*, { )

( ^{1 1}

l n if b

a

l n if a

l n if b b a a

l n if b

a a a D

t t

t t t t

t t

for any vertex a of G(n; ±{1, 2}).

Fix a  V . Then the mirror through the vertex a divides the vertex set of G(n;

±{1, 2}) into two sets S

and S

. Let S

= { a ₁ , a ₂ ,..., a _t ^{}, S}

= { b ₁ , b ₂ ,..., b _t ^} for t=  ^{(  n 1 ) / 2}  ^{. Then} a i ^and b _i ^are images of each other with respect to the mirror through a. See Figure 3.

Figure 3. G (12; {1, 2})

Lemma 2: Let G(n; ±{1, 2}) be an undirected circulant graph where n=4l, 4l+ 2 or 4l + 3, l  1. Let M = {a, a

}. Then any two vertices in

S 1 ⁽ S ₂ ⁾

have distinct M-coordinates.

Proof: Let u , v  S ₁ be such that d(a, u)=d(a, v). Then u and v are conse- cutive vertices on C _n ^{. Let} u  a _i ^,

1

 a _i 

v . Without loss of generality,, let i be odd. The s hort e s t pa ths

a i

a a

aa _{1 3 5} ... ^and aa 2 a 4 ... a _{i } 1 are of

equal length. Then the lengths of shortest paths aa ₂ a ₄ ... a _{i  1} ^and a ₁ a ₂ a ₄ ... a _{i  1} are the same and

d(a

, a

) = d(a, a

) – 1 = d(a,a

) – 1

= d(a

, a

)–1  d(a

, a

).

Thus any two vertices in S

have distinct M-coordinates. The proof is similar for any two vertices in S

.

Theorem 2: Let G(n; ±{1, 2}) be an undirected circulant graph. Then, for l  1 ^,

1.  (G (n; ±{1, 2}) = 3,

when n = 4l, 4l + 2, 4l +3.

2. 2 <  (G (n; ±{1, 2})  4, when n = 4l + 1.

Proof: Let n = 4l. Let a be a vertex of G(4l;±{1,2}). By the structure of G(4l; ±{1, 2}), vertices a _{2  i 1} ^, a _{2 i} ^,

1 2  i

b ^, b _{2 i} ^{, 1  i }  ^{t 2}  are at distance i from a. Note that i < . The vertices at distance  are a*, a _t ^, b _t where a* is a vertex on the mirror through a. Let M={a, a

}. Then by Lemma 2, any two vertices in S

(S

) have distinct M-coor- dinates. Now among all pairs (x, y), x

 S

, y  S

only the pairs ( a _{2 i} ^, b _{2  i 1} ^),

 ²  ¹

1  i  t  , where a ^{2 (}  ^{t 2}  ^{ 1 )} _{= a*,} are equidistant from both a and b. Now, include a ₂ in M. It ready follows that d( a ₄ ^, a ₂ ^)d( b ₃ ^, a ₂ ). Now, for or i>2, d( a _{2 i} ^, a ₂ ^{) = d(} a _{2 i} ^, a ₄ ^{) + d(} a ₄ ^, a ₂ ⁾

= i – 2 + d( a ₄ ^, a ₂ ^).

Figure 4. G (9; {1, 2}) and M = { a , p , q , r } Similarly, for i > 2,

d( b _{2  i 1} ^, a ₂ ^)=d( b _{2  i 1} ^, b ₃ )+d( b ₃ ^, a ₂ ⁾

= i – 2 + d( b ₃ ^, a ₂ ^).

Hence d( a _{2 i} ^, a ₂ ^{)  d(} b 2  i 1 ^, a ₂ ^{), for}

 ²  ¹

1  i  t  . By Theorem 1,  (G(n;±

{1, 2}) > 2. Thus M = {a, a ₁ ^, a ₂ ^{} is a}

minimum metric basis. Proofs for G(n;

±{1,2}) when n=4l+2, 4l+3 or 4l+1 are similar to that of G(n; ±{1, 2}).

It is important to note that the mirror through the vertex a does not pass through any of the diametrically opposite vertices of a when n= 4l +1 or 4l+ 3. See Figure 4 and Figure 5.

Figure 5. G (11; {1, 2}) and

M = { a , p , q }

The following theorems can be proved proceeding along the same lines as in Theorem 2.

Theorem 3: Let G(n; ±{1, 2, 3}), n  8 be a circulant undirected graph. Then

1. 2 <  (G(n, ±{1, 2, 3}))  5, when n = 6l, l  2 ^.

2. 2 <  (G(n, ±{1, 2, 3}))  4, when n = 6l + 2, 6l+ 4, l  1 ^. 3. 2 <  (G(n, ±{1, 2, 3}))  5,

when n = 6l + 1, l  2 ^. 4. 2 <  (G(n, ±{1, 2, 3}))  4,

when n = 6l + 3, 6l + 5, l  1 ^.

Figure 6. G (9; {1, 2, 3}) and M = { p , q , r }

A metric basis of a digraph G(V, E) is a subset M  V such that for each pair of vertices u, v V \ M, there exists a vertex w  M such that d(w, u)

d(w, v). We have the following conjec- ture. See Figure 6.

Conjecture 1: Let G (n; {1, 2… j}

be a directed circulant graph. Then

 (G (n; {1, 2 … j}) = j, 1  j  n  1 .

4. CO NCLUSI ON

We have obtained the minimum metric dimension of undirected circulant graphs G(n; ±{1, 2}), G(n; ±{1, 2, 3}). The problem for G(n; {1, 2 … j}), j > 2 is under investigation.

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