Cmjv01i07p0831 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol. 1(7), 831-840 (2010)

Metric Dimension of Uniform and Quasi-uniform Theta Graphs ¹BHARATI RAJAN, ¹INDRA RAJASINGH and ²VENUGOPAL, P. ¹ Department of Mathematics, Loyola College, Chennai 600 034, India ² Department of Mathematics, SSN College of Engg., Kalavakkam 603 110, India venu_27@rediffmail.com ABSTRACT Let

be an ordered set of vertices in a graph is called the M-

G. Then

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. In this paper, we determine the minimum metric dimension of the class of uniform theta graphs, an upper bound for the minimum metric dimension of quasi uniform theta graphs, series-parallel connection of uniform theta graphs and oriented uniform theta graphs. Keywords: Generalized theta graph, metric basis, minimum metric dimension

1. INTRODUCTION A generalized theta graph consists of a pair of end vertices joined by n internally disjoint paths of lengths , where denote the number of internal vertices in the respective paths. We call the end vertices as North Pole (N) and South Pole (S). A path between the North Pole and the South Pole is called as a longitude and is denoted by L. Figure 1 shows a theta graph with four longitudes.

longitude

S Figure 1: A generalized theta graph with 4 longitudes

A generalized theta graph G with ℓ longitudes is said to be a

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Bharati Rajan, et al., J. Comp. & Math. Sci. Vol. 1(7), 831-840 (2010)

832

graph1 if . Here stands for the number of internal vertices of . A uniform theta graph with â&#x201E;&#x201C; longitudes and k vertices on each longitude is denoted by . Figure 2(a) is a and uniform theta graph with uniform

theta

(a)

(b)

Figure 2: (a) Uniform theta graph Î¸(6,4) (b) Quasiuniform theta graph

A generalized theta graph G with â&#x201E;&#x201C; longitudes is said to be quasiuniform . The if graph in Figure 2(b) is a quasi-uniform theta graph. A quasi-uniform theta graph is said to be even or odd according as | is even or odd. See Figure 3. Clearly a uniform theta graph is an even quasiuniform theta graph. N

(a)

(b)

Figure 3: (a) An odd quasi-uniform theta graph (b) An even quasi-uniform theta graph

2. AN OVERVIEW OF THE PAPER Let be an ordered set of vertices in a graph G. Then 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 12 by . The minimum metric dimension (MMD) problem is to find a minimum metric basis. If M is a metric basis then it is clear that for any two vertices x and y of V \ such that M there is a vertex Khuller et al.13 described the application of this problem in the field of computer science. Further, this concept has wide applications in motion planning in the field of robotics, sonar and loran stations, pharmaceutical chemistry, combinatorial search and optimization. The concept of metric basis and minimum metric basis has appeared in the literature under a different name as early as 1975. Slater in18 and later in19 had called a metric basis and a minimum metric basis as a locating set and a reference set 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. Chartrand et al.11 have called a metric basis and a minimum metric basis as a resolving set and a minimum resolving set. If G has p vertices then it is clear . Harary et al.12 that have shown that for the complete graph ,

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Bharati Rajan, et al., J. Comp. & Math. Sci. Vol. 1(7), 831-840 (2010)

graph

the minimum metric dimensions x

are given by and

and the complete bipartite

the cycle

833

. This problem 14

has been studied for grids , trees, multidimensional grids13, Petersen graphs1, torus networks3, oriented butterfly network5, butterfly derived architectures6, Illiac networks7, enhanced hypercube networks4, circulant networks8, binary tree derived architectures9, De Bruijn graphs and Kautz networks15, Benes and butterfly networks16 and honeycomb networks17. The MMD problem is NP-complete for general graphs10. Recently Manuel et al.16 have proved that this problem is NPcomplete for bipartite graphs by a reduction from 3-SAT, thus narrowing down the gap between the polynomial classes and NPcomplete classes of the MMD problem. 3. MINIMUM METRIC DIMENSION OF UNIFORM THETA GRAPHS We begin this section with an interesting result that provides a lower bound for the minimum metric dimension of the class of uniform theta graphs. Lemma 1: Let G be a uniform theta graph . Let M be a metric basis of G. Then any cycle of G has a member of M. Proof: Suppose there is a cycle, say C in G which has no vertex of M. . Now any two vertices Then x and y of C equidistant from N are equidistant from every vertex of M. See Figure 4. This contradicts the fact that M is a metric basis.

Figure 4: Uniform theta graph with two vertices x and y marked on a cycle C

Since there are distinct cycles , Lemma 1 gives the following

in result. Lemma 2: (Lower bound) .â&#x2013; We next prove that the above lower bound is an upper bound too for Lemma3:

Proof: We exhibit a metric basis of cardinality . Let be the be the vertex neighbours of N and let adjacent to . Let be the internal vertices on the longitude . See Figure 5. We claim that is a metric basis for . N x1

-2

xl-1

xl = y1 a

k -1

S Figure 5 : An example to illustrate different cases in Lemma 3

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Bharati Rajan, et al., J. Comp. & Math. Sci. Vol. 1(7), 831-840 (2010)

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Let . Case (i): u and v are not poles. Suppose . Then we have Now suppose and

If then any if then or j.

, , for . On the other hand , , where t is either i

Case (ii): Suppose u is a pole and v is not. and . Let Then any will apply if except case,

for

, for . The same argument and .

this

Case (iii): Let . Then we have for any i, since . Hence for every pair of , there exists a vertex vertices such that . This means that .■ Lemmas 2 and 3 imply the following theorem. Theorem 1: ■ A simple calculation yields the following result. Theorem 2: .■

When , the graph reduces to a cycle. Thus we deduce the following result of Harary and Melter 12. Corollary 1: If G is a cycle, then .■ When , the graph reduces to a path. Thus we deduce the following result of Khuller et al.13. Corollary 2: The minimum metric dimension of a path is 1.■ 4. MINIMUM METRIC DIMENSION OF QUASI-UNIFORM THETA GRAPHS Theorem 3: Let G be an even quasi-uniform longitudes theta graph with and k vertices, on each , and t vertices on . Then . Proof: We exhibit a metric basis of cardinality . Let , be vertices on the longitude Let , be the vertices on

. . See

Figure 6. N y1

x1j

xi(j

x2 j

x(l

1)j

xi j x

i(j +1 )

S Figure 6 : An even quasi-uniform theta graph with k = 8 and t = 4

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Bharati Rajan, et al., J. Comp. & Math. Sci. Vol. 1(7), 831-840 (2010)

The longitudes in G. Let and

divide

and

where

and induce a cycle be a vertex such that S

Case (ii): Suppose u is a pole and v is not. Let and . -section of

If v belongs to the

into equal segments

then .

and

claim that

. If for

metric basis for G. Now let be any two vertices of G. Case (i): u and v are not poles. Let . Suppose u and v lie on then .

If some

Suppose

Case (iii): Suppose In this case,

with In

this

for

case

then

for . Then since the shortest

path from

to v passes through N.

and

then

is a

and

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. for any

.â&#x2013;

See Figure 7. N P'i

Theorem 4: Let G be an odd quasi-uniform theta graph with longitudes and k vertices, on each and t vertices on . Then .

v x i( j-1 )

x1 j

xi j u L1

xi( j+1) Li

Proof: We exhibit a metric basis of cardinality . As in Theorem 3, let be vertices on the longitude

. be the vertices on

longitudes

and

S Figure 7 : An example to illustrate theorem 3

Suppose If

induce a cycle

then to v passes through u.

and

is an odd cycle, there are two and such

that

. , as the shortest path

from

G. Since vertices

Let . The

. That is S divide

into segments

where the length of say the length of

and

is one less than

. Let

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Bharati Rajan, et al., J. Comp. & Math. Sci. Vol. 1(7), 831-840 (2010)

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Let

Suppose u and v both lie on then .

and

We claim that

or Suppose

and is a metric basis of G for suitably chosen s. See Figure 8.

with

case, N 1

x1 j x2j

xi(j

xi j

x(l

in M, we

observe that S and are equidistant from all these vertices. So we choose in order to distinguish this pair. Pairs of vertices equidistant from North Pole belonging to and are equidistant from . So we so that the length of the path is one less than the of

. Case (i): u and v are not poles.

. . In this case .

the

, then

. If

and if

Figure 4: An odd quasi-uniform theta graph with k = 8 and t = 5

length path

Suppose

choose

for

Case (ii): Suppose u is a pole and v is not. Let u = N and .

Having chosen

. In this

for 2 )s

or Let and

x i(j + 1)

this

for

Let and case

then for

then for some i, 1 ≤ i

≤ . Case (iii): u = N, v = S. In this case, i,

for any

.■

5. MINIMUM METRIC DIMENSION OF SERIES-PARALLEL CONNECTION OF UNIFORM THETA GRAPHS A series-parallel graph represents a network obtained by repeating "series connection" and "parallel connection". The subclass of directed series-parallel graphs plays an important role in computer science.

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Bharati Rajan, et al., J. Comp. & Math. Sci. Vol. 1(7), 831-840 (2010)

A (two-terminal) series-parallel graph is defined recursively as follows20: i. A graph G of a single edge is a series-parallel graph. The ends of the edge are called the and terminals of G and are denoted by and . ii. Let G₁ be a series-parallel graph with terminals and and let G₂ be a series-parallel graph with terminals and . a) A graph obtained from G₁ and G₂ by identifying vertex with vertex is a series-parallel graph whose terminals are and . Such a connection is called a series connection and G is denoted by . See Figure 9.

vs(G) = vs (G1)

vt (G)= vt (G2)

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vs (G) = vs(G1 )= vs(G2)

vt (G) = vt (G1) = vt(G2 ) G2

Figure 6: Parallel connection

Theorem 5: If G1 and G₂ are uniform theta graphs with ℓ longitudes each and if then . Proof: Let be a uniform theta graph with and terminals Let be obtained by identifying with . Then and . and have ℓ longitudes each Suppose and k vertices on each longitude. Define a set for as in Lemma 3. Then any two vertices of are distinguished by at least one member of .

N1 x1

G1 xl-2

a Figure 5: Series connection

b) A graph obtained from G₁ and G₂ by identifying vertex with vertex and identifying with is a seriesparallel graph whose terminals are and . Such a connection is called a parallel connection and G is denoted by . See Figure 10.

u y

S 1= N2 v y

l-2

b G2 S2 Figure 11: Series connection of two uniform theta graphs

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Bharati Rajan, et al., J. Comp. & Math. Sci. Vol. 1(7), 831-840 (2010)

838

Now . Then two vertices of G₂ equidistant from S₁ are equidistant from every vertex of M₁. Hence . M₁ is not a metric basis for To distinguish pairs of vertices of G₂, we define as we did in the case of G₁. See Figure 11. But is not a metric basis for G because the vertices u and v which are neighbours of S₁ lying on the longitudes of G₁ and G₂ not containing any landmarks, are equidistant from every vertex of . This forces us to adjoin either u or v . to Hence .■ Theorem 6: Let , Then .■ If G₁ and G₂ are uniform theta graphs each isomorphic to then is also a uniform theta graph isomorphic to . Thus we have the following result.

Theorem 8: Let G be . Then . Proof: Let , where the neighbours of N. We claim that M is a metric basis for G. Let . Case (i): u and v are not poles. , If then . If and then , k is either i or j. By our orientation, we have for . Case (ii): u is a pole and v is not. Let . If Then for any

N y

Theorem 7: Let

l-2

where each is isomorphic to Then

Ll-2

Ll-1

.■

6. MINIMUM METRIC DIMENSION OF ORIENTED UNIFORM THETA GRAPHS Let graph. Let of G.

be a uniform theta

be the longitudes Orient the longitudes from N to S and the last longitude from S to N. We denote the resulting oriented graph as .

Figure 7: A uniform theta graph in which first longitudes are oriented from the N to S and the last longitude from S to N

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Bharati Rajan, et al., J. Comp. & Math. Sci. Vol. 1(7), 831-840 (2010)

Case (iii): u = N and v = S. By our orientation,

any

we for Therefore, there exists

have any for some

such that . This implies that M is a metric basis and consequently .■ 7. CONCLUSION In this paper, we have determined the minimum metric dimension of uniform theta graphs for both undirected and directed categories. We have obtained an upper bound for the minimum metric dimension of undirected quasi-uniform theta graphs. We have extended our study to the series and parallel connection of uniform theta graphs. The problem of finding the minimum metric dimension of generalized theta graphs θ (s₁, s₂ ... sn) for both undirected and directed is still open. The problem can be extended for series and parallel connection of such theta graphs. BIBLIOGRAPHY 1. Bharati Rajan, Indra Rajasingh, Cynthia J. A., Paul Manuel, On Minimum Metric Dimension, The Indonesia-Japan Conference on Combinatorial Geometry and Graph Theory, Bandung, Indonesia, September 13-16, (2003). 2. Bharati Rajan, Indra Rajasingh, Amutha A., Paul Manuel, Minimum Tree Spanner Problem for Generalized Theta Graphs, Proceedings of the Third National Conference on Mathematical and Computational Models, pages 441447, December 15-16, (2005).

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3. Bharati Rajan, Indra Rajasingh, Chris Monica M., Paul Manuel, Landmarks in Torus Networks, Journal of Discrete Mathematical Sciences and Cryptography, vol. 9, no.2, pages 263 -271, (2006). 4. Bharati Rajan, Indra Rajasingh, Chris Monica M., Paul Manuel, Minimum Metric Dimension of Enhanced Hypercube Networks, Journal of Combinatorial Mathematics and Combinatorial Computation, vol. 67, pages 5-15, (2008). 5. Bharati Rajan, Indra Rajasingh, Venugopal P., Minimum Metric Dimension of Oriented Butterfly Network, 5th Asian Mathematical Conference, AMC 2009, Kuala Lumpur, Malaysia, June 22-26, (2009). 6. Bharati Rajan, Indra Rajasingh, Venugopal P., Chris Monica M., Metric Dimension of Butterfly Derived Architectures, Proceedings of the International Conference on Mathematics and Computer Science, ICMCS 2009, Chennai, India, pages 164-167, (2009). 7. Bharati Rajan, Indra Rajasingh, Venugopal P., Chris Monica M., Minimum Metric Dimension of Illiac Networks, Ars Combinatoria (To appear). 8. Bharati Rajan, Indra Rajasingh, Chris Monica M., Paul Manuel, On Minimum Metric Dimension of Circulant Networks, Journal of Computer and Mathematical Sciences, vol. 1, no. 2, pages 155-162, (2010). 9. Bharati Rajan, Indra Rajasingh, Chris Monica M., Paul Manuel, Landmarks in Binary Tree Derived Architectures, To appear in Ars Combinatoria.

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Bharati Rajan, et al., J. Comp. & Math. Sci. Vol. 1(7), 831-840 (2010)

10. Garey M. R., Johnson D. S., Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Company, New York, (1979). 11. Gary Chartrand, Linda Eroh, Mark A. Johnson, Ortrud R. Oellermann, Resolvability in Graphs and the Metric Dimension of a Graph, Discrete Applied Mathematics, vol. 105, pages 99-113, (2000). 12. Harary F., Melter R. A., On the Metric Dimension of a Graph, Ars Combinatoria, vol. 2, pages 191-195, (1976). 13. Khuller S., Ragavachari B., Rosenfeld A., Landmarks in Graphs, Discrete Applied Mathematics, vol. 70, pages 217-229, (1996). 14. Melter R. A., Tomcscu I., Metric Bases in Digital Geometry, Computer Vision, Graphics and Image Processing, vol. 25, pages 113-121, (1984). 15. Paul Manuel, Bharati Rajan, Indra Rajasingh, Cynthia J. A., NPCompleteness of Minimum Metric Dimension Problem for Directed

16.

17.

18. 19.

20.

Graphs, Proceedings of the International Conference on Computer and Communication Engineering, ICCCE'06, Malaysia, vol. 1, pages 601605, May 9-11, (2006). Paul Manuel, M. I. Abd-El-Barr, Indra Rajasingh, Bharati Rajan, An Efficient Representation of Benes Networks and its Applications, Journal of Discrete Algorithms, vol. 6, no.1, pages 11-19, (2008). Paul Manuel, Bharati Rajan, Indra Rajasingh, Chris Monica M., On Minimum Metric Dimension of Honeycomb networks, Journal of Discrete Algorithms, vol. 6, no. 1, pages 20-27, (2008). Slater P. J., Leaves of Trees, Congr. Numer., vol. 14, pages 549-559, (1975). Slater P. J., Dominating and Reference Sets in a Graph, Journal of Mathematical and Physical Sciences, vol. 22, pages 445-455, (1988). Zhou X., Takao Nishizeki, Multicolorings of Series-Parallel Graphs, Algorithmica, vol. 38, No. 2, pages 271-297, February (2004).

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)