J. Comp. & Math. Sci. Vol.2 (1), 37-46 (2011)
Metric Dimension of Uniform and Quasi-uniform Theta Graphs ¹BHARATI RAJAN, ¹INDRA RAJASINGH and ²VENUGOPAL P. ¹Department of Mathematics, Loyola College, Chennai 600 034, India ²Department of Mathematics, SSN College of Engg., Kalavakkam 603 110, India venu_27@rediffmail.com ABSTRACT Let ଵ , ଶ … be an ordered set of vertices in a graph G. Then , ଵ , , ଶ … , 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. 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 2 , … consists of a pair of end vertices joined by n internally disjoint paths of lengths 1, 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. A generalized theta graph G with ℓ longitudes , … ℓ is said to be a uniform
theta graph1 if | | | | | ℓ |. Here | | stands for the number of internal vertices of . N longitude L1
L2
L3
L4
S Figure 1: A generalized theta graph , , , with 4 longitudes
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
38
Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (1), 37-46 (2011)
A uniform theta graph with â„“ longitudes and k vertices on each longitude is denoted
by â„“, . Figure 2(a) is a uniform theta graph with â„“ 6 and 4.
N
N
S
S
(a)
(b)
Figure 2: (a) Uniform theta graph θ(6,4) (b) Quasi-uniform theta graph
A generalized theta graph G with â„“ longitudes ଵ , ଶ â„“ is said to be quasi-uniform if | ଵ | | ଶ | | â„“ŕŹżŕŹľ | | â„“ |. The graph in Figure 2(b) is a quasi-uniform theta graph.
N
A quasi-uniform theta graph is said to be even or odd according as | â„“ŕŹżŕŹľ | | â„“ | is even or odd.See Figure 3. Clearly a uniform theta graph is an even quasi-uniform theta graph.
N
S
S
(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 by 12. 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 \ M there is a vertex such that , , . 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
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (1), 37-46 (2011)
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 that 1 ! ! " # 1. Harary et al.12 have shown that for the complete graph $ , the cycle % and the complete bipartite graph $ , the minimum metric dimensions are given by $ " # 1, % 2 and $ , ' ( ) # 2. This problem has been studied for grids14, 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.
39
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 â„“, . 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. Then * + \ % . Now any two vertices 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.â– N C x
y
S Figure 4: Uniform theta graph with two vertices x and y marked on a cycle C
Since there are â„“ # 1 distinct cycles in â„“, , Lemma 1 gives the following result. Lemma 2: (Lower bound) â„“, - â„“ # 1. â– We next prove that the above lower bound is an upper bound too for â„“, . Lemma 3: â„“, ! â„“ # 1, â„“ 3, 2. Proof: We exhibit a metric basis of cardinality â„“ # 1. Let , â„“ be the neighbours of N and let / 0 be the vertex
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (1), 37-46 (2011)
40
adjacent to â„“ . Let â„“ , be the internal vertices on the longitude â„“ . See Figure 5. We claim that â‚ , â‚‚ . . . â„“ , / is a metric basis for â„“, .
N x1
x2
xl 2 -
xl
xl = y1 -1 a
y2
y
k -1
y
k
S Figure 5: An example to illustrate different cases in Lemma 3
Let , + \ . Case (i): u and v are not poles. Suppose , , 1 ! 3 ! â„“. Then we have , , . Now suppose and
, 3 4. If 0, 0, , then , , , for any , 5 3, 4. On the other hand if 0,
0, , then , , , where t is either i or j. Case (ii): Suppose u is a pole and v is not. Let 0 and
, 1 ! 4 ! â„“. Then , 0 , , for any , 5 4. The same argument will apply if 6 and
, 1 ! 4 ! â„“ except for
. In this case, /, /, .
Case (iii): Let 0, 6. Then we have , 0 , 6 for any i, since 2. Hence for every pair of vertices , +\ , there exists a vertex such that , , . This means that â„“, ! â„“ # 1. â– Lemmas 2 and 3 imply the following theorem. Theorem 1: â„“, â„“ # 1, â„“ 3, 2. â– A simple calculation yields the following result. Theorem 2: â„“, â„“, â„“ ! 3, ! 2. â– When â„“ 2, the graph â„“, reduces to a cycle. Thus we deduce the following result of Harary and Melter12. Corollary 1: If G is a cycle, then 2. â– When â„“ 1, the graph â„“, 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.â– 4. MINIMUM METRIC DIMENSION OF QUASI-UNIFORM THETA GRAPHS Theorem 3: Let G be an even quasi-uniform theta graph with â„“ longitudes , â„“ , â„“ 4 and k vertices, 3 on each , 1 ! 3 ! â„“ # 1, and t vertices on â„“ , 1 ! 5 ! . Then ! â„“ # 1. Proof: We exhibit a metric basis of cardinality â„“ # 1. Let , 1 ! 4 ! be vertices on the longitude , 1 ! 3 ! â„“ # 1. Let , 1 ! 4 ! 5 be the vertices on â„“ . See Figure 6.
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (1), 37-46 (2011)
N y1
x1j
x i(j
x2j
x(l
1)j
y2
xi j x
L1
1)
L2
i(j +1)
Li
Ll
1
Ll
y
t
S Figure 6: An even quasi-uniform theta graph with k = 8 and t = 4
N P'i
x1j
y1
v x i( j-1) y2
xi j
Ci
u x i( j+1) L1
Li
Ll
Pi y
t
S Figure 7: An example to illustrate theorem 3
The longitudes and ℓ induce a cycle % in G. Let be a vertex such that S and divide % into equal segments 7 and 7 ′ where 7 : ‌ 6 and 7 ′ : 0 6. We claim that , ‌ ℓ is a
41
metric basis for G. Now let , + \ be any two vertices of G. Case (i): u and v are not poles. Let , , 1 ! 3 ! ℓ # 1. Suppose u and v lie on 7 then , , . Suppose 7 and 7 ′ with ,
, . In this case , , for 3. See Figure 7. Suppose , ℓ . If , , 9 : then , : , , as the shortest path from to v passes through u. Case (ii): Suppose u is a pole and v is not. Let 0 and , 1 ! 3 ! ℓ # 1. If v belongs to the , 0 -section of 7 ′ then , : , 0 . If 7 and if , , 0 then , , 0 for 3. If ℓ then for some 9, 1 ! 9 ! 5. Then , 0 : , since the shortest path from to v passes through N. Case (iii): Suppose 0, 6. In this case, , 0 , 6 for any i, 1 ! 3 ! ℓ # 1. ■Theorem 4: Let G be an odd quasi-uniform theta graph with ℓ longitudes
, â„“ , â„“ 4 and k vertices, 3 on each , 1 ! 3 ! â„“ # 1 and t vertices on
â„“ , 1 : 5 ! . Then ! â„“ # 2. Proof: We exhibit a metric basis of cardinality â„“ # 2. As in Theorem 3, let , 1 ! 4 ! be vertices on the longitude , 1 ! 3 ! â„“ # 1. Let , 1 ! 4 ! 5 be the vertices on â„“ . The longitudes and â„“ induce a cycle % in G. Since % is an odd cycle, there are two vertices and such that 6, 6, . That is S and divide % into segments 7
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (1), 37-46 (2011)
42
and 7 ′ where the length of say 7 is one less than the length of 7 ′ . Let 7 : 6 and 7 ′ : 0 6. We claim that , ‌ ℓ , ℓ , ℓ is a metric basis of G for suitably chosen s. See Figure 8.
N y
1
x1j x2j
x i(j
1)
y
2
xi j x i(j + 1) L1
L2
Li
Ll
x(l
2)s
2
Ll
1
Ll
y
t
Suppose u and v both lie on 7 or 7 ′ then , , . Suppose 7 and 7 ′ with , , . In this case, , , for 9 3. Let , , 3 â„“ # 1, â„“ or â„“ and â„“ . In this case , , for 3 â„“ # 2 or â„“ # 3. Let and , 3 9, 1 ! 3, 9 ! â„“ # 2. Suppose , , . In this case , , for 9 3. Case (ii): Suppose u is a pole and v is not. Let u = N and , 1 ! 3 ! â„“ # 2. If 7 ′ , then , : , 0 . If 7 and if , , 0 then , , 0 for 9 3. If â„“ or â„“ then , 0 , for some i, 1 ≤ i ≤ â„“ # 2. Case (iii): u = N, v = S. In this case, , 0 , 6 for any i, 1 ! 3 ! â„“ # 2. â–
Figure 8: An odd quasi-uniform theta graph with k = 8 and t = 5
5. MINIMUM METRIC DIMENSION OF SERIES-PARALLEL CONNECTION OF UNIFORM THETA GRAPHS
Having chosen , ‌ ℓ in M, we observe that S and are equidistant from all these vertices. So we choose ℓ in order to distinguish this pair. Pairs of vertices equidistant from North Pole belonging to ℓ , ℓ and ℓ are equidistant from , 1 ! 3 ! ℓ # 4. So we choose ℓ so that the length of the path ℓ ℓ 0 is one less than the length of the path ℓ ℓ 6 0.
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. A (two-terminal) series-parallel graph is defined recursively as follows20: i. A graph G of a single edge is a seriesparallel graph. The ends and of the edge are called the terminals of G and are denoted by and .
Case (i): u and v are not poles. Let , , 1 ! 3 ! â„“ # 2.
ii.
S
Let Gâ‚ be a series-parallel graph with terminals and and let Gâ‚‚
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (1), 37-46 (2011)
a)
terminals are and . Such a connection is called a series connection and G is denoted by • . See Figure 9.
be a series-parallel graph with terminals and . A graph obtained from Gâ‚ and Gâ‚‚ by identifying vertex with vertex is a series-parallel graph whose
vs(G) = vs (G1)
43
G2
G1
vt (G) = vt (G2)
Figure 9: Series connection ૹ • ŕŤ›
G1 vt (G) = vt (G1 ) = vt (G2)
vs (G) = vs (G1) = vs (G2) G2
Figure 10: Parallel connection ૹ || ŕŤ›
b)
A graph obtained from Gâ‚ and Gâ‚‚ by identifying vertex with vertex and identifying with is a series-parallel graph whose terminals are and . Such a connection is called a parallel connection and G is denoted by
|| . See Figure 10.
Theorem 5:
N1 x1
x2
xl-2
a
y
1
u S 1= N2 v y
y
2
l-2
b
If G₠and G₂ are uniform theta graphs with ℓ longitudes each and if • then ! 2ℓ # 1. Proof: Let be a uniform theta graph with terminals 0 and 6 , 3
1,2. Let • be obtained by identifying 6 with 0 . Then 0 and 6 .
G1
G2 S2 Figure 11: Series connection of two uniform theta graphs
Suppose and have â„“ longitudes each and k vertices on each longitude. Define a
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
44
Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (1), 37-46 (2011)
set , ‌ ℓ , / for as in Lemma 3. Then any two vertices of are distinguished by at least one member of . Now 6 . Then two vertices of G₂ equidistant from S₠are equidistant from every vertex of M₠. Hence M₠is not a metric basis for • . To distinguish pairs of vertices of G₂, we define , ‌ ℓ , = as we did in the case of G₠. See Figure 11. But > is not a metric basis for G because the vertices u and v which are neighbours of S₠lying on the longitudes of G₠and G₂ not containing any landmarks, are equidistant from every vertex of > . This forces us to adjoin either u or v to > . Hence ! 2 ℓ # 1 ( 1 2ℓ # 1. ■Theorem 6: Let • • • , ) 2. Then ! ) ℓ # 1 ( 1. ■If G₠and G₂ are uniform theta graphs each isomorphic to ℓ, then || is also a uniform theta graph isomorphic to 2ℓ, . Thus we have the following result.
Proof: Let , ‌ ℓ , where , ‌ ℓ the neighbours of N. We claim that M is a metric basis for G. Let , + \ . Case (i): u and v are not poles. If , , 1 ! 3 ! ℓ, then , , . If and
, 3 4 then , , , k is either i or j. By our orientation, we have , , for 1 ! 3 ! â„“ # 2. Case (ii): u is a pole and v is not. Let 0. If , 1 ! 3 ! â„“. Then , , for any , 1 ! 3 ! â„“ # 2.
N y
1
y2
y
l-2
L1
Ll-2
L2
Ll-1
Ll
Theorem 7: Let || || || , ) 2 where each is isomorphic to â„“, . Then ! )â„“ # 1. â– 6. MINIMUM METRIC DIMENSION OF ORIENTED UNIFORM THETA GRAPHS Let â„“, be a uniform theta graph. Let , 1 ! 3 ! â„“ be the longitudes of G. Orient the longitudes , 1 ! 3 ! â„“ # 1 from N to S and the last longitude â„“ from S to N. We denote the resulting oriented graph as â„“, , ? . Theorem 8: Let G be â„“, , ? . Then ! â„“ # 2.
S Figure 12: A uniform theta graph in which first longitudes are oriented from the N to S and the last longitude from S to N
Case (iii): u = N and v = S. By our orientation, we have , , for any , 1 ! 3 ! â„“ # 2. Therefore, for any , + \ , there exists some such that , , . This implies that M is a metric basis and consequently ! â„“ # 2. â–
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (1), 37-46 (2011)
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.
5.
6.
7. 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). 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
8.
9.
10.
11.
45
Combinatorial Computation, Vol. 67, pages 5-15, (2008). 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). 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). Bharati Rajan, Indra Rajasingh, Venugopal P., Chris Monica M., Minimum Metric Dimension of Illiac Networks, Ars Combinatoria (To appear). 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). Bharati Rajan, Indra Rajasingh, Chris Monica M., Paul Manuel, Landmarks in Binary Tree Derived Architectures, To appear in Ars Combinatoria. 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). 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).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
46
Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (1), 37-46 (2011)
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 Graphs, Proceedings of the International Conference on Computer and Communication Engineering, ICCCE'06, Malaysia, Vol. 1, pages 601605, May 9-11, (2006).
16. 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). 17. 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. 18. Slater P. J., Leaves of Trees, Congr. Numer., vol. 14, pages 549-559, (1975). 19. Slater P. J., Dominating and Reference Sets in a Graph, Journal of Mathematical and Physical Sciences, Vol. 22, pages 445-455, (1988). 20. Zhou X., Takao Nishizeki, Multicolorings of Series-Parallel Graphs, Algorithmica, Vol. 38, No. 2, pages 271297, February (2004).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)