WALCOM 2010: WALCOM: Algorithms and Computation pp 293-304

Pathwidth and Searching in Parameterized Threshold Graphs

• D. Sai Krishna
• T. V. Thirumala Reddy
• B. Sai Shashank
• C. Pandu Rangan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5942)

Abstract

Treewidth and pathwidth are important graph parameters that represent how close the graph is to trees and paths respectively. We calculate treewidth and pathwidth on parameterized chordal and threshold graphs. We define a chordal + 1v graph as a graph that can be made into a chordal graph by removing a vertex. We give polynomial time algorithms for computing the treewidth of a chordal + 1v graph, pathwidth of a threshold + 1v graph and a threshold + 2e graph. The mixed search number of a graph is the minimum number of cops required to capture a single robber, who is hiding in the graph. We apply the algorithm to compute the pathwidth in order to compute the mixed search number of a threshold + 1v graph.

Keywords

Graph searching treewidth pathwidth parameterization threshold graphs

References

1. 1.
Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Algebraic Discrete Methods 8(2), 277–284 (1987)
2. 2.
Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybernetica 11, 1–21 (1993)
3. 3.
Bodlaender, H.L.: Treewidth: Characterizations, applications, and computations. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 1–14. Springer, Heidelberg (2006)
4. 4.
Cai, L.: Parameterized complexity of vertex colouring. Discrete Appl. Math. 127(3), 415–429 (2003)
5. 5.
Sai Krishna, D., Thirumala Reddy, T.V., Sai Shashank, B., Pandu Rangan, C.: Pathwidth and searching in parameterized threshold graphs (to appear in LNCS) (2010), http://www.cse.iitm.ac.in/~dsaikris/Site/Research_files/psptg_full.pdf
6. 6.
Flocchini, P., Huang, M.J., Luccio, F.L.: Contiguous search in the hypercube for capturing an intruder. IPDPA 01, 62 (2005)Google Scholar
7. 7.
Flum, J., Grohe, M.: Parameterized Complexity Theory, 1st edn. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2006)Google Scholar
8. 8.
Fomin, F.V., Heggernes, P., Mihai, R.: Mixed search number and linear-width of interval and split graphs. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 304–315. Springer, Heidelberg (2007)
9. 9.
Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching. Theor. Comput. Sci. 399(3), 236–245 (2008)
10. 10.
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, vol. 57. North-Holland Publishing Co., Amsterdam (2004)
11. 11.
Heggernes, P., Mihai, R.: Mixed search number of permutation graphs. In: Preparata, F.P., Wu, X., Yin, J. (eds.) FAW 2008. LNCS, vol. 5059, pp. 196–207. Springer, Heidelberg (2008)
12. 12.
LaPaugh, A.S.: Recontamination does not help to search a graph. J. ACM 40(2), 224–245 (1993)
13. 13.
Lozin, V.V., Milanic, M.: Tree-width and optimization in bounded degree graphs. In: Graph-Theoretic Concepts in Computer Science, 32nd International Workshop, WG, Bergen, Norway, June 22-24, Revised Papers, pp. 45–54 (2007)Google Scholar
14. 14.
Mahadev, N.V.R., Peled, U.N.: Threshold graphs and related topics. Annals of Discrete Mathematics, vol. 56. Elsevier Science Publishers B.V., North Holland, Amsterdam (1995)
15. 15.
Mancini, F.: Minimum fill-in and treewidth of split+ke and split+kv graphs. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 881–892. Springer, Heidelberg (2007)
16. 16.
Marx, D.: Parameterized coloring problems on chordal graphs. Theor. Comput. Sci. 351(3), 407–424 (2006)

Authors and Affiliations

• D. Sai Krishna
• 1
• T. V. Thirumala Reddy
• 1
• B. Sai Shashank
• 2
• C. Pandu Rangan
• 1
1. 1.Department of Computer Science and EngineeringIndian Institute of Technology MadrasChennaiIndia
2. 2.Department of Computer Science and EngineeringIndian Institute of Technology GuwahatiGuwahatiIndia