Abstract
In this paper we study properties of intersection graphs of k-bend paths in the rectangular grid. A k-bend path is a path with at most k 90 degree turns. The class of graphs representable by intersections of k-bend paths is denoted by B k -VPG. We show here that for every fixed k, B k -VPG \(\subsetneq\) B k + 1-VPG and that recognition of graphs from B k -VPG is NP-complete even when the input graph is given by a B k + 1-VPG representation. We also show that the class B k -VPG (for k ≥ 1) is in no inclusion relation with the class of intersection graphs of straight line segments in the plane.
Research support by Czech research grants CE-ITI GAČR P202/12/6061 and GraDR EUROGIGA GAČR GIG/11/E023.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Asinowski, A., Cohen, E., Golumbic, M.C., Limouzy, V., Lipshteyn, M., Stern, M.: String graphs of k-bend paths on a grid. Electronic Notes in Discrete Mathematics 37, 141–146 (2011)
Asinowski, A., Cohen, E., Golumbic, M.C., Limouzy, V., Lipshteyn, M., Stern, M.: Vertex Intersection Graphs of Paths on a Grid. Journal of Graph Algorithms and Applications 16(2), 129–150 (2012)
Bandy, M., Sarrafzadeh, M.: Stretching a knock-knee layout for multilayer wiring. IEEE Trans. Computing 39, 148–151 (1990)
Bellantoni, S., Ben-Arroyo Hartman, I., Przytycka, T.M., Whitesides, S.: Grid intersection graphs and boxicity. Discrete Mathematics 114, 41–49 (1993)
Chaplick, S., Cohen, E., Stacho, J.: Recognizing Some Subclasses of Vertex Intersection Graphs of 0-Bend Paths in a Grid. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 319–330. Springer, Heidelberg (2011)
Coury, M.D., Hell, P., Kratochvíl, J., Vyskočil, T.: Faithful Representations of Graphs by Islands in the Extended Grid. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 131–142. Springer, Heidelberg (2010)
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice-Hall (1999)
Golumbic, M.C., Ries, B.: On the intersection graphs of orthogonal line segments in the plane: characterizations of some subclasses of chordal graphs. To Appear in Graphs and Combinatorics
Kratochvíl, J.: String graphs II, Recognizing string graphs is NP-hard. J. Comb. Theory, Ser. B 52, 67–78 (1991)
Kratochvíl, J., Matoušek, J.: String graphs requiring exponential representations. J. Comb. Theory, Ser. B 53, 1–4 (1991)
Kratochvíl, J.: A Special Planar Satisfiability Problem and a Consequence of Its NP-completeness. Discrete Applied Mathematics 52, 233–252 (1994)
Kratochvíl, J., Matoušek, J.: Intersection Graphs of Segments. J. Comb. Theory, Ser. B 62, 289–315 (1994)
Molitor, P.: A survey on wiring. EIK Journal of Information Processing and Cybernetics 27, 3–19 (1991)
Schaefer, M., Sedgwick, E., Stefankovic, D.: Recognizing string graphs in NP. J. Comput. Syst. Sci. 67, 365–380 (2003)
Sinden, F.: Topology of thin film circuits. Bell System Tech. J. 45, 1639–1662 (1966)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chaplick, S., Jelínek, V., Kratochvíl, J., Vyskočil, T. (2012). Bend-Bounded Path Intersection Graphs: Sausages, Noodles, and Waffles on a Grill. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2012. Lecture Notes in Computer Science, vol 7551. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34611-8_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-34611-8_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34610-1
Online ISBN: 978-3-642-34611-8
eBook Packages: Computer ScienceComputer Science (R0)