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.
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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)
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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
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DOI: https://doi.org/10.1007/978-3-642-34611-8_28
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