On the Size of Outer-String Representations
Outer-string graphs, i.e., graphs that can be represented as intersection of curves in 2D, all of which end in the outer-face, have recently received much interest, especially since it was shown that the independent set problem can be solved efficiently in such graphs. However, the run-time for the independent set problem depends on N, the number of segments in an outer-string representation, rather than the number n of vertices of the graph. In this paper, we argue that for some outer-string graphs, N must be exponential in n. We also study some special string graphs, viz. monotone string graphs, and argue that for them N can be assumed to be polynomial in n. Finally we give an algorithm for independent set in so-called strip-grounded monotone outer-string graphs that is polynomial in n.
string graph
outer-string graph
size of representation
independent set
Theory of computation~Computational geometry
Mathematics of computing~Graph theory
10:1-10:14
Regular Paper
Therese
Biedl
Therese Biedl
Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada
Supported by NSERC.
Ahmad
Biniaz
Ahmad Biniaz
Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada
Supported by NSERC Postdoctoral Fellowship.
Martin
Derka
Martin Derka
School of Computer Science, Carleton University, Ottawa, Canada
Supported by NSERC Vanier fellowship while author was a student at University of Waterloo.
10.4230/LIPIcs.SWAT.2018.10
Pankaj K. Agarwal and Nabil H. Mustafa. Independent set of intersection graphs of convex objects in 2D. Comput. Geom., 34(2):83-95, 2006.
Sergio Cabello and Miha Jejcic. Refining the hierarchies of classes of geometric intersection graphs. Electronic Notes in Discrete Mathematics, 54:223-228, 2016. URL: http://dx.doi.org/10.1016/j.endm.2016.09.039.
http://dx.doi.org/10.1016/j.endm.2016.09.039
Jean Cardinal, Stefan Felsner, Tillmann Miltzow, Casey Tompkins, and Birgit Vogtenhuber. Intersection graphs of rays and grounded segments. In Hans L. Bodlaender and Gerhard J. Woeginger, editors, Graph-Theoretic Concepts in Computer Science - 43rd International Workshop, WG 2017, Eindhoven, The Netherlands, June 21-23, 2017, Revised Selected Papers, volume 10520 of Lecture Notes in Computer Science, pages 153-166. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-68705-6_12.
http://dx.doi.org/10.1007/978-3-319-68705-6_12
Jérémie Chalopin and Daniel Gonçalves. Every planar graph is the intersection graph of segments in the plane: extended abstract. In Michael Mitzenmacher, editor, Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 631-638. ACM, 2009. URL: http://dx.doi.org/10.1145/1536414.1536500.
http://dx.doi.org/10.1145/1536414.1536500
Peter Damaschke. The hamiltonian circuit problem for circle graphs is np-complete. Inf. Process. Lett., 32(1):1-2, 1989. URL: http://dx.doi.org/10.1016/0020-0190(89)90059-8.
http://dx.doi.org/10.1016/0020-0190(89)90059-8
Martin Derka. Restricted String Representations. PhD thesis, David R. Cheriton School of Computer Science, 2017. URL: https://uwspace.uwaterloo.ca/handle/10012/12253.
https://uwspace.uwaterloo.ca/handle/10012/12253
Gideon Ehrlich, Shimon Even, and Robert Endre Tarjan. Intersection graphs of curves in the plane. J. Comb. Theory, Ser. B, 21(1):8-20, 1976. URL: http://dx.doi.org/10.1016/0095-8956(76)90022-8.
http://dx.doi.org/10.1016/0095-8956(76)90022-8
Jacob Fox and János Pach. Separator theorems and Turán-type results for planar intersection graphs. Adv. Math., 219:1070-1080, 2008.
Jacob Fox and János Pach. A separator theorem for string graphs and its applications. Combinatorics, Probability & Computing, 19(3):371-390, 2010. URL: http://dx.doi.org/10.1017/S0963548309990459.
http://dx.doi.org/10.1017/S0963548309990459
Jacob Fox and János Pach. Computing the independence number of intersection graphs. In Dana Randall, editor, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 1161-1165. SIAM, 2011. URL: http://dx.doi.org/10.1137/1.9781611973082.87.
http://dx.doi.org/10.1137/1.9781611973082.87
M. R. Garey and David S. Johnson. The rectilinear steiner tree problem in NP complete. SIAM Journal of Applied Mathematics, 32:826-834, 1977.
M. R. Garey, David S. Johnson, G. L. Miller, and Christos H. Papadimitriou. The complexity of coloring circular arcs and chords. SIAM J. Matrix Analysis Applications, 1(2):216-227, 1980. URL: http://dx.doi.org/10.1137/0601025.
http://dx.doi.org/10.1137/0601025
Sariel Har-Peled and Kent Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. In Nikhil Bansal and Irene Finocchi, editors, Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, volume 9294 of Lecture Notes in Computer Science, pages 717-728. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_60.
http://dx.doi.org/10.1007/978-3-662-48350-3_60
J. Mark Keil, Joseph S. B. Mitchell, Dinabandhu Pradhan, and Martin Vatshelle. An algorithm for the maximum weight independent set problem on outerstring graphs. Comput. Geom., 60:19-25, 2017. Appeared also in the Proceedings of CCCG 2015.
Jan Kratochvíl. String graphs. II. recognizing string graphs is np-hard. J. Comb. Theory, Ser. B, 52(1):67-78, 1991. URL: http://dx.doi.org/10.1016/0095-8956(91)90091-W.
http://dx.doi.org/10.1016/0095-8956(91)90091-W
Jan Kratochvíl and Jiří Matoušek. String graphs requiring exponential representations. J. Comb. Theory, Ser. B, 53(1):1-4, 1991. URL: http://dx.doi.org/10.1016/0095-8956(91)90050-T.
http://dx.doi.org/10.1016/0095-8956(91)90050-T
James R. Lee. Separators in region intersection graphs. In Innovations in Theoretical Computer Science, ITCS'17, 2017.
Jiří Matoušek. Near-optimal separators in string graphs. CoRR, abs/1302.6482, 2013. URL: http://arxiv.org/abs/1302.6482.
http://arxiv.org/abs/1302.6482
Matthias Middendorf and Frank Pfeiffer. The max clique problem in classes of string-graphs. Discrete Mathematics, 108(1-3):365-372, 1992. URL: http://dx.doi.org/10.1016/0012-365X(92)90688-C.
http://dx.doi.org/10.1016/0012-365X(92)90688-C
Matthias Middendorf and Frank Pfeiffer. Weakly transitive orientations, hasse diagrams and string graphs. Discrete Mathematics, 111(1-3):393-400, 1993. URL: http://dx.doi.org/10.1016/0012-365X(93)90176-T.
http://dx.doi.org/10.1016/0012-365X(93)90176-T
János Pach and Géza Tóth. Recognizing string graphs is decidable. Discrete & Computational Geometry, 28(4):593-606, 2002. URL: http://dx.doi.org/10.1007/s00454-002-2891-4.
http://dx.doi.org/10.1007/s00454-002-2891-4
Marcus Schaefer, Eric Sedgwick, and Daniel Štefankovič. Recognizing string graphs is in NP. Journal of Computer and System Sciences, 67(2):365-380, 2003.
Marcus Schaefer and Daniel Stefankovic. Decidability of string graphs. In Jeffrey Scott Vitter, Paul G. Spirakis, and Mihalis Yannakakis, editors, Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6-8, 2001, Heraklion, Crete, Greece, pages 241-246. ACM, 2001. URL: http://dx.doi.org/10.1145/380752.380807.
http://dx.doi.org/10.1145/380752.380807
F. W. Sinden. Topology of thin film rc-circuits. Bell System Technical Journal, 45:1639-1662, 1966. URL: http://dx.doi.org/10.1002/j.1538-7305.1966.tb01713.x.
http://dx.doi.org/10.1002/j.1538-7305.1966.tb01713.x
Andrew Suk. Coloring intersection graphs of x-monotone curves in the plane. Combinatorica, 34(4):487-505, 2014. URL: http://dx.doi.org/10.1007/s00493-014-2942-5.
http://dx.doi.org/10.1007/s00493-014-2942-5
Therese Biedl, Ahmad Biniaz, and Martin Derka
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode