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Intersections and circuits in sets of line segments

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Abstract

Sets of straight line segments with special structures and properties appear in various applications of geometric modeling, such as scientific visualization, computer-aided design, and medical image processing. In this paper, we derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. In particular, we consider sets of segments whose underlying planar graphs are Halin graphs, cactus graphs, maximal planar graphs, and triangle-free planar graphs, as well as randomly produced segment sets.

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Notes

  1. A graph is maximal outerplanar if it has a plane embedding in which all vertices belong to the outer face, and adding any edge to the graph causes it to no longer have this property.

  2. We use this nomenclature instead of triangle-free graph in order to avoid confusion between geometric and graph theoretic triangles.

  3. The nomenclature is derived from Buffon’s Needle Problem, which investigates the probability that a needle will fall on a line when dropped on a sheet with equally spaced lines.

References

  • Balaban IJ (1995) An optimal algorithm for finding segment intersections. In Proceedings of the eleventh annual symposium on computational geometry, pp. 211–219

  • Ben-Moshe B, Dvir A, Segal M, Tamir A (2012) Centdian computation in cactus graphs. J Gr Algorithms Appl 16(2):199–224

    Article  MathSciNet  Google Scholar 

  • Bentley JL, Ottmann TA (1979) Algorithms for reporting and counting geometric intersections. IEEE Trans Computers 9:643–647

    Article  Google Scholar 

  • Bose P, Maheshwari A, Morin P, Morrison J, Smid M, Vahrenhold J (2007) Space-efficient geometric divide-and-conquer algorithms. Comput Geom 37(3):209–227

    Article  MathSciNet  Google Scholar 

  • Brévilliers M, Chevallier N, Schmitt D (2007) Triangulations of line segment sets in the plane. In International conference on foundations of software technology and theoretical computer science, pp. 388–399

  • Brimkov B (2017) On sets of line segments featuring a cactus structure. In International workshop on combinatorial image analysis, pp. 30–39

  • Brimkov B, Hicks IV (2017) Memory efficient algorithms for cactus graphs and block graphs. Discrete Appl Math 216:393–407

    Article  MathSciNet  Google Scholar 

  • Brimkov VE (2013) Approximability issues of guarding a set of segments. Int J Computer Math 90(8):1653–1667

    Article  Google Scholar 

  • Brimkov VE, Leach A, Mastroianni M, Wu J (2011) Guarding a set of line segments in the plane. Theor Computer Sci 412(15):1313–1324

    Article  MathSciNet  Google Scholar 

  • Brimkov VE, Leach A, Wu J, Mastroianni M (2012) Approximation algorithms for a geometric set cover problem. Discrete Appl Mathematics 160:1039–1052

    Article  MathSciNet  Google Scholar 

  • Chan TM, Chen EY (2010) Optimal in-place and cache-oblivious algorithms for 3-d convex hulls and 2-d segment intersection. Comput Geom 43(8):636–646

    Article  MathSciNet  Google Scholar 

  • Chazelle BM (1983) Reporting and counting arbitrary planar intersections. Report CS-83-16, Department of Computer Science, Brown University, Providence, RI, USA

  • Chen EY, Chan TM (2003) A space-efficient algorithm for line segment intersection. In: Proceedings of the 15th Canadian conference on computational geometry, pp. 68–71

  • Cornuéjols G, Naddef D, Pulleyblank WR (1983) Halin graphs and the travelling salesman problem. Math Program 26(3):287–294

    Article  MathSciNet  Google Scholar 

  • Dujmović V, Eppstein D, Suderman M, Wood DR (2007) Drawings of planar graphs with few slopes and segments. Comput Geom 38(3):194–212

    Article  MathSciNet  Google Scholar 

  • de Castro N, Cobos FJ, Dana JC, Márquez A, Noy M (2002) Triangle-free planar graphs and segment intersection graphs. J Gr Algorithms Appl 6(1):7–26

    Article  MathSciNet  Google Scholar 

  • Durocher S, Mondal D (2018) Drawing plane triangulations with few segments. Comput Geom 77:40–45

    MathSciNet  MATH  Google Scholar 

  • Eppstein D (2016) Simple recognition of Halin graphs and their generalizations. J Gr Algorithms Appl 20(2):323–346

    Article  MathSciNet  Google Scholar 

  • Francis MC, Kratochvíl J, Vyskočil T (2012) Segment representation of a subclass of co-planar graphs. Discrete Math 312(10):1815–1818

    Article  MathSciNet  Google Scholar 

  • Green B, Tao T (2013) On sets defining few ordinary lines. Discrete Comput Geom 50(2):409–468

    Article  MathSciNet  Google Scholar 

  • Harary F, Uhlenbeck G (1953) On the number of Husimi trees: I. Proc Nat Acad Sci 39:315–322

    Article  MathSciNet  Google Scholar 

  • Horton SB, Parker RG (1995) On Halin subgraphs and supergraphs. Discrete Appl Math 56(1):19–35

    Article  MathSciNet  Google Scholar 

  • Husimi K (1950) Note on Mayers’ theory of cluster integrals. J Chem Phys 18:682–684

    Article  MathSciNet  Google Scholar 

  • Igamberdiev, A, Meulemans W, Schulz A (2015) Drawing planar cubic 3-connected graphs with few segments: algorithms and experiments. In International symposium on graph drawing and network visualization, pp. 113–124

  • Kára J, Kratochvíl J (2006) Fixed parameter tractability of independent set in segment intersection graphs. In International workshop on parameterized and exact computation, pp. 166–174

  • Kariv O, Hakimi SL (1979) An algorithmic approach to network location problems. I: the p-center. SIAM J Appl Math 37(3):513–538

    Article  MathSciNet  Google Scholar 

  • Koontz WLG (1980) Economic evaluation of loop feeder relief alternatives. Bell Syst Tech J 59(3):277–281

    Article  Google Scholar 

  • Kumar CP (2019) On the regions formed by the diagonals of a convex polygon. arXiv:1904.04065

  • Mairson HG, Stolfi J (1988) Reporting and counting intersections between two sets of line segments. In: Earnshaw RA (ed) Theoretical foundations of computer graphics and CAD. Springer, Berlin, pp 307–325

    Chapter  Google Scholar 

  • Oliveros D, O’Neill C, Zerbib S (2020) The geometry and combinatorics of discrete line segment hypergraphs. Discrete Math 343(6):111825

    Article  MathSciNet  Google Scholar 

  • Poonen B, Rubinstein M (1998) The number of intersection points made by the diagonals of a regular polygon. SIAM J Discrete Math 11(1):135–156

    Article  MathSciNet  Google Scholar 

  • Preparata F, Shamos MI (2012) Computational geometry: an introduction. Springer, New York

    MATH  Google Scholar 

  • Rappaport D, Imai H, Toussaint GT (1990) Computing simple circuits from a set of line segments. Discrete Comput Geom 5(3):289–304

    Article  MathSciNet  Google Scholar 

  • Samee MAH, Alam MJ, Adnan MA, Rahman MS (2008) Minimum segment drawings of series-parallel graphs with the maximum degree three. In International symposium on graph drawing, pp. 408–419

  • Sysło MM, Proskurowski A (1983) On Halin graphs. In: Graph theory, Springer, Berlin, pp 248–256

  • Tiernan JC (1970) An efficient search algorithm to find the elementary circuits of a graph. Commun ACM 13:722–726

    Article  MathSciNet  Google Scholar 

  • Vahrenhold J (2007) Line-segment intersection made in-place. Comput Geom 38:213–230

    Article  MathSciNet  Google Scholar 

  • Wagner K (1936) Bemerkungen zum Vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung 46:26–32

    MATH  Google Scholar 

Download references

Acknowledgements

We thank two anonymous referees for their helpful suggestions which helped improve the paper. We thank Edinah Gnang for suggesting the study of Buffon segments and Stephen Hartke for several useful discussions. This work is partially supported by NSF-DMS Grants 1603823 and 1604458.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Slippery Rock University, Slippery Rock, PA, 16057, USA

    Boris Brimkov

  2. Department of Mathematics, San Jose State University, San Jose, CA, 95192, USA

    Jesse Geneson

  3. Department of Mathematics and Computer Science, Susquehanna University, Selinsgrove, PA, 17870, USA

    Alathea Jensen

  4. Department of Mathematics and Statistics, Washington State University, Pullman, WA, 99164, USA

    Jordan Broussard

  5. Broad Institute of MIT and Harvard, Cambridge, MA, 02142, USA

    Pouria Salehi Nowbandegani

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  1. Boris Brimkov
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  2. Jesse Geneson
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  3. Alathea Jensen
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  4. Jordan Broussard
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  5. Pouria Salehi Nowbandegani
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Correspondence to Boris Brimkov.

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Brimkov, B., Geneson, J., Jensen, A. et al. Intersections and circuits in sets of line segments. J Comb Optim 44, 2302–2323 (2022). https://doi.org/10.1007/s10878-021-00731-3

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