Unsolved problems in visibility graphs of points, segments, and polygons

In this survey article, we present open problems and conjectures on visibility graphs of points, segments, and polygons along with necessary backgrounds for understanding them.

[1]  Carlos M. Nicolas The Empty Hexagon Theorem , 2007, Discret. Comput. Geom..

[2]  Micha Sharir,et al.  Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes , 2010, SIAM J. Comput..

[3]  Stephen K. Wismath,et al.  Characterizing bar line-of-sight graphs , 1985, SCG '85.

[4]  Hazel Everett,et al.  Negative Results on Characterizing Visibility Graphs , 1995, Comput. Geom..

[5]  T. R. Riley,et al.  The Absence of Efficient Dual Pairs of Spanning Trees in Planar Graphs , 2006, Electron. J. Comb..

[6]  Jim Lawrence,et al.  Oriented matroids , 1978, J. Comb. Theory B.

[7]  Hossam Ahmed Elgindy,et al.  Hierarchical decomposition of polygons with applications , 1985 .

[8]  Ileana Streinu,et al.  Non-stretchable pseudo-visibility graphs , 2005, CCCG.

[9]  J. Bólyai,et al.  SOME APPLICATIONS OF GRAPH THEORY AND COMBINATORIAL METHODS TO NUMBER THEORY AND GEOMETRY , 1978 .

[10]  Christina Koch,et al.  Obstacle Numbers of Graphs , 2010, Discret. Comput. Geom..

[11]  Emo WELZL,et al.  Constructing the Visibility Graph for n-Line Segments in O(n²) Time , 1985, Inf. Process. Lett..

[12]  Subhash Suri,et al.  Reconstructing visibility graphs with simple robots , 2012, Theor. Comput. Sci..

[13]  Hazel Everett,et al.  Stabbing Information of a Simple Polygon , 1999, Discret. Appl. Math..

[14]  Hazel Everett,et al.  Planar segment visibility graphs , 2000, Comput. Geom..

[15]  Stefan Felsner,et al.  Thickness of Bar 1-Visibility Graphs , 2006, Graph Drawing.

[16]  Günter M. Ziegler,et al.  Oriented Matroids , 2017, Handbook of Discrete and Computational Geometry, 2nd Ed..

[17]  Kathleen Romanik Directed Rectangle-visibility Graphs Have Unbounded Dimension , 1997, Discret. Appl. Math..

[18]  Yonutz Stanchescu Planar sets containing no three collinear points and non-averaging sets of integers , 2002, Discret. Math..

[19]  Jorge Urrutia Open Problems in Computational Geometry , 2002, LATIN.

[20]  G. Freiman Foundations of a Structural Theory of Set Addition , 2007 .

[21]  Stephen K. Wismath,et al.  Orthogonal polygon reconstruction from stabbing information , 2002, Comput. Geom..

[22]  James Abello,et al.  Visibility Graphs and Oriented Matroids , 1994, GD.

[23]  Leonidas J. Guibas,et al.  Visibility of disjoint polygons , 2005, Algorithmica.

[24]  Subir Kumar Ghosh,et al.  Approximation Algorithms for Art Gallery Problems in Polygons and Terrains , 2010, WALCOM.

[25]  Csaba D. Tóth,et al.  Segment endpoint visibility graphs are Hamiltonian , 2001, Comput. Geom..

[26]  Deniz Sariöz,et al.  Convex obstacle numbers of outerplanar graphs and bipartite permutation graphs , 2011, ArXiv.

[27]  V. Soltan,et al.  The Erdos-Szekeres problem on points in convex position – a survey , 2000 .

[28]  Subhash Suri,et al.  On the Limitations of Combinatorial Visibilities , 2009 .

[29]  D. T. Lee,et al.  Computational complexity of art gallery problems , 1986, IEEE Trans. Inf. Theory.

[30]  David M. Mount,et al.  An Output Sensitive Algorithm for Computing Visibility Graphs , 1987, FOCS.

[31]  Heiko Harborth Konvexe Fünfecke in ebenen Punktmengen. , 1978 .

[32]  Yahya Ould Hamidoune,et al.  Representing a planar graph by vertical lines joining different levels , 1983, Discret. Math..

[33]  Jirí Matousek Blocking Visibility for Points in General Position , 2009, Discret. Comput. Geom..

[34]  Sue Whitesides,et al.  Rectangle Visibility Graphs: Characterization, Construction, and Compaction , 2003, STACS.

[35]  Noushin Saeedi Bidokhti On fully characterizing terrain visibility graphs , 2012 .

[36]  János Komlós,et al.  Almost tight bounds forɛ-Nets , 1992, Discret. Comput. Geom..

[37]  Esther E. Klein,et al.  On some extremum problems in elementary geometry , 2006 .

[38]  Thomas C. Shermer,et al.  Computing the maximum clique in the visibility graph of a simple polygon , 2007, J. Discrete Algorithms.

[39]  Anil Maheshwari,et al.  Characterizing and Recognizing Weak Visibility Polygons , 1993, Comput. Geom..

[40]  Hazel Everett,et al.  Recovery of Convex Hulls From External Visibility Graphs , 1993, CCCG.

[41]  Stephen G. Hartke,et al.  Further Results on Bar k-Visibility Graphs , 2007, SIAM J. Discret. Math..

[42]  共立出版株式会社 コンピュータ・サイエンス : ACM computing surveys , 1978 .

[43]  David G. Kirkpatrick,et al.  Improved Approximation for Guarding Simple Galleries from the Perimeter , 2010, Discret. Comput. Geom..

[44]  David P. Dobkin,et al.  Searching for empty convex polygons , 1988, SCG '88.

[45]  Tobias Gerken Empty Convex Hexagons in Planar Point Sets , 2008, Discret. Comput. Geom..

[46]  Stephan Eidenbenz,et al.  MAXIMUM CLIQUE and MINIMUM CLIQUE PARTITION in Visibility Graphs , 2000, IFIP TCS.

[47]  David Rappaport,et al.  Computing simple circuits from a set of line segments , 1987, SCG '87.

[48]  Yann Disser,et al.  A polygon is determined by its angles , 2011, Comput. Geom..

[49]  David R. Wood,et al.  On Visibility and Blockers , 2009, J. Comput. Geom..

[50]  Joseph O'Rourke,et al.  Vertex-edge pseudo-visibility graphs: characterization and recognition , 1997, SCG '97.

[51]  Thomas C. Shermer,et al.  Hiding people in polygons , 1989, Computing.

[52]  G. Tóth,et al.  The Erdős – Szekeres Theorem : Upper Bounds and Related Results , 2004 .

[53]  Collette R. Coullard,et al.  Distance visibility graphs , 1991, SCG '91.

[54]  Stephan Johannes Eidenbenz,et al.  (In-)Approximability of visibility problems on polygons and terrains , 2000 .

[55]  David R. Wood,et al.  On the Chromatic Number of the Visibility Graph of a Set of Points in the Plane , 2005, Discret. Comput. Geom..

[56]  David Eppstein,et al.  Drawings of planar graphs with few slopes and segments , 2007, Comput. Geom..

[57]  Jeremy P. Spinrad,et al.  Visibility Graphs of Towers , 1997, Comput. Geom..

[58]  Rolf Klein,et al.  A new upper bound for the VC-dimension of visibility regions , 2011, SoCG '11.

[59]  Fabrizio Luccio,et al.  A Visibility Problem in VLSI Layout Compaction , 1984 .

[60]  Thomas C. Shermer,et al.  On Rectangle Visibility Graphs. III. External Visibility and Complexity , 1996, CCCG.

[61]  Shiou-Yi Lin,et al.  Planar Visibility Graphs , 1994, CCCG.

[62]  Mark H. Overmars Finding Sets of Points without Empty Convex 6-Gons , 2003, Discret. Comput. Geom..

[63]  H. Warren Lower bounds for approximation by nonlinear manifolds , 1968 .

[64]  Prosenjit Bose,et al.  On Rectangle Visibility Graphs , 1996, GD.

[65]  Andranik Mirzaian Hamiltonian Triangulations and Circumscribing Polygons of Disjoint Line Segments , 1992, Comput. Geom..

[66]  Frank Plumpton Ramsey,et al.  On a Problem of Formal Logic , 1930 .

[67]  Stephan Eidenbenz,et al.  Inapproximability Results for Guarding Polygons and Terrains , 2001, Algorithmica.

[68]  Jian Shen,et al.  Characterization of [1, k]-Bar Visibility Trees , 2006, Electron. J. Comb..

[69]  Subir Kumar Ghosh On Recognizing and Characterizing Visibility Graphs of Simple Polygons , 1988, SWAT.

[70]  Michael T. Goodrich,et al.  Almost optimal set covers in finite VC-dimension , 1995, Discret. Comput. Geom..

[71]  Jan Mycielski Sur le coloriage des graphs , 1955 .

[72]  Steven Skiena,et al.  Complexity aspects of visibility graphs , 1995, Int. J. Comput. Geom. Appl..

[73]  Sariel Har-Peled Geometric Approximation Algorithms , 2011 .

[74]  Raimund Seidel,et al.  Constructing arrangements of lines and hyperplanes with applications , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[75]  Subir Kumar Ghosh,et al.  Some results on point visibility graphs , 2015, Theor. Comput. Sci..

[76]  Sung Yong Shin,et al.  Characterizing and recognizing the visibility graph of a funnel-shaped polygon , 1995, Algorithmica.

[77]  Joan P. Hutchinson,et al.  Rectangle-visibility Representations of Bipartite Graphs , 1994, Discret. Appl. Math..

[78]  János Pach,et al.  A note on blocking visibility between points , 2009 .

[79]  Noga Alon,et al.  Can visibility graphs Be represented compactly? , 1993, SCG '93.

[80]  Jorge Urrutia,et al.  Art Gallery and Illumination Problems , 2000, Handbook of Computational Geometry.

[81]  David Haussler,et al.  Learnability and the Vapnik-Chervonenkis dimension , 1989, JACM.

[82]  Leonidas J. Guibas,et al.  The power of geometric duality , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[83]  János Pach,et al.  Midpoints of segments induced by a point set , 2003 .

[84]  Hazel Everett,et al.  Recognizing Visibility Graphs of Spiral Polygons , 1990, J. Algorithms.

[85]  Chak-Kuen Wong,et al.  A note on visibility graphs , 1987, Discret. Math..

[86]  T. C. Shermer,et al.  Recent results in art galleries (geometry) , 1992, Proc. IEEE.

[87]  Roberto Tamassia,et al.  A unified approach to visibility representations of planar graphs , 1986, Discret. Comput. Geom..

[88]  David Eppstein,et al.  Spanning Trees and Spanners , 2000, Handbook of Computational Geometry.

[89]  Ömer Egecioglu,et al.  Visibility graphs of staircase polygons and the weak Bruhat order, I: From visibility graphs to maximal chains , 1995, Discret. Comput. Geom..

[90]  Subhash Suri,et al.  Simple Robots with Minimal Sensing: From Local Visibility to Global Geometry , 2007, Int. J. Robotics Res..

[91]  David R. Wood,et al.  On the Connectivity of Visibility Graphs , 2011, Discret. Comput. Geom..

[92]  G. Kalai,et al.  Guarding galleries where every point sees a large area , 1997 .

[93]  Richard Pollack,et al.  Semispaces of Configurations, Cell Complexes of Arrangements , 1984, J. Comb. Theory, Ser. A.

[94]  Yann Disser,et al.  Mapping a polygon with holes using a compass , 2014, Theor. Comput. Sci..

[95]  BORIS ARONOV,et al.  Small-size ε-nets for axis-parallel rectangles and boxes , 2009, STOC '09.

[96]  Mark de Berg,et al.  Computational geometry: algorithms and applications, 3rd Edition , 1997 .

[97]  J. O'Rourke Art gallery theorems and algorithms , 1987 .

[98]  Yann Disser Mapping Polygons , 2011 .

[99]  Ellen Gethner,et al.  Unit Bar-Visibility Layouts of Triangulated Polygons , 2004, GD.

[100]  P. Valtr Guarding galleries where no point sees a small area , 1998 .

[101]  Subir Kumar Ghosh,et al.  Visibility Algorithms in the Plane , 2007 .

[102]  John Hershberger,et al.  An optimal visibility graph algorithm for triangulated simple polygons , 1989, Algorithmica.

[103]  Ömer Egecioglu,et al.  Visibility graphs of staircase polygons with uniform step length , 1993, Int. J. Comput. Geom. Appl..

[104]  Géza Tóth,et al.  The Erdo"s-Szekeres Theorem: upper Bounds and Related Results , 2007 .

[105]  Robert E. Tarjan,et al.  Rectilinear planar layouts and bipolar orientations of planar graphs , 1986, Discret. Comput. Geom..

[106]  Derek G. Corneil,et al.  Visibility graph recognition , 1990 .

[107]  T. Shermer Recent Results in Art Galleries , 1992 .

[108]  Joseph O'Rourke,et al.  Open Problems in the Combinatorics of Visibility and Illumination , 1998 .

[109]  Joseph O'Rourke,et al.  Two Segment Classes with Hamiltonian Visibility Graphs , 1994, Comput. Geom..

[110]  János Pach,et al.  Research problems in discrete geometry , 2005 .

[111]  Robert M. Haralick,et al.  Decomposition of Two-Dimensional Shapes by Graph-Theoretic Clustering , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[112]  János Pach,et al.  Graphs with Large Obstacle Numbers , 2010, WG.

[113]  Mamoru Watanabe,et al.  On a Counterexample to a Conjecture of Mirzaian , 1992, Comput. Geom..

[114]  Ellen Gethner,et al.  Bar k-Visibility Graphs , 2007, J. Graph Algorithms Appl..

[115]  Jeremy P. Spinrad,et al.  Efficient graph representations , 2003, Fields Institute monographs.

[116]  Mark de Berg,et al.  Computational Geometry: Algorithms and Applications, Second Edition , 2000 .

[117]  David G. Kirkpatrick,et al.  Determining Bar-representability for Ordered Weighted Graphs , 1996, Comput. Geom..

[118]  Thomas Andreae Some Results on Visibility Graphs , 1992, Discret. Appl. Math..

[119]  Michel Pocchiola,et al.  Topologically sweeping visibility complexes via pseudotriangulations , 1996, Discret. Comput. Geom..

[120]  David R. Wood,et al.  Blocking coloured point sets , 2010 .

[121]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[122]  Imre Bárány,et al.  Problems and Results around the Erdös-Szekeres Convex Polygon Theorem , 2000, JCDCG.

[123]  James King VC-Dimension of Visibility on Terrains , 2008, CCCG.

[124]  David M. Mount,et al.  An output sensitive algorithm for computing visibility graphs , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[125]  A new necessary condition for the vertex visibility graphs of simple polygons , 1994, Discret. Comput. Geom..

[126]  Frank Harary,et al.  Graph Theory , 2016 .

[127]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[128]  Alice M. Dean,et al.  Unit Bar-visibility Graphs , 2022 .

[129]  David Avis,et al.  Computing the largest empty convex subset of a set of points , 1985, SCG '85.

[130]  V. A. Koshelev On the Erdös-Szekeres problem in combinatorial geometry , 2007, Electron. Notes Discret. Math..

[131]  Stephan Eidenbenz,et al.  Inapproximability of finding maximum hidden sets on polygons and terrains , 2002, Comput. Geom..

[132]  Der-Tsai Lee Proximity and reachability in the plane. , 1978 .

[133]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[134]  Richard Pollack,et al.  Upper bounds for configurations and polytopes inRd , 1986, Discret. Comput. Geom..

[135]  Tomás Lozano-Pérez,et al.  An algorithm for planning collision-free paths among polyhedral obstacles , 1979, CACM.

[136]  Godfried T. Toussaint,et al.  On computing simple circuits on a set of line segments , 1986, SCG '86.

[137]  Vladimir Vapnik,et al.  Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities , 1971 .

[138]  Prosenjit Bose,et al.  Every Large Point Set contains Many Collinear Points or an Empty Pentagon , 2009, CCCG.

[139]  David Avis,et al.  A combinational approach to polygon similarity , 1983, IEEE Trans. Inf. Theory.

[140]  Florian Pfender Visibility Graphs of Point Sets in the Plane , 2008, Discret. Comput. Geom..

[141]  Petra Mutzel,et al.  The Thickness of Graphs: A Survey , 1998, Graphs Comb..

[142]  Donald E. Knuth,et al.  Axioms and Hulls , 1992, Lecture Notes in Computer Science.

[143]  Csaba D. Tóth,et al.  Alternating paths through disjoint line segments , 2003, Inf. Process. Lett..

[144]  Subir Kumar Ghosh,et al.  Approximation algorithms for art gallery problems in polygons , 2010, Discret. Appl. Math..

[145]  J. Horton Sets with No Empty Convex 7-Gons , 1983, Canadian Mathematical Bulletin.

[146]  Micha Sharir,et al.  On shortest paths in polyhedral spaces , 1986, STOC '84.

[147]  Prosenjit Bose,et al.  Every Set of Disjoint Line Segments Admits a Binary Tree , 1994, ISAAC.

[148]  G. Szekeres,et al.  A combinatorial problem in geometry , 2009 .