Refining the Hierarchies of Classes of Geometric Intersection Graphs

We analyse properties of geometric intersection graphs to show the strict containment between some natural classes of geometric intersection graphs. In particular, we show the following properties: - A graph $G$ is outerplanar if and only if the 1-subdivision of $G$ is outer-segment. - For each integer $k\ge 1$, the class of intersection graphs of segments with $k$ different lengths is a strict subclass of the class of intersection graphs of segments with $k+1$ different lengths. - For each integer $k\ge 1$, the class of intersection graphs of disks with $k$ different sizes is a strict subclass of the class of intersection graphs of disks with $k+1$ different sizes. - The class of outer-segment graphs is a strict subclass of the class of outer-string graphs.

[1]  Andrzej Schinzel,et al.  On Pythagorean triangles , 1998 .

[2]  David G. Kirkpatrick,et al.  Unit disk graph recognition is NP-hard , 1998, Comput. Geom..

[3]  William T. Trotter,et al.  Triangle-free intersection graphs of line segments with large chromatic number , 2012, J. Comb. Theory, Ser. B.

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

[5]  Daniel Král,et al.  On Intersection Graphs of Segments with Prescribed Slopes , 2001, Graph Drawing.

[6]  Sergio Cabello,et al.  Refining the Hierarchies of Classes of Geometric Intersection Graphs , 2017 .

[7]  Marcus Schaefer,et al.  Decidability of string graphs , 2001, STOC '01.

[8]  J. Kratochvil,et al.  Intersection Graphs of Segments , 1994, J. Comb. Theory, Ser. B.

[9]  Jan Kratochvíl,et al.  String graphs requiring exponential representations , 1991, J. Comb. Theory, Ser. B.

[10]  Marcus Schaefer,et al.  Recognizing string graphs in NP , 2003, J. Comput. Syst. Sci..

[11]  F. Sinden Topology of thin film RC circuits , 1966 .

[12]  F. McMorris,et al.  Topics in Intersection Graph Theory , 1987 .

[13]  Jiří Matoušek,et al.  Geometry, Structure and Randomness in Combinatorics , 2014, Centro di Ricerca Matematica Ennio De Giorgi Series.

[14]  Stefan Felsner,et al.  Grid Intersection Graphs and Order Dimension , 2015, Order.

[15]  Svante Janson,et al.  On String Graph Limits and the Structure of a Typical String Graph , 2014, J. Graph Theory.

[16]  Jean Cardinal,et al.  Intersection Graphs of Rays and Grounded Segments , 2016, WG.

[17]  Jirí Matousek,et al.  Near-Optimal Separators in String Graphs , 2013, Combinatorics, Probability and Computing.

[18]  Svante Janson,et al.  Thresholds for classes of intersection graphs , 1992, Discret. Math..

[19]  Colin McDiarmid,et al.  Integer realizations of disk and segment graphs , 2011, J. Comb. Theory, Ser. B.

[20]  Jirí Matousek String graphs and separators , 2014, Geometry, Structure and Randomness in Combinatorics.

[21]  Jérémie Chalopin,et al.  Every planar graph is the intersection graph of segments in the plane: extended abstract , 2009, STOC '09.

[22]  I. Bárány LECTURES ON DISCRETE GEOMETRY (Graduate Texts in Mathematics 212) , 2003 .

[23]  Marcus Schaefer,et al.  Complexity of Some Geometric and Topological Problems , 2009, GD.

[24]  Charles J. Colbourn,et al.  Unit disk graphs , 1991, Discret. Math..

[25]  Colin McDiarmid,et al.  The number of disk graphs , 2014, Eur. J. Comb..

[26]  Jan Kratochvíl,et al.  String graphs. II. recognizing string graphs is NP-hard , 1991, J. Comb. Theory, Ser. B.

[27]  Mark A. Buckingham,et al.  Circle Graphs , 2015 .

[28]  D. Shanks Solved and Unsolved Problems in Number Theory , 1964 .

[29]  Jan Kratochvíl,et al.  String graphs. I. The number of critical nonstring graphs is infinite , 1991, J. Comb. Theory, Ser. B.

[30]  Andrew Suk Coloring intersection graphs of x-monotone curves in the plane , 2014, Comb..