Algorithms and Discrete Applied Mathematics

Some of the nicest characterizations of graph families are stated in terms of obstructions – forbidden induced subgraphs or other substructures. A typical example characterizes interval graphs by the absence of asteroidal triples and induced cycles of length greater than three. I will discuss similar obstruction characterizations for classes of digraphs. The obstructions are novel, but similar in spirit to asteroidal triples. Surprisingly, these obstructions permit new characterizations even for undirected graphs. In particular, I will describe the first obstruction characterization of circular arc graphs, and a corresponding certifying polynomial time recognition algorithm for this graph class. The digraph results are joint with Arash Rafiey, Jing Huang, and Tomás Feder, and the circular arc graph characterization and algorithm results are joint with Juraj Stacho and Mathew Francis. Probabilistic Arguments in Graph Coloring

[1]  H. M. Mulder.  The interval function of a graph , 1980 .

[2]  Prosenjit Bose,et al.  On Structural and Graph Theoretic Properties of Higher Order Delaunay Graphs , 2009, Int. J. Comput. Geom. Appl..

[3]  Alexander Pilz,et al.  Blocking Delaunay triangulations , 2010, CCCG.

[4]  Ambat Vijayakumar,et al.  ON THE MEDIAN AND THE ANTIMEDIAN OF A COGRAPH , 2010 .

[5]  Michael B. Dillencourt,et al.  Toughness and Delaunay triangulations , 1987, SCG '87.

[6]  Edward Minieka,et al.  Anticenters and antimedians of a network , 1983, Networks.

[7]  Prosenjit Bose,et al.  Some properties of k-Delaunay and k-Gabriel graphs , 2013, Comput. Geom..

[8]  Henry Martyn Mulder,et al.  A tight axiomatization of the median procedure on median graphs , 2013, Discret. Appl. Math..

[9]  Nicolas Bonichon,et al.  Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces , 2010, WG.

[10]  Rakesh V. Vohra,et al.  An axiomatic characterization of some locations in trees , 1996 .

[11]  Maria Saumell,et al.  10-Gabriel graphs are Hamiltonian , 2014, Inf. Process. Lett..

[12]  Henry Martyn Mulder,et al.  Axiomization of the center function on trees , 2008, Australas. J Comb..

[13]  Michael B. Dillencourt,et al.  A Non-Hamiltonian, Nondegenerate Delaunay Triangulation , 1987, Inf. Process. Lett..

[14]  Chuan Yi Tang,et al.  20-relative Neighborhood Graphs Are Hamiltonian , 1990, J. Graph Theory.

[15]  Chuan Yi Tang,et al.  Solving the Euclidean Bottleneck Biconnected Edge Subgraph Problem by 2-Relative Neighborhood Graphs , 1992, Discret. Appl. Math..

[16]  Jeremy P. Spinrad,et al.  Bipartite permutation graphs , 1987, Discret. Appl. Math..

[17]  Michiel H. M. Smid,et al.  On the Stretch Factor of Convex Delaunay Graphs , 2008, ISAAC.

[18]  Boris Aronov,et al.  Witness Gabriel graphs , 2010, Comput. Geom..

[19]  Chuan Yi Tang,et al.  Solving the Euclidean bottleneck matching problem byk-relative neighborhood graphs , 2005, Algorithmica.

[20]  Paul Chew,et al.  There are Planar Graphs Almost as Good as the Complete Graph , 1989, J. Comput. Syst. Sci..