Simple Geometrical Intersection Graphs

A graph G = (V, E) is said to be an intersection graph if and only if there is a set of objects such that each vertex v in V corresponds to an object Ov and {u, v} ∈ E if and only if Ov and Ou have a nonempty intersection. Interval graphs are typical intersection graph class, and widely investigated since they have simple structures and many hard problems become easy on the graphs. In this paper, we survey known results and investigate (unit) grid intersection graphs, which is one of natural generalized interval graphs. We show that the graph class has so rich structure that some typical problems are still hard on the graph class.

[1]  J. Mark Keil.  Finding Hamiltonian Circuits in Interval Graphs , 1985, Inf. Process. Lett..

[2]  P. Hell,et al.  Interval bigraphs and circular arc graphs , 2004 .

[3]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[4]  Gottfried Tinhofer,et al.  The Isomorphism Problem For Directed Path Graphs and For Rooted Directed Path Graphs , 1996, J. Algorithms.

[5]  Kellogg S. Booth,et al.  A Linear Time Algorithm for Deciding Interval Graph Isomorphism , 1979, JACM.

[6]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[7]  Shin-Ichi Nakano,et al.  A New Approach to Graph Recognition and Applications to Distance-Hereditary Graphs , 2007, Journal of Computer Science and Technology.

[8]  Ryuhei Uehara,et al.  Generalized Hi-Q is NP-complete , 1990 .

[9]  M. Golumbic.  Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57) , 2004 .

[10]  Yota Otachi,et al.  Relationships between the class of unit grid intersection graphs and other classes of bipartite graphs , 2007, Discret. Appl. Math..

[11]  Haiko Müller,et al.  Recognizing Interval Digraphs and Interval Bigraphs in Polynomial Time , 1997, Discret. Appl. Math..

[12]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[13]  Ryuhei Uehara.  Canonical Data Structure for Interval Probe Graphs , 2004, ISAAC.

[14]  Ryuhei Uehara,et al.  Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs , 2005, Discret. Appl. Math..

[15]  Charles J. Colbourn,et al.  On testing isomorphism of permutation graphs , 1981, Networks.

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

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

[18]  Ján Plesník,et al.  The NP-Completeness of the Hamiltonian Cycle Problem in Planar Digraphs with Degree Bound Two , 1979, Inf. Process. Lett..