An implicit representation of chordal comparability graphs in linear time

Ma and Spinrad have shown that every transitive orientation of a chordal comparability graph is the intersection of four linear orders. That is, chordal comparability graphs are comparability graphs of posets of dimension four. Among other uses, this gives an implicit representation of a chordal comparability graph using O(n) integers so that, given two vertices, it can be determined in O(1) time whether they are adjacent, no matter how dense the graph is. We give a linear time algorithm for finding the four linear orders, improving on their bound of O(n^2).

[1]  Moni Naor,et al.  Implicit representation of graphs , 1992, STOC '88.

[2]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[3]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[4]  William T. Trotter,et al.  The dimension of cycle-free orders , 1992 .

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

[6]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[7]  Peter L. Hammer,et al.  Discrete Applied Mathematics , 1993 .

[8]  Jeremy P. Spinrad,et al.  Modular decomposition and transitive orientation , 1999, Discret. Math..

[9]  Robert E. Tarjan,et al.  A linear-time algorithm for a special case of disjoint set union , 1983, J. Comput. Syst. Sci..

[10]  Wen-Lian Hsu,et al.  Fast and Simple Algorithms for Recognizing Chordal Comparability Graphs and Interval Graphs , 1999, SIAM J. Comput..

[11]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[12]  Wojciech A. Trybulec.  Partially Ordered Sets , 1990 .

[13]  Clemente Izurieta,et al.  An Implicit Representation of Chordal Comparabilty Graphs in Linear-Time , 2006, WG.

[15]  J. Spinrad,et al.  Cycle-free partial orders and chordal comparability graphs , 1991 .

[16]  Ben Dushnik,et al.  Partially Ordered Sets , 1941 .

[17]  M. Yannakakis.  The Complexity of the Partial Order Dimension Problem , 1982 .