The Roberts characterization of proper and unit interval graphs

In this note, a constructive proof that the classes of proper interval graphs and unit interval graphs coincide is given, a result originally established by Fred S. Roberts. Additionally, the proof yields a linear-time and space algorithm to compute a unit interval representation, given a proper interval graph as input.

[1]  S. Olariu,et al.  Optimal greedy algorithms for indifference graphs , 1992, Proceedings IEEE Southeastcon '92.

[2]  Xiaotie Deng,et al.  Linear-Time Representation Algorithms for Proper Circular-Arc Graphs and Proper Interval Graphs , 1996, SIAM J. Comput..

[3]  Stephan Olariu,et al.  An Optimal Greedy Heuristic to Color Interval Graphs , 1991, Inf. Process. Lett..

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

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

[6]  Patrick Suppes,et al.  Foundational aspects of theories of measurement , 1958, Journal of Symbolic Logic.

[7]  Stephan Olariu,et al.  Simple Linear Time Recognition of Unit Interval Graphs , 1995, Inf. Process. Lett..

[8]  Celina M. H. de Figueiredo,et al.  A Linear-Time Algorithm for Proper Interval Graph Recognition , 1995, Inf. Process. Lett..

[9]  C. P. Rangan,et al.  A Unified Approach to Domination Problems on Interval Graphs , 1988, Inf. Process. Lett..

[10]  Derek G. Corneil,et al.  A simple 3-sweep LBFS algorithm for the recognition of unit interval graphs , 2004, Discret. Appl. Math..

[11]  N. Sloane,et al.  Proof Techniques in Graph Theory , 1970 .

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

[13]  Peter L. Hammer,et al.  Difference graphs , 1990, Discret. Appl. Math..

[14]  Douglas B. West,et al.  A short proof that 'proper = unit' , 1998, Discret. Math..

[15]  Bhawani Sankar Panda,et al.  A linear time recognition algorithm for proper interval graphs , 2003, Inf. Process. Lett..