Interval Graph
A graph 
is an interval graph if it captures the intersection relation for some set of intervals
on the real line (Harary and Palmer 1973, p. 264).
Formally, 
is an interval graph provided that one can assign to each 
an interval 
such that 
is nonempty precisely when 
.
Given a list of intervals represented by Interval objects, this construction can be expressed in the Wolfram Language using RelationGraph with the relation that two distinct intervals have nonempty IntervalIntersection.
Star graphs are interval graphs, but cycle graphs (for 
)
are not (Skiena 1990, p. 164). Determining if a graph is an interval graph and
realizing it can be done in 
time (Booth and Lueker 1976; Skiena 1990, p. 164).
A graph 
is an interval graph iff the vertices of 
can be ordered 
, ..., 
such that 
adj 
implies 
adj 
whenever 
(West 2000, p. 346).
Every induced subgraph of an interval graph is itself an interval graph (Jacobson et al. 1991; West 2000, p. 226).
See also
Explore with Wolfram|Alpha
References
Booth, K. S. and Lueker, G. S. "Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity using PQ-Tree Algorithms." J. Comput. System Sci. 13, 335-379, 1976.Fishburn, P. C. Interval Orders and Interval Graphs: A Study of Partially Ordered Sets. New York: Wiley, 1985.Gilmore, P. C. and Hoffman, A. J. "A Characterization of Comparability Graphs and of Interval Graphs." Canad. J. Math. 16, 539-548, 1964.Harary, F. and Palmer, E. M. "A Survey of Graphical Enumeration Problems." In A Survey of Combinatorial Theory (Ed. J. N. Srivastava). Amsterdam, Netherlands: North-Holland, pp. 259-275, 1973.Jacobson, M. S.; McMorris, F. R.; and Mulder, H. M. "Tolerance Intersection Graphs." In Proc. Kalamazoo 1988 (Ed. Y. Alavi, G. Chartrand, O. R. Oellermann, and A. J. Schwenk). New York: Wiley, pp. 705-724, 1991.Lekkerkerker, C. G. and Boland, J. C. "Representation of a Finite Graph by a Set of Intervals on the Real Line." Fund. Math. 51, 45-64, 1962.Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 163-164, 1990.West, D. B. Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 195-196 and 346, 2000.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Interval Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IntervalGraph.html