In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals.
Interval graphs are chordal graphs and perfect graphs. They can be recognized in linear time, and an optimal graph coloring or maximum clique in these graphs can be found in linear time. The interval graphs include all proper interval graphs, graphs defined in the same way from a set of unit intervals.
These graphs have been used to model food webs, and to study scheduling problems in which one must select a subset of tasks to be performed at non-overlapping times. Other applications include assembling contiguous subsequences in DNA mapping, and temporal reasoning.
An interval graph is an undirected graph formed from a family of intervals
Three vertices form an asteroidal triple (AT) in a graph if, for each two, there exists a path containing those two but no neighbor of the third. A graph is AT-free if it has no asteroidal triple. The earliest characterization of interval graphs seems to be the following:
- A graph is an interval graph if and only if its maximal cliques can be ordered such that each vertex that belongs to two of these cliques also belongs to all cliques between them in the ordering. That is, for every with , it is also the case that whenever .
- A graph is an interval graph if and only if it does not contain the cycle graph as an induced subgraph and is the complement of a comparability graph.
Various other characterizations of interval graphs and variants have been described.
Efficient recognition algorithmEdit
Determining whether a given graph is an interval graph can be done in time by seeking an ordering of the maximal cliques of that is consecutive with respect to vertex inclusion. Many of the known algorithms for this problem work in this way, although it is also possible to recognize interval graphs in linear time without using their cliques.
The original linear time recognition algorithm of Booth & Lueker (1976) is based on their complex PQ tree data structure, but Habib et al. (2000) showed how to solve the problem more simply using lexicographic breadth-first search, based on the fact that a graph is an interval graph if and only if it is chordal and its complement is a comparability graph. A similar approach using a 6-sweep LexBFS algorithm is described in Corneil, Olariu & Stewart (2009).
Related families of graphsEdit
By the characterization of interval graphs as AT-free chordal graphs, interval graphs are strongly chordal graphs and hence perfect graphs. Their complements belong to the class of comparability graphs, and the comparability relations are precisely the interval orders.
From the fact that a graph is an interval graph if and only if it is chordal and its complement is a comparability graph, it follows that graph and its complement are both interval graphs if and only if the graph is both a split graph and a permutation graph.
The interval graphs that have an interval representation in which every two intervals are either disjoint or nested are the trivially perfect graphs.
A graph has boxicity at most one if and only if it is an interval graph; the boxicity of an arbitrary graph is the minimum number of interval graphs on the same set of vertices such that the intersection of the edges sets of the interval graphs is .
The intersection graphs of arcs of a circle form circular-arc graphs, a class of graphs that contains the interval graphs. The trapezoid graphs, intersections of trapezoids whose parallel sides all lie on the same two parallel lines, are also a generalization of the interval graphs.
Proper interval graphsEdit
Proper interval graphs are interval graphs that have an interval representation in which no interval properly contains any other interval; unit interval graphs are the interval graphs that have an interval representation in which each interval has unit length. A unit interval representation without repeated intervals is necessarily a proper interval representation. Not every proper interval representation is a unit interval representation, but every proper interval graph is a unit interval graph, and vice versa. Every proper interval graph is a claw-free graph; conversely, the proper interval graphs are exactly the claw-free interval graphs. However, there exist claw-free graphs that are not interval graphs.
An interval graph is called -proper if there is a representation in which no interval is contained by more than others. This notion extends the idea of proper interval graphs such that a 0-proper interval graph is a proper interval graph. An interval graph is called -improper if there is a representation in which no interval contains more than others. This notion extends the idea of proper interval graphs such that a 0-improper interval graph is a proper interval graph. An interval graph is -nested if there is no chain of length of intervals nested in each other. This is a generalization of proper interval graphs as 1-nested interval graphs are exactly proper interval graphs.
The mathematical theory of interval graphs was developed with a view towards applications by researchers at the RAND Corporation's mathematics department, which included young researchers—such as Peter C. Fishburn and students like Alan C. Tucker and Joel E. Cohen—besides leaders—such as Delbert Fulkerson and (recurring visitor) Victor Klee. Cohen applied interval graphs to mathematical models of population biology, specifically food webs.
Interval graphs are used to represent resource allocation problems in operations research and scheduling theory. In these applications, each interval represents a request for a resource (such as a processing unit of a distributed computing system or a room for a class) for a specific period of time. The maximum weight independent set problem for the graph represents the problem of finding the best subset of requests that can be satisfied without conflicts. See interval scheduling for more information.
An optimal graph coloring of the interval graph represents an assignment of resources that covers all of the requests with as few resources as possible; it can be found in polynomial time by a greedy coloring algorithm that colors the intervals in sorted order by their left endpoints.
Other applications include genetics, bioinformatics, and computer science. Finding a set of intervals that represent an interval graph can also be used as a way of assembling contiguous subsequences in DNA mapping. Interval graphs also play an important role in temporal reasoning.
Interval completions and pathwidthEdit
If is an arbitrary graph, an interval completion of is an interval graph on the same vertex set that contains as a subgraph. The parameterized version of interval completion (find an interval supergraph with k additional edges) is fixed parameter tractable, and moreover, is solvable in parameterized subexponential time.
The pathwidth of an interval graph is one less than the size of its maximum clique (or equivalently, one less than its chromatic number), and the pathwidth of any graph is the same as the smallest pathwidth of an interval graph that contains as a subgraph.
- Lekkerkerker & Boland (1962).
- Fulkerson & Gross (1965); Fishburn (1985)
- Gilmore & Hoffman (1964).
- McKee & McMorris (1999); Brandstädt, Le & Spinrad (1999)
- Hsu (1992).
- Fishburn (1985); Golumbic (1980)
- Fishburn (1985).
- Eckhoff (1993).
- Roberts (1969); Gardi (2007)
- Faudree, Flandrin & Ryjáček (1997), p. 89.
- Proskurowski & Telle (1999).
- Beyerl & Jamison (2008).
- Klavík, Otachi & Šejnoha (2019).
- Cohen (1978), pp. ix–10.
- Cohen (1978), pp. 12–33.
- Bar-Noy et al. (2001).
- Cormen et al. (2001). sfnp error: no target: CITEREFCormenLeisersonRivestStein2001 (help)
- Zhang et al. (1994).
- Golumbic & Shamir (1993).
- Villanger et al. (2009).
- Bliznets et al. (2014).
- Bodlaender (1998).
- Bar-Noy, Amotz; Bar-Yehuda, Reuven; Freund, Ari; Naor, Joseph (Seffi); Schieber, Baruch (2001), "A unified approach to approximating resource allocation and scheduling", Journal of the ACM, 48 (5): 1069–1090, CiteSeerX 10.1.1.124.9886, doi:10.1145/502102.502107, S2CID 12329294
- Beyerl, Jeffery J.; Jamison, Robert E. (2008), "Interval graphs with containment restrictions", Proceedings of the Thirty-Ninth Southeastern International Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium, 191, pp. 117–128, arXiv:1109.6675, MR 2489816
- Bliznets, Ivan; Fomin, Fedor V.; Pilipczuk, Marcin; Pilipczuk, Michał (2014), "A subexponential parameterized algorithm for proper interval completion", in Schulz, Andreas S.; Wagner, Dorothea (eds.), Proceedings of the 22nd Annual European Symposium on Algorithms (ESA 2014), Wroclaw, Poland, September 8–10, 2014, Lecture Notes in Computer Science, 8737, Springer-Verlag, pp. 173–184, arXiv:1402.3473, doi:10.1007/978-3-662-44777-2_15, ISBN 978-3-662-44776-5, S2CID 12385294
- Bodlaender, Hans L. (1998), "A partial k-arboretum of graphs with bounded treewidth", Theoretical Computer Science, 209 (1–2): 1–45, doi:10.1016/S0304-3975(97)00228-4, hdl:1874/18312
- Booth, K. S.; Lueker, G. S. (1976), "Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms", Journal of Computer and System Sciences, 13 (3): 335–379, doi:10.1016/S0022-0000(76)80045-1
- Brandstädt, A.; Le, V.B.; Spinrad, J.P. (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 978-0-89871-432-6
- Cohen, Joel E. (1978), Food webs and niche space, Monographs in Population Biology, 11, Princeton, NJ: Princeton University Press, pp. 1–189, ISBN 978-0-691-08202-8, PMID 683203
- Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001) , Introduction to Algorithms (2nd ed.), MIT Press and McGraw-Hill, ISBN 0-262-03293-7
- Corneil, Derek; Olariu, Stephan; Stewart, Lorna (2009), "The LBFS structure and recognition of interval graphs", SIAM Journal on Discrete Mathematics, 23 (4): 1905–1953, doi:10.1137/S0895480100373455
- Eckhoff, Jürgen (1993), "Extremal interval graphs", Journal of Graph Theory, 17 (1): 117–127, doi:10.1002/jgt.3190170112
- Faudree, Ralph; Flandrin, Evelyne; Ryjáček, Zdeněk (1997), "Claw-free graphs — A survey", Discrete Mathematics, 164 (1–3): 87–147, doi:10.1016/S0012-365X(96)00045-3, MR 1432221
- Fishburn, Peter C. (1985), Interval orders and interval graphs: A study of partially ordered sets, Wiley-Interscience Series in Discrete Mathematics, New York: John Wiley & Sons
- Fulkerson, D. R.; Gross, O. A. (1965), "Incidence matrices and interval graphs", Pacific Journal of Mathematics, 15 (3): 835–855, doi:10.2140/pjm.1965.15.835
- Gardi, Frédéric (2007), "The Roberts characterization of proper and unit interval graphs", Discrete Mathematics, 307 (22): 2906–2908, doi:10.1016/j.disc.2006.04.043
- Gilmore, P. C.; Hoffman, A. J. (1964), "A characterization of comparability graphs and of interval graphs", Canadian Journal of Mathematics, 16: 539–548, doi:10.4153/CJM-1964-055-5
- Golumbic, Martin Charles (1980), Algorithmic Graph Theory and Perfect Graphs, Academic Press, ISBN 978-0-12-289260-8
- Golumbic, Martin Charles; Shamir, Ron (1993), "Complexity and algorithms for reasoning about time: a graph-theoretic approach", Journal of the ACM, 40 (5): 1108–1133, CiteSeerX 10.1.1.35.528, doi:10.1145/174147.169675, S2CID 15708027
- Habib, Michel; McConnell, Ross; Paul, Christophe; Viennot, Laurent (2000), "Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition, and consecutive ones testing", Theoretical Computer Science, 234 (1–2): 59–84, doi:10.1016/S0304-3975(97)00241-7
- Hsu, Wen-Lian (1992), "A simple test for interval graphs", in Mayr, Ernst W. (ed.), Graph-Theoretic Concepts in Computer Science, 18th International Workshop, WG '92, Wiesbaden-Naurod, Germany, June 19–20, 1992, Proceedings, Lecture Notes in Computer Science, 657, Springer, pp. 11–16, doi:10.1007/3-540-56402-0_31
- Klavík, Pavel; Otachi, Yota; Šejnoha, Jiří (2019), "On the classes of interval graphs of limited nesting and count of lengths", Algorithmica, 81 (4): 1490–1511, arXiv:1510.03998, doi:10.1007/s00453-018-0481-y, MR 3936165
- Lekkerkerker, C. G.; Boland, J. C. (1962), "Representation of a finite graph by a set of intervals on the real line", Fundamenta Mathematicae, 51: 45–64, doi:10.4064/fm-51-1-45-64
- McKee, Terry A.; McMorris, F. R. (1999), Topics in Intersection Graph Theory, SIAM Monographs on Discrete Mathematics and Applications, ISBN 978-0-89871-430-2
- Proskurowski, Andrzej; Telle, Jan Arne (1999), "Classes of graphs with restricted interval models", Discrete Mathematics & Theoretical Computer Science, 3 (4): 167–176, CiteSeerX 10.1.1.39.9532
- Roberts, F. S. (1969), "Indifference graphs", in Harary, Frank (ed.), Proof Techniques in Graph Theory, New York, NY: Academic Press, pp. 139–146, ISBN 978-0123242600, OCLC 30287853
- Villanger, Yngve; Heggernes, Pinar; Paul, Christophe; Telle, Jan Arne (2009), "Interval completion is fixed parameter tractable", SIAM Journal on Computing, 38 (5): 2007–2020, CiteSeerX 10.1.1.73.8999, doi:10.1137/070710913
- Zhang, Peisen; Schon, Eric A.; Fischer, Stuart G.; Cayanis, Eftihia; Weiss, Janie; Kistler, Susan; Bourne, Philip E. (1994), "An algorithm based on graph theory for the assembly of contigs in physical mapping of DNA", Bioinformatics, 10 (3): 309–317, doi:10.1093/bioinformatics/10.3.309, PMID 7922688