Simple linear time recognition of unit interval graphs

Corneil, Derek G.; Kim, Hiryoung; Natarajan, Sridhar; Olariu, Stephan; Sprague, Alan P.

doi:10.1016/0020-0190(95)00046-f

Cited by 114 publications

(123 citation statements)

References 6 publications

Supporting

Mentioning

122

Contrasting

Unclassified

Order By: Relevance

“…Case 1: An Unlocated Component. Since there are no indistinguishable vertices, we compute in time O(n + m) using the algorithm of [4] any ordering ⊳ for which we want to produce a representation as small as possible.…”

Section: Structural Resultsmentioning

confidence: 99%

Extending Partial Representations of Subclasses of Chordal Graphs

Klavík

Kratochvı́l

Otachi

et al. 2012

Algorithms and Computation

View full text Add to dashboard Cite

Chordal graphs are intersection graphs of subtrees of a tree T . We investigate the complexity of the partial representation extension problem for chordal graphs. A partial representation specifies a tree T ′ and some pre-drawn subtrees of T ′ . It asks whether it is possible to construct a representation inside a modified tree T which extends the partial representation (i.e, keeps the pre-drawn subtrees unchanged).We consider four modifications of T ′ and get vastly different problems. In some cases, it is interesting to consider the complexity even if just T ′ is given and no subtree is pre-drawn. Also, we consider three well-known subclasses of chordal graphs: Proper interval graphs, interval graphs and path graphs. We give an almost complete complexity characterization.We further study the parametrized complexity of the problems when parametrized by the number of pre-drawn subtrees, the number of components and the size of the tree T ′ . We describe an interesting relation with integer partition problems. The problem 3-Partition is used for all NP-completeness reductions. The extension of interval graphs when the space in T ′ is limited is "equivalent" to the BinPacking problem.

show abstract

Section: Structural Resultsmentioning

confidence: 99%

Extending Partial Representations of Subclasses of Chordal Graphs

Klavík

Kratochvı́l

Otachi

et al. 2012

Algorithms and Computation

View full text Add to dashboard Cite

show abstract

“…Given a clique path P of an interval graph G and two vertices u and v of G, we say that u transcends v in P if f P (u) ≤ f P (v) and l P (u) > l P (v). The following theorem [18], already implicit from several earlier works on proper interval graphs [6,8,23], shows that proper interval graphs do have a unique clique path. It will be used heavily in our proofs.…”

Section: Theorem 21 ([20])mentioning

confidence: 92%

“…Theorem 2.2 ( [6,8,18,23]). A connected chordal graph is a proper interval graph if and only if it has a unique clique path P , and no vertex transcends any other vertex in P .…”

Section: Theorem 21 ([20])mentioning

confidence: 99%

Computing Role Assignments of Proper Interval Graphs in Polynomial Time

Heggernes

Hof

Paulusma

2011

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Further information on publisher's website:http://dx.doi.org/10. 1016/j.jda.2011.12.004 Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Journal of discrete algorithms. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be re ected in this document. Changes may have been made to this work since it was submitted for publication. A de nitive version was subsequently published in Journal of discrete algorithms, 14, 2012, 10.1016/j.jda.2011.12.004 Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractAn R-role assignment of a graph G is a locally surjective homomorphism from G to graph R. For a fixed graph R, the R-Role Assignment problem is to decide, for an input graph G, whether G has an R-role assignment. When both graphs G and R are given as input, the problem is called Role Assignment. In this paper, we study the latter problem. It is known that R-Role Assignment is NP-complete already when R is a path on three vertices. In order to obtain polynomial time algorithms for Role Assignment, it is therefore necessary to put restrictions on G. So far, the only known non-trivial case for which this problem is solvable in polynomial time is when G is a tree. We present an algorithm that solves Role Assignment in polynomial time when G is a proper interval graph. Thus we identify the first graph class other than trees on which the problem is tractable. As a complementary result, we show that Role Assignment is Graph Isomorphism-hard on chordal graphs, a superclass of proper interval graphs and trees.

show abstract

“…A proper interval ordering together with the leftmost neighbour of each vertex is an O(n)-space representation of proper interval graphs, that can be computed in linear time [3,7]. Other vertex ordering characterisations of proper interval graphs formulate conditions on neighbourhoods or maximal cliques [10].…”

Section: Introductionmentioning

confidence: 99%

“…Other recognition algorithms also apply BFS strategies but with a different approach: for every connected component, find a vertex of special kind and run BFS starting with this vertex. A graph is then proper interval if and only if the BFS levels are cliques and the neighbourhoods between consecutive levels satisfy the so called chain property [3,14]. On the representation side, these latter algorithms compute an ordered vertex partition and verify neighbourhood properties.…”

Section: Introductionmentioning

confidence: 99%