Graphclass: interval

Definition:

A graph is an interval graph if it has an intersection model consisting of intervals on a straight line.

Even though the weighted clique and colourability problems are polynomial time solvable on both interval and co-interval graphs, the weighted clique partitioning problem [PCliqW] is NP-hard for interval graphs

[1484]
D. Gijswijt, V. Jost, M. Queyranne
Clique partitioning of interval graphs with submodular costs on the cliques
RAIRO Operations Research 41 275-287 (2007)
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References

[453]
M.C. Golumbic
Algorithmic Graph Theory and Perfect Graphs
Academic Press, New York 1980

;

[119]
H.L. Bodlaender
A tourist guide through treewidth
Acta Cybernetica 11 1993 1--23
[127]
K.S. Booth, G.S. Lueker
Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms.
J. Comput. Syst. Sci. 13, 335-379 (1976). [ISSN 0022-0000]
[499]
M. Habib, R. McConnell, C. Paul, L. Viennot
Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing.
Theor. Comput. Sci. 234, No.1-2, 59-84 (2000). [ISSN 0304-3975]
[595]
W.--L. Hsu
On--line recognition of interval graphs in ${\cal O}(m+n\log n)$ time
Lecture Notes in Comp. Sci. 1120 1995 27--38
[687]
N. Korte, R.H. M\"ohring
An incremental linear--time algorithm for recognizing interval graphs
SIAM J. Computing 18 1989 68--81
[1143]
M.S. Chang
Efficient algorithms for the domination problems on interval and circular-arc graphs.
SIAM J. Comput. 27, No.6, 1671-1694 (1998)
[1529]
J.M. Keil
Finding hamiltonian circuits in interval graphs
Information Proc. Lett. 20 201-206 (1985)
[1574]
P. Festa, P.M. Pardalos, M.G.C. Resende
Feedback set problems
in: D.Z. Du, P.M. Pardalos, Handbook of Combinatorial Optimization, Supplement vol. A, Kluwer Academic Publishers, 209-259 (2000)
[1577]
C.L. Lu, C.Y. Tang
A linear-time algorithm for the weighted feedback vertex problem on interval graphs
Information Proc. Lett. 61 107-111 (1997)

Equivalent classes

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Complement classes

Inclusions

The map shows the inclusions between the current class and a fixed set of landmark classes. Minimal/maximal is with respect to the contents of ISGCI. Only references for direct inclusions are given. Where no reference is given, check equivalent classes or use the Java application. To check relations other than inclusion (e.g. disjointness) use the Java application, as well.

Map

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Minimal superclasses

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Maximal subclasses

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Problems

Problems in italics have no summary page and are only listed when ISGCI contains a result for the current class.

3-Colourability
[?]
Input: A graph G in this class.
Output: True iff each vertex of G can be assigned one colour out of 3 such that whenever two vertices are adjacent, they have different colours.
Linear [+]Details
Clique
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of pairwise adjacent vertices, with |S| >= k.
Polynomial [+]Details
Clique cover
[?]
Input: A graph G in this class and an integer k.
Output: True iff the vertices of G can be partitioned into k sets Si, such that whenever two vertices are in the same set Si, they are adjacent.
Linear [+]Details
Cliquewidth
[?]
Whether the cliquewidth of the graphs in this class is bounded by a constant k .
The cliquewidth of a graph is the number of different labels that is needed to construct the graph using the following operations:
  • creation of a vertex with label i,
  • disjoint union,
  • renaming labels i to label j,
  • connecting all vertices with label i to all vertices with label j.
Unbounded [+]Details
Cliquewidth expression
[?]
Input: A graph G in this class.
Output: An expression that constructs G according to the rules for cliquewidth, using only a constant number of labels.
Undefined if this class has unbounded cliquewidth.
Unbounded or NP-complete [+]Details
Colourability
[?]
Input: A graph G in this class and an integer k.
Output: True iff each vertex of G can be assigned one colour out of k such that whenever two vertices are adjacent, they have different colours.
Linear [+]Details
Cutwidth
[?]
Input: A graph G in this class and an integer k.
Output: True iff the cutwidth of G is at most k (see bounded cutwidth).
Unknown to ISGCI [+]Details
Domination
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of vertices, with |S| <= k, such that every vertex in G is either in S or adjacent to a vertex in S.
Linear [+]Details
Feedback vertex set
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of vertices, with |S| <= k, such that every cycle in G contains a vertex from S.
Linear [+]Details
Graph isomorphism
[?]
Input: Graphs G and H in this class
Output: True iff G and H are isomorphic.
Linear [+]Details
Hamiltonian cycle
[?]
Input: A graph G in this class.
Output: True iff G has a simple cycle that goes through every vertex of the graph.
Linear [+]Details
Hamiltonian path
[?]
Input: A graph G in this class.
Output: True iff G has a simple path that goes through every vertex of the graph.
Polynomial [+]Details
Independent dominating set
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of pairwise non-adjacent vertices, with |S| <= k, such that every vertex in G is either in S or adjacent to a vertex in S.
Linear [+]Details
Independent set
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of pairwise non-adjacent vertices, such that |S| >= k.
Linear [+]Details
Maximum cut
[?]
(decision variant)
Input: A graph G in this class and an integer k.
Output: True iff the vertices of G can be partitioned into two sets A,B such that there are at least k edges in G with one endpoint in A and the other endpoint in B.
Unknown to ISGCI [+]Details
Recognition
[?]
Input: A graph G.
Output: True iff G is in this graph class.
Linear [+]Details
Treewidth
[?]
Input: A graph G in this class and an integer k.
Output: True iff the treewidth of G is at most k (see bounded treewidth).
Polynomial [+]Details
Weighted clique
[?]
Input: A graph G in this class with weight function on the vertices and a real k.
Output: True iff G contains a set S of pairwise adjacent vertices, such that the sum of the weights of the vertices in S is at least k.
Polynomial [+]Details
Weighted feedback vertex set
[?]
Input: A graph G in this class with weight function on the vertices and a real k.
Output: True iff G contains a set S of vertices, such that the sum of the weights of the vertices in S is at most k and every cycle in G contains a vertex from S.
Linear [+]Details
Weighted independent dominating set
[?]
Input: A graph G in this class with weight function on the vertices and a real k.
Output: True iff G contains a set S of pairwise non-adjacent vertices, with the sum of the weights of the vertices in S at most k, such that every vertex in G is either in S or adjacent to a vertex in S.
Polynomial [+]Details
Weighted independent set
[?]
Input: A graph G in this class with weight function on the vertices and a real k.
Output: True iff G contains a set S of pairwise non-adjacent vertices, such that the sum of the weights of the vertices in S is at least k.
Linear [+]Details
Weighted maximum cut
[?]
(decision variant)
Input: A graph G in this class with weight function on the edges and a real k.
Output: True iff the vertices of G can be partitioned into two sets A,B such that the sum of weights of the edges in G with one endpoint in A and the other endpoint in B is at least k.
NP-complete [+]Details