Graphclass: perfect

Definition:

A graph is perfect if for all induced subgraphs H: \chi(H) = \omega(H), where \chi is the chromatic number and \omega is the size of a maximum clique.

References

[88]
C. Berge
Les probl\`emes de colorations en th\'eorie des graphes
{\sl Publ. Inst. Stat. Univ. Paris, 9} 1960 123--160
[89]
C. Berge
F\"arbung von Graphen deren saemtliche bzw. deren ungeraden Kreise starr sind
Wiss. Zeitschr. Martin-Luther-Univ. Halle-Wittenberg 114 1961
[95]
C. Berge
Motivations and history of some of my conjectures
Discrete Math. 165/166 61--70 1997
[96]
C. Berge, V. Chv\'atal (eds.)
Topics on perfect graphs
Annals of Discrete Math. 21 1984
[453]
M.C. Golumbic
Algorithmic Graph Theory and Perfect Graphs
Academic Press, New York 1980
[737]
L. Lov\'asz
Perfect graphs
{\sl Selected Topics in Graph Theory} 2, {\sc L.W. Beineke, R.J. Wilson}eds.,Academic Press, New York 1983 55--87
[1026]
B. Toft
Coloring, stable sets and perfect graphs
In: {\sc R. Graham, M. Gr\"otschel, L. Lov\'asz}, eds.,Handbook of Combinatorics, Vol. I, North-Holland, 1995 0 233--288

;

[476]
M. Gr\"otschel, L. Lov\'asz, A. Schrijver
The ellipsoid method and its consequences in combinatorial optimization
Combinatorica 1 169--197, 1981 Corrigendum
[1226]
M. Chudnovsky, G. Cornuejols, X. Liu, P. Seymour, K. Vuskovic
Cleaning for Bergeness
manuscript 2003
[1227]
M. Chudnovsky, G. Cornuejols, X. Liu, P. Seymour, K. Vuskovic
Recognizing Berge graphs
manuscript 2003
[1228]
M. Chudnovsky, G. Cornuejols, X. Liu, P. Seymour, K. Vuskovic
A polynomial algorithm for recognizing Berge graphs
manuscript 2003
[1588]
The decycling number of graphs
S. Bau, L.W. Beineke
Australas. J. Comb. 25 285-298 (2002)
[1589]
R.M. Karp
Reducibility among combinatorial problems
Complexity of Computer Computations (R.E. Miller, J.W. Thatcher, ed.), Plenum Press, New York-London 85-103 (1972)

Equivalent classes

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Complement classes

self-complementary

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

[+]Details

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.
Polynomial [+]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.
Polynomial [+]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.
Polynomial [+]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).
NP-complete [+]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.
NP-complete [+]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.
NP-complete [+]Details
Graph isomorphism
[?]
Input: Graphs G and H in this class
Output: True iff G and H are isomorphic.
GI-complete [+]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.
NP-complete [+]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.
NP-complete [+]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.
NP-complete [+]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.
Polynomial [+]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.
NP-complete [+]Details
Recognition
[?]
Input: A graph G.
Output: True iff G is in this graph class.
Polynomial [+]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).
NP-complete [+]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.
NP-complete [+]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.
NP-complete [+]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.
Polynomial [+]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