Recognizing Berge Graphs

  title={Recognizing Berge Graphs},
  author={M. Chudnovsky and G{\'e}rard Cornu{\'e}jols and Xinming Liu and Paul D. Seymour and Kristina Vuskovic},
A graph is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. In this paper we give an algorithm to test if a graph G is Berge, with running time O(|V (G)|9). This is independent of the recent proof of the strong perfect graph conjecture. 
Odd holes in bull-free graphs
This paper shows that testing whether a graph contains an induced odd cycle of length at least five can be done in polynomial time if the input graph has no induced subgraph isomorphic to the bull (a triangle with two disjoint pendant edges).
Even pairs in square-free Berge graphs
1 9 O ct 2 01 4 Colouring graphs with no odd holes
  • 2014
Induced subgraphs of graphs with large chromatic number. I. Odd holes
Colouring graphs with no odd holes
An odd hole in a graph is an induced subgraph which is a cycle of odd length at least five. In 1985, A. Gyárfás made the conjecture that for all t there exists n such that every graph with no Kt
Coloring square-free Berge graphs
Detecting wheels
It is proved that the problem whose instance is a graph G and whose question is “does G contains a wheel as an induced subgraph” is NP-complete.
Even pairs in square-free Berge graphs with no odd prism
We consider the class of Berge graphs that contain no odd prism and no square (cycle on four vertices). We prove that every graph G in this class either is a clique or has an even pair, as
Induced Subgraphs of Graphs with Large Chromatic Number. III. Long Holes
We prove a 1985 conjecture of Gyárfás that for all k, ℓ, every graph with sufficiently large chromatic number contains either a clique of cardinality more than k or an induced cycle of length more
Graphs with no induced wheel or antiwheel
It is shown here that graphs that contain no wheel and no antiwheel have a very simple structure and consequently can be recognized in polynomial time.


About Skew Partitions in Minimal Imperfect Graphs
This paper proves the conjecture that a minimal imperfect graph G cannot have a skew cutset S decomposable into disjoint sets A and B joined by all possible edges in the particular case where at least one of A andB is a stable set.
Even‐hole‐free graphs part II: Recognition algorithm
An algorithm that determines in polytime whether a graph contains an even hole is presented, based on a decomposition theorem for even‐hole‐free graphs obtained in Part I of this work.
Testing balancedness and perfection of linear matrices
This paper gives polynomial algorithms to test whether a linear matrix is balanced or perfect, based on decomposition results previously obtained by the authors.
An Optimal Algorithm to Detect a Line Graph and Output Its Root Graph
Given a graph H, a graph G is sought such that H is the line graph of G, if G exists, and the algorithm obtains H within the order of E + O + N steps.
Decomposition of oddholefree graphs by double star cutsets and 2joins ” , preprint (
  • 2001
A MAX{m, n} Algorithm for Determining the Graph H from Its Line Graph C
Färbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind
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  • 1961
At least one of c 3 , . . . , c 2n−1 belongs to
  • At least one of c 3 , . . . , c 2n−1 belongs to