Even‐hole‐free graphs part I: Decomposition theorem

Conforti, Michele; Cornuéjols, Gérard; Kapoor, Ajai; Vušković, Kristina

doi:10.1002/jgt.10006

Cited by 57 publications

(79 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A graph G is even-hole free if any induced subgraph of G is not a cycle consisting of an even number of vertices [6]. This graph class includes several well-known classes, such as trees, interval graphs and chordal graphs.…”

Section: Even-hole-free Graphsmentioning

confidence: 99%

Reconfiguration of Vertex Covers in a Graph

Ito

Hiroyuki

Zhou

2016

IEICE Trans. Inf. & Syst.

View full text Add to dashboard Cite

SUMMARY Suppose that we are given two vertex covers C 0 and C t of a graph G, together with an integer threshold k ≥ max{|C 0 |, |C t |}. Then, the vertex cover reconfiguration problem is to determine whether there exists a sequence of vertex covers of G which transforms C 0 into C t such that each vertex cover in the sequence is of cardinality at most k and is obtained from the previous one by either adding or deleting exactly one vertex. This problem is PSPACE-complete even for planar graphs. In this paper, we first give a linear-time algorithm to solve the problem for even-hole-free graphs, which include several well-known graphs, such as trees, interval graphs and chordal graphs. We then give an upper bound on k for which any pair of vertex covers in a graph G has a desired sequence. Our upper bound is best possible in some sense.

show abstract

Section: Even-hole-free Graphsmentioning

confidence: 99%

Reconfiguration of Vertex Covers in a Graph

Ito

Hiroyuki

Zhou

2016

IEICE Trans. Inf. & Syst.

View full text Add to dashboard Cite

show abstract

“…They posed the question [22] of the existence of a polynomialtime algorithm for the problem, which is answered in the affirmative here. We note that (even hole)-free graphs can be recognized in polynomial time [8,9].…”

Section: Background and Previous Workmentioning

confidence: 98%

“…some L i has a true partner in R (equivalently, some R i has a true partner in L), 4. some L i has a false partner in L ∪ R, 5. some R j has a false partner in L ∪ R, 6. for each i, if L i has no false partner in R, then L i has a true partner in L, 7. for each i, if R i has no false partner in L, then R i has a true partner in R, 8. if the set F of parts in L that have no true partners in R is not empty, then there is a part R j that is a false partner of all parts in F, 9. if the set H of parts in R that have no true partners in L is not empty, then there is a part L i that is a false partner of all parts in H, 10. if the set U of parts of L∪R that have no false partners in L∪R is not empty, then the parts in U must have the clique structure, each of them has a true partner in L and in R, and the two parts in U that are not true partners (if they exist) must belong to L ∩ R. Output: A set of at most 2k instances {Φ 1 , Φ 2 , .…”

Section: Properties (A) and (B) Ensure That Each A I Has Either A Trumentioning

confidence: 99%

The Complexity of the List Partition Problem for Graphs

Cameron¹,

Eschen²,

Hoàng³

et al. 2008

SIAM J. Discrete Math.

View full text Add to dashboard Cite

Abstract. The k-partition problem is as follows: Given a graph G and a positive integer k, partition the vertices of G into at most k parts A 1 , A 2 , . . . , A k , where it may be specified that A i induces a stable set, a clique, or an arbitrary subgraph, and pairs A i , A j (i = j) be completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list k-partition problem generalizes the k-partition problem by specifying for each vertex x, a list L(x) of parts in which it is allowed to be placed. Many well-known graph problems can be formulated as list k-partition problems: e.g., 3-colorability, clique cutset, stable cutset, homogeneous set, skew partition, and 2-clique cutset. We classify, with the exception of two polynomially equivalent problems, each list 4-partition problem as either solvable in polynomial time or NP-complete. In doing so, we provide polynomial-time algorithms for many problems whose polynomial-time solvability was open, including the list 2-clique cutset problem. This also allows us to classify each list generalized 2-clique cutset problem and list generalized skew partition problem as solvable in polynomial time or NP-complete.

show abstract

“…The first major structural study of even-hole-free graphs was done by Conforti, Cornuéjols, Kapoor and Vušković in [45] and [46]. They were focused on showing that even-hole-free graphs can be recognized in polynomial time (a problem that at that time was not even known to be in NP), and their primary motivation was to develop techniques which can then be used in the study of perfect graphs.…”

Section: Even-hole-free Graphsmentioning

confidence: 99%

The world of hereditary graph classes viewed through Truemper configurations

Vušković¹

2013

Surveys in Combinatorics 2013

Self Cite

View full text Add to dashboard Cite

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few specific types, that we will call Truemper configurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices).The configurations that Truemper identified in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper configurations, focusing on the algorithmic consequences, trying to understand what structurally enables efficient recognition and optimization algorithms.

show abstract

Even‐hole‐free graphs part I: Decomposition theorem

Abstract: We prove a decomposition theorem for even-hole-free graphs. The decompositions used are 2-joins and star, double-star and triple-star cutsets. This theorem is used in the second part of this paper to obtain a polytime recognition algorithm for even-hole-free graphs.

Cited by 57 publications

References 13 publications

Reconfiguration of Vertex Covers in a Graph

Reconfiguration of Vertex Covers in a Graph

The Complexity of the List Partition Problem for Graphs

The world of hereditary graph classes viewed through Truemper configurations

Contact Info

Product

Resources

About