Minimal Imperfect Graphs Research Articles

Structure of Cubic Lehman Matrices

A pair $(A,B)$ of square $(0,1)$-matrices is called a Lehman pair if $AB^T=J+kI$ for some integer $k\in\{-1,1,2,3,\ldots\}$. In this case $A$ and $B$ are called Lehman matrices. This terminology arises because Lehman showed that the rows with the fewest ones in any non-degenerate minimally nonideal (mni) matrix $M$ form a square Lehman submatrix of $M$. Lehman matrices with $k=-1$ are essentially equivalent to partitionable graphs (also known as $(\alpha,\omega)$-graphs), so have been heavily studied as part of attempts to directly classify minimal imperfect graphs. In this paper, we view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite graph, focusing in particular on the case where the graph is cubic. From this perspective, we identify two constructions that generate cubic Lehman graphs from smaller Lehman graphs. The most prolific of these constructions involves repeatedly replacing suitable pairs of edges with a particular $6$-vertex subgraph that we call a $3$-rung ladder segment. Two decades ago, Lütolf & Margot initiated a computational study of mni matrices and constructed a catalogue containing (among other things) a listing of all cubic Lehman matrices with $k =1$ of order up to $17 \times 17$. We verify their catalogue (which has just one omission), and extend the computational results to $20 \times 20$ matrices. Of the $908$ cubic Lehman matrices (with $k=1$) of order up to $20 \times 20$, only two do not arise from our $3$-rung ladder construction. However these exceptions can be derived from our second construction, and so our two constructions cover all known cubic Lehman matrices with $k=1$.

Claw‐free circular‐perfect graphs

AbstractThe circular chromatic number of a graph is a well‐studied refinement of the chromatic number. Circular‐perfect graphs form a superclass of perfect graphs defined by means of this more general coloring concept. This article studies claw‐free circular‐perfect graphs. First, we prove that if G is a connected claw‐free circular‐perfect graph with χ(G)>ω(G), then min{α(G), ω(G)}=2. We use this result to design a polynomial time algorithm that computes the circular chromatic number of claw‐free circular‐perfect graphs. A consequence of the strong perfect graph theorem is that minimal imperfect graphs G have min{α(G), ω(G)}=2. In contrast to this result, it is shown in Z. Pan and X. Zhu [European J Combin 29(4) (2008), 1055–1063] that minimal circular‐imperfect graphs G can have arbitrarily large independence number and arbitrarily large clique number. In this article, we prove that claw‐free minimal circular‐imperfect graphs G have min{α(G), ω(G)}≤3. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 163–172, 2010

New near-factorizations of finite groups

The attempts to construct a counterexample to the Strong Perfect Graph Conjecture yielded the notion of partitionable graphs as minimal imperfect graphs; then near-factorizations of finite groups gained some interest since from any near-factorization some partitionable graphs can be constructed in a natural way. Recently, the proof of SPGC was declared by Chudnovsky, Robertson, Seymour and Thomas [3], but near-factorizations remain interesting on their own rights as (i) rare objects being “close” to factorizations of groups; and (ii) they yield graphs with surprising properties.

A Berge-keeping operation for graphs

We prove that a certain simple operation does not create odd holes or odd antiholes in a graph unless there are already some. In order to apply it, we need a vertex whose neighborhood has a coloring where the union of any two color classes is a connected graph; the operation is the shrinking of each of the color classes. Odd holes and antiholes do have such a vertex, and this property of minimal imperfect graphs implies the strong perfect graph theorem through the results of the paper. Conceivably, this property may be a target in the search for a proof of the strong perfect graph theorem different from the monumental achievement of Chudnovsky, Robertson, Seymour, and Thomas.

Quasi-star-cutsets and some consequences

This paper introduces a structural property of minimal imperfect graphs, no minimal imperfect graph has a quasi-star-cutset, before describing some consequences. First, a large class of perfect graphs is obtained from the class of all bipartite graphs and their line-graphs; this class contains several well-known classes of perfect graphs. The second consequence deals with a composition operation of graphs which preserves perfection and which encompasses the most part of the compositions known to preserve perfection.

No odd pairs in minimal imperfect NP5 graphs

We prove that a minimal imperfect graph containing a vertex which is not on any induced P 5 has no odd pair.

About Skew Partitions in Minimal Imperfect Graphs

V. Chvátal conjectured in 1985 that a minimal imperfect graph G cannot have a skew cutset (i.e., a cutset S decomposable into disjoint sets A and B joined by all possible edges). We prove here the conjecture in the particular case where at least one of A and B is a stable set.

On minimal cutsets in P5-free minimal imperfect graphs

Let G be a minimal imperfect P 5-free graph (i.e. a minimal imperfect graph not containing a path on 5 vertices as induced subgraph) and let S be a minimal cutset of G. In this paper we study several properties of such cutsets, in particular we prove that the subgraph induced by S is connected, contains a P 4, cannot induce a bipartite subgraph of G,… . As a by-product we also give a structural characterization of such graphs.

On vertex neighborhood in minimal imperfect graphs

Lubiw (J. Combin. Theory Ser. B 51 (1991) 24) conjectures that in a minimal imperfect Berge graph, the neighborhood graph N(v) of any vertex v must be connected; this conjecture implies a well known Chvátal's conjecture (Chvátal, First Workshop on Perfect Graphs, Princeton, 1993) which states that N(v) must contain a P 4 . In this note we will prove an intermediary conjecture for some classes of minimal imperfect graphs. It is well known that a graph is P 4 -free if, and only if, every induced subgraph with at least two vertices either is disconnected or its complement is disconnected; this characterization implies that P 4 -free graphs can be constructed by complete join and disjoint union from isolated vertices. We propose to replace P 4 -free graphs by a similar construction using bipartite graphs instead of isolated vertices.

Even pairs in Artemis graphs

Everett et al. (Discrete Math. 165–166 (1997) 233–252) defined a stretcher to be a graph whose edge set can be partitioned into two disjoint triangles and three vertex disjoint paths, each with an endpoint in both triangles. They also conjectured that graphs with no odd hole, no antihole and no stretcher (called Artemis graphs) may be reduced to a clique by successive contractions of even pairs. To date, no proof exists that Artemis graphs really have even pairs. We enquire here about sufficient conditions for a non-even pair of vertices to extend to a stretcher and deduce two results: the first one is a property of minimal imperfect graphs, the second one guarantees the existence of an even pair in certain Artemis graphs.

On minimal imperfect graphs without induced P5

In this paper, we summarize previous known results on P 5-free minimal imperfect graphs (i.e. minimal imperfect graphs not containing a path on 5 vertices as induced subgraph) and we introduce two new classes of graphs (defined by a local property) that contain P 5-free graphs. Next, we show that most of the results concerning P 5-free graphs can be extended to these classes. Moreover, we present a structural characterization of these graphs which leads to some new results. In particular, we prove that the Strong Perfect Graph Conjecture holds true for P 5-free and F-free graphs where F is any connected configuration on 5 vertices not containing an induced 2 K 2.

P4-domination in minimal imperfect graphs

Numerous studies concerning the number and disposition of chordless paths on four vertices in graphs proved that there is a strong relation between these paths and the perfectness of graphs. In this paper, we further investigate this relation by introducing the notion of P 4-domination and formulating two conjectures, which are used both to find an equivalent version of the odd pair conjecture and to frame it. In the last section, we prove some particular cases of these conjectures.

On minimal imperfect graphs with circular symmetry

Results of Lovász and Padberg entail that the class of so-called partitionable graphs contains all the potential counterexamples to Berge's famous Strong Perfect Graph Conjecture, which asserts that the only minimal imperfect graphs are the odd chordless cycles with at least five vertices (“odd holes”) and their complements (“odd antiholes”). Only two constructions (due to Chvátal, Graham, Perold, and Whitesides) are known for making partitionable graphs. The first one does not produce any counterexample to Berge's Conjecture, as shown by Sebö. Here we prove that the second one does not produce any counterexample either. This second construction is given by near-factorizations of cyclic groups based on the so-called “British number systems” introduced by De Bruijn. All the partitionable graphs generated by this second construction (in particular odd holes and odd antiholes) have circular symmetry. No other partitionable graph with circular symmetry is known, and we conjecture that none exists; in this direction we prove that any counterexample must contain a clique and a stable set with at least six vertices each. Also we prove that every minimal imperfect graph with circular symmetry must have an odd number of vertices. © 1998 John Wiley & Sons, Inc. J Graph Theory 29: 209–225, 1998

On minimal imperfect graphs with circular symmetry

Results of Lovász and Padberg entail that the class of so-called partitionable graphs contains all the potential counterexamples to Berge's famous Strong Perfect Graph Conjecture, which asserts that the only minimal imperfect graphs are the odd chordless cycles with at least five vertices (“odd holes”) and their complements (“odd antiholes”). Only two constructions (due to Chvátal, Graham, Perold, and Whitesides) are known for making partitionable graphs. The first one does not produce any counterexample to Berge's Conjecture, as shown by Sebö. Here we prove that the second one does not produce any counterexample either. This second construction is given by near-factorizations of cyclic groups based on the so-called “British number systems” introduced by De Bruijn. All the partitionable graphs generated by this second construction (in particular odd holes and odd antiholes) have circular symmetry. No other partitionable graph with circular symmetry is known, and we conjecture that none exists; in this direction we prove that any counterexample must contain a clique and a stable set with at least six vertices each. Also we prove that every minimal imperfect graph with circular symmetry must have an odd number of vertices. © 1998 John Wiley & Sons, Inc. J Graph Theory 29: 209–225, 1998

On transversals in minimal imperfect graphs

Chvátal (1984) proved that no minimal imperfect graph has a small transversal, that is, a set of vertices of cardinality at most α + ω − 1 which meets every ω-clique and every α-stable set. In this paper we prove that a slight generalization of this notion of small transversal leads to a conjecture which is as strong as Berge's strong perfect graph conjecture for a very large class of graphs, namely for those graphs whose diameter does not exceed 6.

Path parity and perfection

Two nonadjacent vertices x and y in a graph G form an even pair if every induced path between them has an even number of edges. For a given pair { x, y} in a graph G, we denote by G xy the graph obtained from G by contracting x and y. In 1982, Fonlupt and Uhry proved that if G is perfect then so is G xy . In 1987, Meyniel used this fact to prove that no minimal imperfect graph contains an even pair. In the last eight years, even pairs have become an important tool for proving that certain classes of graphs are perfect and for designing optimization algorithms on special classes of perfect graphs. This paper surveys results of these types. It also discusses numerous related concepts including odd pairs.

Some properties of minimal imperfect graphs

The Even Pair Lemma, proved by Meyniel, states that no minimal imperfect graph contains a pair of vertices such that all chordless paths joining them have even lengths. This Lemma has proved to be very useful in the theory of perfect graphs. The Odd Pair Conjecture, with ‘even’ replaced by ‘odd’, is the natural analogue of the Even Pair Lemma. We prove a partial result for this conjecture, namely: no minimal imperfect graph G contains a three-pair, i.e. two nonadjacent vertices u 1, u 2 such that all chordless paths of G joining u 1 to u 2 contain precisely three edges. As a by-product, we obtain short proofs of two previously known theorems: the first one is a well-known theorem of Meyniel (a graph is perfect if each of its odd cycles with at least five vertices contains at least two chords), the second one is a theorem of Olariu (a graph is perfect if it contains no odd antihole, no P 5 and no extended claw as induced subgraphs).

The connectivity of minimal imperfect graphs

fr ABSTRACT We prove that partitionable graphs are 2w - 2-connected, that this bound is sharp, and prove some structural properties of cutsets of cardinality 2w - 2. The proof of the con- nectivity result is a simple linear algebraic proof. 0 1996 John Wiley & Sons, Inc.

Minimal imperfect graphs: A simple approach

We provide a very short proof of the following theorem of Lovasz, and of its consequences:A graph is perfect if and only if in every induced subgraph the number of vertices does not exceed the product of the stability and clique numbers of the subgraph.

On Critical Edges in Minimal Imperfect Graphs

An edge of a graph is called critical , if deleting it the stability number of the graph increases, and a nonedge is called co-critical , if adding it to the graph the size of the maximum clique increases. We prove in this paper, that the minimal imperfect graphs containing certain configurations of two critical edges and one co-critical nonedge are exactly the odd holes or antiholes. Then we deduce some reformulations of the strong perfect graph conjecture and prove its validity for some particular cases. Among the consequences we prove that the existence in every minimal imperfect graph G of a maximum clique Q , for which G − Q has one unique optimal coloration, is equivalent to the strong perfect graph conjecture, as well as the existence of a vertex v in V ( G ) such that the (uniquely colorable) perfect graph G − v has a “combinatorially forced” color class. These statements contain earlier results involving more critical edges, of Markossian, Gasparian and Markossian, and those of Bacsó and they also imply that a class of partitionable graphs constructed by Chvátal, Graham, Perold, and Whitesides does not contain counterexamples to the strong perfect graph conjecture.