Imperfect Graphs Research Articles

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.

An induced subgraph characterization of domination perfect graphs

AbstractLet γ(G) ι(G) be the domination number and independent domination number of a graph (G), respectively. A graph (G) is called domination perfect if γ(H) = ι(H), for every induced subgraph H of (G). There are many results giving a partial characterization of domination perfect graphs. In this paper, we present a finite induced subgraph characterization of the entire class of domination perfect graphs. The list of forbidden subgraphs in the charcterization consists of 17 minimal domination imperfect graphs. Moreover, the dominating set and independent dominating set problems are shown to be both NP‐complete on some classes of graphs. © 1995 John Wiley & Sons, Inc.

Discs in unbreakable graphs

We show that every vertex in an unbreakable graph is in a disc, where a disc is a chordless cycle, or the complement of a chordless cycle, with at least five vertices. A corollary is that every vertex in a minimal imperfect graph is in a disc.

Near-perfect matrices

A 0, 1 matrixA isnear-perfect if the integer hull of the polyhedron {x⩾0: Ax⩽\(\bar 1\)} can be obtained by adding one extra (rank) constraint. We show that in general, such matrices arise as the cliquenode incidence matrices of graphs. We give a colouring-like characterization of the corresponding class of near-perfect graphs which shows that one need only check integrality of a certain linear program for each 0, 1, 2-valued objective function. This in contrast with perfect matrices where it is sufficient to check 0, 1-valued objective functions. We also make the following conjecture: a graph is near-perfect if and only if sequentially lifting any rank inequality associated with a minimally imperfect graph results in the rank inequality for the whole graph. We show that the conjecture is implied by the Strong Perfect Graph Conjecture. (It is also shown to hold for graphs with no stable set of size eleven.) Our results are used to strengthen (and give a new proof of) a theorem of Padberg. This results in a new characterization of minimally imperfect graphs: a graph is minimally imperfect if and only if both the graph and its complement are near-perfect.

Complete Multi-partite Cutsets in Minimal Imperfect Graphs

We show that a minimal imperfect graph G cannot contain a cutset C which induces a complete multi-partite graph unless C is a stable set and G is an odd hole. This generalizes a result of Tucker, who proved that the only minimal imperfect graphs containing stable cutsets are the odd holes. It is also a special case of a conjecture of Chvátal.

Locally perfect graphs

A new property of minimal imperfect graphs is given. This leads to a way to add a new vertex to a perfect graph so that the resulting graph remains perfect. It is shown that the same holds for strongly perfect graphs. Some colouring properties are also considered. In particular we define a new kind of colouring, the locally perfect colouring, and we show that the graphs having such a colouring for any subgraph are perfect. Such graphs are called locally perfect. The triangulated graphs, the parity graphs, and the perfect graphs without a clique with more than three vertices are shown to be locally perfect. Finally the graphs obtained by some rules of construction are studied.

A generalization of chvátal's star-cutset lemma

A graph is perfect if for each of its induced subgraphs H, the chromatic number χ( H) of H equals the largest number ω( H) or pairwise adjacent vertices in H. We present a new property of minimal imperfect graphs which implies Chvátals well-known Star-Cutset Lemma, and which gives a new characterization of perfect graphs.

A new conjecture about minimal imperfect graphs

H. Meyniel proved that in every minimal imperfect graph, every pair of vertices is joined by a chordless path containing an odd number of edges. We conjectured that in every minimal imperfect graph, every pair of vertices is joined by a path containing an even number of edges. We give an equivalent version of this new conjecture.

No antitwins in minimal imperfect graphs

It is customary to call vertices x and y twins if every vertex distinct from x and y is adjacent either to both of them or to neither of them. By analogy, we shall call vertices x and y antitwins if every vertex distinct from x and y is adjacent to precisely one of them. Lovász proved that no minimal imperfect graph has twins. The purpose of this note is to prove the analogous statement for antitwins.

Bull-free Berge graphs are perfect

Abull is the (self-complementary) graph with verticesa, b, c, d, e and edgesab, ac, bc, bd, ce; a graphG is calledBerge if neitherG not its complement contains a chordless cycle whose length is odd and at least five. We prove that bull-free Berge graphs are perfect; a part of our argument relies on a new property of minimal imperfect graphs.

Decomposition of perfect graphs

We describe composition and decomposition schemes for perfect graphs, which generalize all recent results in this area, e.g., the amalgam and the 2-amalgam split. Our approach is based on the consideration of induced cycles and their complements in perfect graphs (as opposed to the consideration of cycles for defining biconnected or 3-connected graphs). Our notion of 1-inseparable graphs is “parallel” to that of biconnected graphs in that different edges in different inseparable components of a graph are not contained in any induced cycle or any complement of an induced cycle. Furthermore, in a special case which generalizes the join operation, this definition is self-complementary in a natural fashion. Our 2-composition operation, which only creates even induced cycles in the composed graphs, is based on two perfection-preserving vertex merge operations on perfect graphs. As a by-product, some new properties of minimally imperfect graphs are presented.

A New Property of Critical Imperfect Graphs and some Consequences

We prove a new property of critical imperfect graphs. As a consequence, we define a new class of perfect graphs. This class contains perfectly orderable graphs and graphs in which that every odd cycle has two chords.

A note on even pairs

We generalize a theorem of Meyniel about minimal imperfect graphs to partitionable graphs. The general result follows easily from a theorem of Bland, Huang, and Trotter.

Alternating orientation and alternating colouration of perfect graphs

We establish a property of minimal imperfect graphs, and use this property to generate two classes of perfect graphs. The first class contains all comparability graphs, all triangulated graphs, and two other classes of perfect graphs. The second class contains all triangulated graphs and all line-graphs of bipartite graphs.