On the strong perfect graph conjecture and critical graphs

In this paper we obtain some properties of graphs which are critical with respect to perfectness. Some alternate forms of Berge's [2] strong perfect graph conjecture are also given.     

Norms and perfect graphs

The weak Berge hypothesis states that a graph is perfect if and only if its complement is perfect. Previous proofs of this hypothesis have used combinatorial or polyhedral methods.In this paper, the concept of norms related to graphs is used to provide an alternative proof for the weak Berge hypothesis.This is a written account of an invited lecture delivered by the second author on occasion of the 12. Symposium on Operations Research, Passau, 9.–11. 9. 1987.     

Line perfect graphs

The concept of line perfection of a graph is defined so that a simple graph is line perfect if and only if its line graph is perfect in the usual sense. Line perfect graphs are characterized as those which contain no odd cycles of size larger than 3. Two well-know theorems of König for bipartite graphs are shown to hold also for line perfect graphs; this extension provides a reinterpretation of the content of these theorems.Supported by National Science Foundation Grants GK-42095 and ENG 76-09936.     

On critically perfect graphs

A perfect graph is critical, if the deletion of any edge results in an imperfect graph. We give examples of such graphs and prove some basic properties. We relate critically perfect graphs to well-known classes of perfect graphs, investigate the structure of the class of critically perfect graphs, and study operations preserving critical perfectness. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 394–404, 1999     

Trivially perfect graphs

An undirected graph is trivially perfect if for every induced subgraph the stability number equals the number of (maximal) cliques. We characterize the trivially perfect graphs as a proper subclass of the triangulated graphs (thus disproving a claim of Buneman [3]), and we relate them to some well-known classes of perfect graphs.     

Coloring planar perfect graphs by decomposition

This paper describes a decomposition scheme for coloring perfect graphs. Based on this scheme, one need only concentrate on coloring highly connected (at least 3-connected) perfect graphs. This idea is illustrated on planar perfect graphs, which yields a straightforward coloring algorithm. We suspect that, under appropriate definition, highly connected perfect graphs might possess certain regular properties that are amenable to coloring algorithms. This research has been supported in part by National Science Foundation under grant ECS—8105989 to Northwestern University.     

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.     

Recognizing bull-free perfect graphs

A bull is the graph obtained from a triangle by adding two pendant vertices adjacent to distinct vertices of the triangle. Chvátal and Sbihi showed that the strong perfect graph conjecture holds for Bull-free graphs. We give a polynomial time recognition algorithm for Bull-free perfect graphs.     

Square-free perfect graphs

We prove that square-free perfect graphs are bipartite graphs or line graphs of bipartite graphs or have a 2-join or a star cutset.     

Uniquely colorable perfect graphs

This paper defines the concept of sequential coloring. If G or its complement is one of four major types of perfect graphs, G is shown to be uniquely k-colorable it and only if it is sequentially k-colorable. It is conjectured that this equivalence is true for all perfect graphs. A potential role for sequential coloring in verifying the Strong Perfect Graph Conjecture is discussed. This conjecture is shown to be true for strongly sequentially colorable graphs.     

Meyniel graphs are strongly perfect

Meyniel (Discrete Math.16 (1976), 339–342) proved that a graph is perfect whenever each of its odd cycles of length at least five has at least two chords. This result is strengthened by proving that every graph satisfying Meyniel's condition is strongly perfect (i.e., each of its induced subgraphs H contains a stable set which meets all the maximal cliques in H).     

Upper domination and upper irredundance perfect graphs

Let β(G), Γ(G) and IR(G) be the independence number, the upper domination number and the upper irredundance number, respectively. A graph G is called Γ-perfect if β(H) = Γ(H), for every induced subgraph H of G. A graph G is called IR-perfect if Γ(H) =IR(H), for every induced subgraph H of G. In this paper, we present a characterization of Γ-perfect graphs in terms of a family of forbidden induced subgraphs, and show that the class of Γ-perfect graphs is a subclass of IR-perfect graphs and that the class of absorbantly perfect graphs is a subclass of Γ-perfect graphs. These results imply a number of known theorems on Γ-perfect graphs and IR-perfect graphs. Moreover, we prove a sufficient condition for a graph to be Γ-perfect and IR-perfect which improves a known analogous result.     

Progress on perfect graphs

 A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved efficiently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph. In the first part of the paper we survey the main aspects of perfect graphs and their relevance. In the second part we outline our recent proof of the Strong Perfect Graph Conjecture of Berge from 1961, the following: a graph is perfect if and only if it has no induced subgraph isomorphic to an odd cycle of length at least five, or the complement of such an odd cycle. Received: December 19, 2002 / Accepted: April 29, 2003 Published online: May 28, 2003 Key words. Berge graph – perfect graph – skew partition Mathematics Subject Classification (1991): 05C17     

Random perfect graphs

We investigate the asymptotic structure of a random perfect graph P_n sampled uniformly from the set of perfect graphs on vertex set . Our approach is based on the result of Prömel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number and clique number is close to a concentrated distribution L(n) which plays an important role in our generation method. We also prove that the probability that P_n contains any given graph H as an induced subgraph is asymptotically 0 or or 1. Further we show that almost all perfect graphs are 2‐clique‐colorable, improving a result of Bacsó et al. from 2004; they are almost all Hamiltonian; they almost all have connectivity equal to their minimum degree; they are almost all in class one (edge‐colorable using Δ colors, where Δ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon .     

Star-cutsets and perfect graphs

We first establish a certain property of minimal imperfect graphs and then use it to generate large classes of perfect graphs.     

On-line coloring of perfect graphs

Lovász, Saks, and Trotter showed that there exists an on-line algorithm which will color any on-linek-colorable graph onn vertices withO(nlog^(2k–3) n/log^(2k–4) n) colors. Vishwanathan showed that at least (log ^k–1 n/k ^k ) colors are needed. While these remain the best known bounds, they give a distressingly weak approximation of the number of colors required. In this article we study the case of perfect graphs. We prove that there exists an on-line algorithm which will color any on-linek-colorable perfect graph onn vertices withn ^10k/loglogn colors and that Vishwanathan's techniques can be slightly modified to show that his lower bound also holds for perfect graphs. This suggests that Vishwanathan's lower bound is far from tight in the general case.Research partially supported by Office of Naval Research grant N00014-90-J-1206.     

s-strongly perfect cartesian product of graphs

The study of perfectness, via the strong perfect graph conjecture, has given rise to numerous investigations concerning the structure of many particular classes of perfect graphs. In “Perfect Product Graphs” (Discrete Mathematics, Vol. 20, 1977, pp. 177--186), G. Ravindra and K. R. Parthasarathy tried, but unfortunately without success, to characterize the perfectness of the cartesian product of graphs (see also MR No. 58--10567, 1979). In this paper we completely characterize the graphs that are both nontrivial cartesian products and s-strongly perfect.     

Perfect codes in graphs

The classical problem of the existence of perfect codes is set in a vector space. In this paper it is shown that the natural setting for the problem is the class of distance-transitive graphs. A general theory is developed that leads to a simple criterion for the existence of a perfect code in a distance-transitive graph, and it is shown that this criterion implies Lloyd's theorem in the classical case.     

A characterization of domination perfect graphs

Let γ(G) and i(G) be the domination number and independent domination number of a graph G, respectively. Sumner and Moore [8] define a graph G to be domination perfect if γ(H) = i(H), for every induced subgraph H of G. In this article, we give a finite forbidden induced subgraph characterization of domination perfect graphs. Bollobás and Cockayne [4] proved an inequality relating γ(G) and i(G) for the class of K^1,k -free graphs. It is shown that the same inequality holds for a wider class of graphs.     

On the structure of bull-free perfect graphs

A bull is a graph obtained by adding a pendant vertex at two vertices of a triangle. Chvátal and Sbihi showed that the Strong Perfect Graph Conjecture holds for bull-free graphs. We show that bull-free perfect graphs are quasi-parity graphs, and that bull-free perfect graphs with no antihole are perfectly contractile. Our proof yields a polynomial algorithm for coloring bull-free strict quasi-parity graphsPartially supported by CNPq, grant 30 1160/91.0