## 1. Introduction

For any polyhedron *P*, let *P _{I}* denote the convex hull of all integer points in

*P*. Chvátal (1973) (and implicitly Gomory, 1958) introduced a method to obtain approximations of

*P*outgoing from

_{I}*P*as follows. If Σ

*a*

_{i}x_{i}*b*is a valid inequality for

*P*and has integer coefficients only, then Σ

*a*⌊

_{i}x_{i}*b*⌋ is a Chvátal–Gomory cut for

*P*and valid for

*P*. Define

_{I}*P*′ to be the set of points satisfying all Chvátal–Gomory cuts for

*P*, and let

*P*

^{0}=

*P*and

*P*

^{t+1}=(

*P*)′ for non-negative integers

^{t}*t*. Obviously

*P*⊆

^{I}*P*⊆

^{t}*P*holds for every

*t*. An inequality Σ

*a*

_{i}x_{i}*b*is said to have Chvátal rank at most

*t*if it is a valid inequality for the polytope

*P*. Chvátal showed that for each polyhedron

^{t}*P*there exists a finite

*t*0 with

*P*=

^{t}*P*; the smallest such

^{I}*t*is the

*Chvátal rank*of

*P*.

In this paper, we consider the Chvátal rank of two different linear relaxations of the stable set polytope STAB(*G*) of a graph *G*, outgoing from the system of inequalities associated with all edges and with all cliques of *G*, respectively.

Given a graph *G*=(*V*, *E*), a stable set in *G* is a set of mutually nonadjacent nodes. Accordingly, the *stable set polytope* STAB(*G*) is defined as the convex hull of the incidence vectors of all stable sets in *G*. Finding a stable set of maximum cardinality in *G* is an NP-hard problem, which can be stated as the following integer program:

The optimal value of this problem is denoted by *α*(*G*), the *stability number* or *independence number* of *G*. A natural linear relaxation of (1) gives rise to the *edge constraint stable set polytope*:

A clique *Q*⊆*G* is a set of mutually adjacent nodes. It is easy to see that edges are special cliques, and that any stable set can contain at most one node of a clique. Hence, *clique constraints*

are valid for STAB(*G*) and can be used to strengthen the last relaxation, leading to the *clique constraint stable set polytope*

Further generalizations of clique constraints are 0/1-constraints associated with arbitrary induced subgraphs *G*′⊆*G*, the *rank constraints*

Using these, the *rank constraint stable set polytope* is defined by

Clearly,

holds for any graph *G*. One central question is to identify and characterize classes of graphs that satisfy some of these inclusions with equality. Here we take a slightly different approach and study for some prominent graph classes, how many applications of the Chvátal–Gomory procedure are required to arrive at STAB(*G*), starting from one of the relaxations mentioned above. More precisely, for , define *the Chvátal rank of G w.r.t*. as

We will consider the values of *cr _{E}*(

*G*) and

*cr*(

_{Q}*G*) for several graph classes. The clique constraint stable set polytope has been widely studied due to its interesting polyhedral properties; for instance, QSTAB(

*G*) is related to the stable set polytope of its complement through anti-blocker theory (Fulkerson, 1971, 1972). From a practical point of view, however, it is more natural to consider ESTAB(

*G*) as an initial stage for Chvátal–Gomory rounding, as QSTAB(

*G*) might contain an exponentially large number of inequalities even for simple graph classes (see Section 2.2 for an example), whereas ESTAB(

*G*) has only

*O*(|

*V*|

^{2}) for any graph. Moreover, the separation of clique inequalities is NP-hard in general (Grötschel et al., 1981), but trivial for edge inequalities.

STAB(*G*) and ESTAB(*G*) coincide if and only if *G* is *bipartite* (i.e., if *V* can be partitioned into two stable sets), and thus *cr _{E}*(

*G*)=0 holds for any bipartite graph

*G*. It is well known that ESTAB(

*G*)′ has, besides edge constraints, only rank inequalities associated with all chordless odd cycles

*C*

_{2k+1}⊆

*G*, the

*odd hole constraints*

Chvátal called a graph *G t-perfect* if STAB(*G*) and ESTAB(*G*)′ coincide, i.e. if *cr _{E}*(

*G*)=1 holds. More generally, a graph

*G*is

*t*if STAB(

^{i}-perfect*G*)=ESTAB(

*G*)

*, i.e. if*

^{i}*cr*(

_{E}*G*)=

*i*holds. Then bipartite (resp.

*t*-perfect) graphs are exactly the

*t*

^{0}-perfect (resp.

*t*

^{1}-perfect) graphs.

STAB(*G*) and QSTAB(*G*) coincide if and only if *G* is *perfect* (Chvátal, 1975; Padberg, 1974), that is if for all induced subgraphs *G*′⊆*G*, the node set of *G*′ can be partitioned into as many stable sets as the size *ω*(*G*′) of a largest clique in *G*′. Thus, *cr _{Q}*(

*G*)=0 holds if and only if

*G*is perfect. According to a recent characterization achieved by Chudnovsky et al. (2006), perfect graphs are precisely those graphs without any odd holes

*C*

_{2k+1}with

*k*2, or their complements, the odd antiholes . Odd holes are

*t*-perfect and, if

*G*is either an odd hole or an odd antihole, then

*cr*(

_{Q}*G*)=1.

A graph *G* is said to be *rank-perfect* if STAB(*G*) equals RSTAB(*G*), see Wagler (2004a). By definition, all perfect, *t*-perfect and *h-perfect* graphs are rank-perfect (the latter are these graphs *G* where STAB(*G*) is given by clique and odd hole constraints). Further examples of rank-perfect graphs are odd antiholes (Padberg, 1974) and antiwebs (Wagler, 2004b).

Let [*n*] denote the integer set {0, … , *n*−1}. An *antiweb* is a graph with node set [*n*] and edges *ij* if and only if , where *n*2*k*+2. Antiwebs include all cliques , odd antiholes , and odd holes . The clique constraints and rank constraints associated with induced (sub)antiwebs, together with non-negativity constraints, provide a complete linear description of the stable set polytope of antiwebs (Wagler, 2004b).

A graph *G* is *a-perfect* if STAB(*G*) is given by non-negativity inequalities and rank constraints associated with induced cliques and antiwebs in *G*. The class of *a*-perfect graphs is a subclass of rank-perfect graphs and a common superclass of *h*-perfect graphs and antiwebs. Thus, we have *cr _{R}*(

*G*)=0 for all such graphs. In the following, we focus our attention on the Chvátal rank of ESTAB(

*G*) and QSTAB(

*G*) for some fundamental graph classes (cliques, odd holes, odd antiholes and antiwebs), as well as the implications for related subclasses of

*a*-perfect graphs.

According to Chvátal et al. (1989), for any monotone polytope *P* in [0, 1]* ^{n}*, a valid inequality

*a*

^{T}x*b*of

*P*with integer coefficients can be obtained by applying the Chvátal–Gomory procedure at most times. As ESTAB(

_{I}*G*) and QSTAB(

*G*) are monotone polytopes in the 0/1-cube, implying for rank-perfect graphs

*G*that

holds, and hence their Chvátal rank is at most of order *O*(|*G*|). As the main result of this paper, we will provide an upper bound for the Chvátal rank of *a*-perfect graphs in terms of their clique and stability number (instead of their number of nodes).