# On the Chvátal rank of linear relaxations of the stable set polytope

## Abstract

We study the Chvátal rank of two linear relaxations of the stable set polytope, the edge constraint and the clique constraint stable set polytope. For some classes of graphs whose stable set polytope is given by 0/1-valued constraints only, we present either the exact value of the Chvátal rank or upper bounds (of the order of their largest cliques and stable sets), which improve the bounds previously known from the literature (of the order of the graph itself).

## 1. Introduction

For any polyhedron P, let PI denote the convex hull of all integer points in P. Chvátal (1973) (and implicitly Gomory, 1958) introduced a method to obtain approximations of PI outgoing from P as follows. If Σaixib is a valid inequality for P and has integer coefficients only, then Σaixib⌋ is a Chvátal–Gomory cut for P and valid for PI. Define P′ to be the set of points satisfying all Chvátal–Gomory cuts for P, and let P0=P and Pt+1=(Pt)′ for non-negative integers t. Obviously PIPtP holds for every t. An inequality Σaixib is said to have Chvátal rank at most t if it is a valid inequality for the polytope Pt. Chvátal showed that for each polyhedron P there exists a finite t0 with Pt=PI; the smallest such 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=(VE), 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:

(1)

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 QG 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 crE(G) and crQ(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 crE(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 C2k+1G, the odd hole constraints

Chvátal called a graph G t-perfect if STAB(G) and ESTAB(G)′ coincide, i.e. if crE(G)=1 holds. More generally, a graph G is ti-perfect if STAB(G)=ESTAB(G)i, i.e. if crE(G)=i holds. Then bipartite (resp. t-perfect) graphs are exactly the t0-perfect (resp. t1-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, crQ(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 C2k+1 with k2, or their complements, the odd antiholes . Odd holes are t-perfect and, if G is either an odd hole or an odd antihole, then crQ(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 n2k+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 crR(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 aTxb of PI with integer coefficients can be obtained by applying the Chvátal–Gomory procedure at most times. As ESTAB(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).

## 2. Chvátal ranks of inequalities and graphs

In order to determine the Chvátal rank of STAB(G) w.r.t. some linear relaxation , we need to know when the facet-defining inequalities for STAB(G) are valid for . More precisely, for an inequality aTxb being valid for STAB(G), we define by

its Chvátal rank w.r.t. the relaxation . Clearly, we have that

In the following, we discuss for some classes of rank-perfect graphs their Chvátal rank w.r.t. QSTAB(G) and ESTAB(G) by determining the Chvátal rank of the facets of STAB(G).

We say that a subgraph G′⊆G induces a facet aTxb of STAB(G) if

holds. For a family of facet-inducing subgraphs of G, we say that a facet aTxb of STAB(G) is generated by if aTxb can be obtained by applying (one round of) the Chvátal–Gomory procedure to inequalities induced by the subgraphs in .

### 2.1. Cliques

Clique constraints are of particular interest as they are required to describe the stable set polytope of any graph (Padberg, 1974). We clearly have crQ(χQ, 1)=0 for every clique Q, and from Chvátal (1973) it is known that . Our purpose is to determine the exact values for crE(χQ, 1) and crE(Kn).

Lemma 1.Let n3. A clique Kn and the corresponding clique constraint have Chvátal rank

(2)

Proof. For qn, let Qq be the family of all subcliques of Kn having size q. As any induced subgraph of Kn is a clique, we have . Moreover, every node of Kn belongs to exactly cliques in Qq. Hence, adding up all constraints corresponding to cliques from Qq yields

This is the required clique constraint if and only if we have . The smallest integer satisfying this condition is and thus

(3)

We will prove (2) by induction on the value of n. The condition can be easily verified for n=3, as crE(K3)=1. Now assume (2) holds for all integers strictly smaller than n. Then it follows from (3) that

If n is even, then , and the proof is completed. Otherwise, let , which means . Since n is odd, holds strictly and hence implies . This finishes the proof, as . ▪

As clique constraints suffice to describe the stable set polytope of perfect graphs, we have:

Corollary 2.A perfect graph G withhas.

For any h-perfect graph G, the stable set polytope is given by clique and odd hole constraints. Odd hole constraints can be generated from edges by Chvátal (1973), and to generate a constraint x(Q)1 associated with a clique QG, only subcliques of Q can be used; thus, we obtain:

Corollary 3.Every imperfect h-perfect graph G withhas crQ(G)=1 and.

### 2.2. Odd antiholes

For every odd antihole with k2 the only nontrivial facets of are rank constraints associated with maximal cliques and the graph itself (Padberg, 1974). The full rank constraint can be obtained by applying the Chvátal–Gomory procedure to the maximum clique constraints, i.e. holds. However, the number of inclusion-wise maximal cliques contained in grows exponentially with respect to k, as the next lemma shows.

Lemma 4.The odd antiholecontains exactly

inclusion-wise maximal cliques of size l, with.

Proof. Assume and ijE if and only if . Let be a maximal clique in , with and . Observe that Q may be uniquely represented by the vector

where . In the following, we call i the start node and d0, … , dl−1 the distance components.

Since Q is a clique, every distance component must be larger than one. Moreover, if dr4 holds for some distance component, then is a clique, and hence Q is not maximal. Thus, we have , for all 0jl−1. The maximality of Q also implies that , with i=2 only if dl−1=3. Conversely, it is straightforward to see that any such vector defines a maximal clique.

Let a, b∈Z+ denote the number of distance components equal to 2 and 3, respectively. These numbers must satisfy the following system:

It follows that a=3l−(2k+1), b=(2k+1)−2l and hence 2l2k+13l, or equivalently, .

Let be the set of indices corresponding to distance components with value 3, and observe that the pair (iIb) uniquely determines the vector and hence the clique Q. There are possible choices for Ib, and each of them can be completed to a valid pair by choosing i∈{0, 1}. For i=2, the restriction dl−1=3 has to be considered, which is equivalent to asking l−1∈Ib. There are then possible choices for Ib. Adding together these quantities and substituting b=(2k+1)−2l, the assertion of the lemma follows. ▪

To obtain all facets of starting from edge constraints, note that we clearly have

since and holds. The next lemma shows that also non-maximum cliques can suffice to generate the rank constraint associated with an odd antihole.

Lemma 5.For any odd antihole, we have

with.

Proof. For , let be an inclusion-wise maximal clique of size and consider the family of maximal cliques defined by

with all sums taken modulo 2k+1. Observe that each node of is contained in exactly cliques from F. Hence, adding up the 2k+1 clique inequalities we obtain

It follows that F generates the odd antihole constraint if and only if . The smallest integer satisfying this inequalities is and therefore .

Since is completely described by the full rank facet and clique inequalities, it also follows from the last observation that

To complete the proof, note that for all k4 we have . Thus, 2(q−2)>k−2 and . The result then follows as a consequence of Lemma 1. Finally, if k∈{2, 3}, one can verify that k=q holds.▪

Remark 6. For some values of k, we have that holds, since

In these cases, the exact value of can be obtained from . This is true, for instance, for , and . In fact, we conjecture that holds in general.

We call a graph m-perfect if its stable set polytope is given by non-negativity inequalities and rank constraints associated with cliques and minimally imperfect subgraphs (i.e., odd holes and odd antiholes). As odd hole constraints can be generated from edges by Chvátal (1973) and rank constraints associated with cliques and odd antiholes from subcliques of appropriate size, we obtain as a consequence of (3) in Lemma 1 and Lemma 5:

Corollary 7.An imperfect m-perfect graph G withand largest odd antiholehasandwith.

### 2.3. Antiwebs

We now turn to stable set polytopes of antiwebs. An antiweb is called prime if k+1 and n are relatively prime. Note that all cliques , odd antiholes , and odd holes are examples of prime antiwebs. Trotter (1975) showed that the rank constraint associated with an antiweb is facet-defining if and only if the antiweb is prime. Following Wagler (2004b), the stable set polytope of an antiweb is completely described by non-negativity inequalities and rank constraints associated with induced prime subantiwebs (including clique constraints and the full rank constraint if is prime).

Our aim is to determine and for all antiwebs. We first note which antiwebs are t0-perfect, then we characterize the t1-perfect antiwebs, and present an algorithm that determines an upper bound of for general antiwebs. Finally, we show how to adapt the procedure to obtain upper bounds for .

The t0-perfect graphs are exactly the bipartite graphs, i.e., the perfect graphs with clique number 2. According to Trotter (1975), the perfect antiwebs are exactly

• the cliques ,
• the antiholes with even n,
• the matchings for all k0.

(Recall that we excluded the stable sets with by definition.) Hence, by , an antiweb is t0-perfect if and only if holds.

We next determine all t1-perfect antiwebs. Besides and the odd holes for all k1, which are all antiwebs that do not induce the full rank facet and contain edges, triangles, and odd holes as only facet-inducing subgraphs. In other words, they are antiwebs with clique number <4 whose only prime subantiwebs are for k0.

Trotter (1975) determined necessary and sufficient conditions for an antiweb to be contained as an induced subgraph in a larger antiweb . Since these conditions will be used throughout the section, we state them here as a separate lemma.

Lemma 8. (Trotter, 1975).The antiwebis an induced subgraph ofif and only if we have n′<n, k′<k, and

(4)

We next provide the characterization of all non-bipartite t-perfect antiwebs. The term Moebius ladders refers to the antiwebs with k2.

Theorem 9.A non-bipartite antiwebis t-perfect if and only if it is

• an odd holewith k1,
• a Moebius ladderwith odd k3, or
• equal to.

Proof. In order to characterize the t-perfect antiwebs, we shall determine all antiwebs having no other prime subantiwebs than triangles and odd holes. More precisely, for all antiwebs different from those listed in the theorem, we either exhibit at least one of the following

• the clique with ,
• the odd antihole with ,
• a Moebius ladder with even k2, which has as it is prime,

or show that itself is prime. We first observe:

Claim 10.No antiweb with clique number at least 4 is t-perfect.

Hence, we restrict our attention to antiwebs with clique number 2 or 3. By , which are exactly the antiwebs with .

Claim 11.The only t-perfect antiwebs with clique number 3 are.

Proof. For fixed k, an antiweb has clique number 3 if . For k=0, this is just which is obviously t-perfect. For k=1, we have to consider and . The first one equals , is perfect, and has by Lemma 1. The latter one is the odd antihole and has . For k2, we show that all these antiwebs but contain either or the even Moebius ladder as an induced subgraph, or are prime.

From Lemma 8, if and only if

holds. Observe that the right-most inequality is always satisfied, as we have , for all k1. Hence, no antiweb with is t-perfect.

Furthermore, contains if and only if

holds. By for all k2, no antiweb with n4k can be t-perfect.

Combining the two observations, all possible candidates for t-perfect antiwebs must satisfy

It is straightforward to see that this inequality system has no solution for k5. For 2k4, the only antiwebs fulfilling this condition are , , and . Among them, , and are prime and, therefore, not t-perfect. does not induce the full rank facet and has K3 and C5 as only facet-inducing subgraphs; hence, is indeed t-perfect. ▪

Claim 12.The only non-bipartite t-perfect antiwebs with clique number 2 are odd holeswith k1 and Moebius ladderswith odd k3.

Proof. For fixed k, an antiweb has clique number 2 if . As the antiwebs are bipartite for all k0, and odd holes for all k1, we start with the Moebius ladders for k2.

Any Moebius ladder is 3-regular (as, by definition, node i has exactly as neighbors). Hence, all their subantiwebs are either 1-regular matchings or 2-regular odd holes. Whether or not a Moebius ladder is t-perfect depends, therefore, on whether or not it induces the full rank facet. This is the case if k is even, but not if k is odd (as then both k+1 and n=2k+4 are even). Hence, all odd Moebius ladders are indeed t-perfect. On the other hand, it is well known that even Moebius ladders are t2-perfect.

It remains to show that all other antiwebs with clique number 2 are not t-perfect (note that they satisfy and k3). For that, we either exhibit an even Moebius ladder as an induced subgraph, or show that itself is prime.

From Lemma 8, if and only if , and

Since , the last condition can be shown to be equivalent to

(5)

Moreover, for the same reason and hold, and hence (5) suffices to imply . There is an satisfying this condition if and only if

With , this is equivalent to

(6)

Now we prove that, with the only two exceptions and , all antiwebs having k3 and satisfy (6). Let and observe that and k3 imply r1. Moreover, from it follows

(7)

We distinguish the following three cases:

• If r3, then and from (7) we obtain
The last inequality holds by , and (6) yields for some .
• If r=2 then (7) implies
Observe that the right-hand side is smaller than 2 for any x1, and hence (6) is again satisfied. For x=0, the definition of r implies that
Hence (6) holds for k∈{5, 6}, but not for k=7, which yields the exception .
• If r=1 the inequality (7) leads to
and implies (6) for any x2. Considering again the definition of r, if x=0 the only possible values for k are 3 and 4, while if x=1 then k may take values between 3 and 5. Moreover, by manually checking the condition (6) in those cases, it is straightforward to see that it is satisfied for all values except x=0 and k=4, which leads to as the second exception.

As both and are prime and different from odd holes, the assertion follows.

Combining all three claims finally verifies the assertion of the theorem. ▪

We now consider the Chvátal rank of not t-perfect antiwebs. As observed at the beginning of the section, for a general graph G we have

As for antiwebs all non-trivial facets are rank constraints associated with prime subantiwebs, we focus on the generation of such inequalities.

Let G:=(VE) be an induced subgraph in an antiweb . For any i∈[n] we denote by Gi the subgraph of induced by the set , with all sums taken modulo n. Observe that by circular symmetry of the antiweb, the graphs G and Gi are isomorphic. In the following, F(G) will refer to the family containing the n isomorphic subgraphs of defined by

The next result provides a necessary and sufficient condition for F(G) to generate the full rank inequality of .

Lemma 13.Let G be any induced subgraph of. The family F(G) generates the full rank inequalityif and only if

(8)

where n(G) is the number of nodes of G.

Proof. Observe that each node of is contained in exactly n(G) graphs from F(G). Hence, adding the rank inequalities corresponding to all graphs , we obtain

Now F(G) generates the full rank inequality of if and only if

The left inequality in the chain holds trivially for any subgraph , as the full rank inequality of an antiweb is always tight. The right inequality is equivalent to (8). ▪

Combining the results from Lemma 8 and Lemma 13, it is possible to determine when the full rank inequality of an antiweb can be generated from subantiwebs.

Lemma 14.For any integers n>n′, n>k′, the antiwebcontains antiwebsthat generate the full rank inequality if and only if

(9)
(10)

Proof. Assume for some n′<n and k′<n. Observe that (4) in Lemma 8 implies the following two inequalities:

On the other hand, substituting in (8) from Lemma 13 we obtain the other condition

Conversely, it is straightforward to verify that (9) and (10) together imply the conditions for Lemma 8 and Lemma 13, and this completes the proof. ▪

A clique of size q is isomorphic to an antiweb , as observed at the beginning of this subsection. Hence, substituting and in (9) we obtain that the full rank constraint of an antiweb can be generated by such cliques if and only if

holds. Moreover, it is straightforward to check that provides such a value for any , and the next result follows.

Corollary 15.For k>0 and,

Lemma 13 and Lemma 14 can be exploited algorithmically for computing an upper bound for the Chvátal rank of .

Lemma 16.Algorithm 1 ends in a finite number of steps and returns an upper boundfor the Chvátal rank of the edge constraint stable set polytope of an antiweb.

Proof. The main algorithm contains, besides the call of subprocedure FACET_RANK in line 32, only finite loops and simple assignments. Moreover, observe that in each new recursive call of the subprocedure, we have k″<k′ and hence the condition k′=0 in line 2 of Algorithm 2 becomes true after a finite number of calls, which guarantees that the recursion terminates.

 Algorithm 1. A recursive algorithm for computing Input:n, k Output: 3: {Check for t0-perfect antiwebs} ifthen return 0 6: {Check for t-perfect antiwebs, use Theorem 9} if () or ( and k odd) or () then return 1 9: {Check for t2-perfect Moebius ladders} if and k even then return 2 12: {Check for cliques, use Lemma 1} ifk=0 then return 15: {Check for antiholes, note: n6 has already been processed} ifk=1 then {Even antiholes are perfect, use Corollary 2} 18: ifn is even then return {For odd antiholes, use Lemma 5} 21: ifn is odd then return 24: {Otherwise, collect all prime subantiwebs} fordo 27:  fordo ifk′+1 and n′ are relatively prime then 30: {Compute maximal Chvátal-rank of a facet} r+←0 fordo 33: r′←FACET_RANK ifr′>rthen r←r′ 36: returnr

First, the algorithm detects if is either t0-perfect (i.e., a matching ), non-bipartite t-perfect (from Theorem 9), or an even Moebius ladder (which are known to be t2-perfect), and returns the corresponding value of . The exact Chvátal rank is also computed for when is a clique, using Lemma 1. Then antiholes with n7 are considered, as antiholes with n6 belong to one of the classes mentioned above. Even antiholes are perfect, and Corollary 2 provides an exact value for . For odd antiholes, Lemma 5 is applied for obtaining the upper bound , which is returned in line 23.

In all other cases, the algorithm first constructs a set L containing all facet-defining subantiwebs , including possibly itself. For each of these antiwebs, an upper bound for the Chvátal rank of the corresponding rank constraint is calculated, and the algorithm then returns the maximum of these values.

 Algorithm 2. Subprocedure FACET_RANK {Finish recursion if is a clique} ifk′=0 then 3: return r←∞ {Check all proper subantiwebs} 6: fordo fordo {Determine if can generate full rank facet} 9:  ifthen r′←FACET_RANK ifr′

Computation of is done in the subprocedure FACET_RANK. In each call, all proper subantiwebs that can generate the facet are examined using (8) from Lemma 13, and upper bounds for the Chvátal ranks of the corresponding rank inequalities are determined via recursive calls. The procedure keeps track of the minimum r of these values, and finally returns r+1. The recursion ends whenever the procedure is called with k′=0, in which case is a clique and Lemma 1 is used again to obtain .▪

Note that the algorithm can be easily modified to compute an upper bound for and as follows:

• the return values in line 14 of the main algorithm and in line 3 of the subprocedure are both set to zero, as holds trivially for all cliques; and
• the return value for odd antiholes is set to 1 in line 23, and for even antiholes is set to 0 in line 19.

Remark that holds for all t0-perfect antiwebs, for all t-perfect antiwebs, and for all t2-perfect Moebius ladders as all of them, except for , have . Moreover, , since implies that this antiweb is not perfect.

Table 1 shows the values of obtained by our algorithm for and , and Table 2 reports the corresponding upper bounds for .

Table 1.
Upper bounds for , for k32 and n2k+2
n\k01234567891011121314151617181920212223242526272829303132
31
420
521
6310
7321
83220
93311
1042210
1143221
12432220
13433221
144332210
154433221
1643322220
1744332221
18533332210
19543332221
205443323220
215433322221
2254433322210
2354433323221
24544343322220
25544433323221
265443333232210
275544333322221
2854443333232220
2955443433232221
30544434332332210
31554443333232221
325444434332322220
335544434333332221
3464444343332322210
3565444443332332221
36654443433333322220
37655444343332322221
386554443433323222210
396544443433323322221
4065544444433333222220
4165544444433323322221
42655444444333333232210
43655444434333333322221
446554444343333233222220
456555444444333333322221
4665544444443333333222210
4765554444443333233232221
48655444444443332332222220
49655544443443333333232221
506555444444433333333222210
516655444444433333333222221
5265554444444333333332322220
5366554444444433333333222221
54655554444444333333332322210
55665544444344343333333322221
566555544444444333333332322220
576655544444444333323332322221
5865555444444443333333332222210
5966555444444443433333332322221
60655554444444433333333333222220
61665554444444443333333332322221
626555544444444433333333323222210
636655544544444434333333333322221
6465555444444444433333333323222220
6566555544444444433333333323222221
66755555444444444343333333323222210
Table 2.
Upper bounds for , for k32 and n2k+2
n\k01234567891011121314151617181920212223242526272829303132
30
400
501
6000
7011
80020
90111
1000110
1101121
12001220
13011121
140011210
150112221
1600112220
1701111221
18001122210
19011122221
200011223220
210111112221
2200111222210
2301111223221
24001122222220
25011112223221
260011112232210
270111112222221
2800111122232220
2901111222232221
30001112222332210
31011111122232221
320011112222322220
330111112222332221
3400111122222322210
3501111222222332221
36001111212223322220
37011111122222322221
380011111222223222210
390111111222223322221
4000111122222233222220
4101111122222223322221
42001111222222233232210
43011111112222233322221
440011111122222233222220
450111111222222233322221
4600111112222222333222210
4701111112222222233232221
48001111222222222332222220
49011111121222222333232221
500011111122222223333222210
510111111122222222333222221
5200111111222222223332322220
5301111111222222223333222221
54001111112222222223332322210
55011111122122232223333322221
560011111222222222233332322220
570111111112222222223332322221
5800111111122222222233332222210
5901111111122222322233332322221
60001111111222222222333333222220
61011111112222222222233332322221
620011111122222222222333323222210
630111111222222223222333333322221
6400111111212222222222333323222220
6501111111112222222222333323222221
66001111111122222232223333323222210

In general, the values of tend to increase as n increases and to decrease as k increases. There are, however, exceptions to this rule, the smallest one being . On the other hand, it is straightforward to verify that coincides with the actual values of for cliques, t0-perfect, and t-perfect antiwebs.

The same idea used in Algorithm 1 leads to the following general upper bound for :

Corollary 17.For any antiweb,

whereandare the cardinalities of the maximum stable set and the maximum clique in, respectively.

Proof. Consider a prime subantiweb whose corresponding facet-defining rank constraint achieves the maximal value for . We know that this inequality can be generated by a family of cliques, or by a family of proper subantiwebs. Repeating this process for at most k′−1 times reveals a subantiweb whose rank inequality can be generated by cliques. Moreover, these cliques have sizes no larger than and hence,

The last inequality follows from the fact that every clique in (resp. every stable set in ) is also a clique (resp. a stable set) in . Applying now Lemma 1, the assertion of the corollary follows. ▪

Table 3 shows the values obtained from the last corollary for antiwebs with and . These values coincide with the output of our algorithm for cliques and some odd antiholes, but they become considerably larger as k increases, which suggests that the bound can still be improved.

Table 3.
Upper bounds obtained from Corollary 17
n\k01234567891011121314151617181920212223242526272829303132
31
421
521
6322
7322
83323
93333
1043334
1143334
12444445
13444445
144444456
154444556
1644455567
1744455567
18545556678
19545556678
205555666789
215555667789
22555566778910
23555566778910
2455566778891011
2555566778891011
265556677889101112
275556677899101112
28555667889910111213
29555667889910111213
30556677889101011121314
31556677889101011121314
3255667789910101112131415
3355667789910111112131415
346566778991011111213141516
356566778991011111213141516
366666788910101112121314151617
376666788910101112121314151617
38666678891010111212131415161718
39666678891010111213131415161718
4066677889101111121313141516171819
4166677889101111121313141516171819
426667789910111112131414151617181920
436667789910111112131414151617181920
44666778991011121213141415161718192021
45666778991011121213141515161718192021
4666677899101112121314151516171819202122
4766677899101112121314151516171819202122
4866677891010111213131415161617181920212223
4966677891010111213131415161617181920212223
506667889101011121313141516161718192021222324
516667889101011121313141516171718192021222324
52666788910101112131414151617171819202122232425
53666788910101112131414151617171819202122232425
5466778891011111213141415161718181920212223242526
5566778891011111213141415161718181920212223242526
566677889101111121314151516171818192021222324252627
576677889101111121314151516171819192021222324252627
58667788910111112131415151617181919202122232425262728
59667788910111112131415151617181919202122232425262728
6066778991011121213141516161718192020212223242526272829
6166778991011121213141516161718192020212223242526272829
626677899101112121314151616171819202021222324252627282930
636677899101112121314151616171819202121222324252627282930
64667789910111212131415161717181920212122232425262728293031
65667789910111212131415161717181920212122232425262728293031
6676778991011121313141516171718192021222223242526272829303132

Remark 18. Since for any a-perfect graph G the stable set polytope is given by non-negativity constraints and rank inequalities associated to induced cliques and antiwebs, we have as a direct consequence from the previous corollary

## 3. Conclusions

In this paper, we focus on the Chvátal rank of the linear relaxation ESTAB(G) of STAB(G) for several classes of graphs.

We first consider the rank of cliques and clique constraints, as they are required to describe the stable set polytope of any graph. We provide the exact value

for all n3. As a consequence, we further obtain the exact values of crE(G) whenever G is perfect or h-perfect, showing that crE(G) is of order .

Next, we consider the rank of odd antiholes and show that

holds with . As , this again provides an upper bound for crE (G) of order for odd antiholes and all m-perfect graphs G.

Finally, we turn to the general case of antiwebs . We characterize the t-perfect antiwebs and provide a procedure to determine an upper bound for , based on the generation of rank facets associated with antiwebs using prime subantiwebs only. It is open whether this procedure already determines the exact value of . This would be the case if no other inequalities being valid for than rank constraints associated with prime subantiwebs are required to generate the full rank constraint of .

In any case, we have an upper bound

with and α=k+1. We infer that crE(G) is at most of order for all antiwebs and, thus, for all a-perfect graphs. Figure 1 shows the inclusion relations of the classes of a-perfect graphs that were considered so far.

In addition, there are several further interesting graph classes known to be a-perfect. Wagler (2005) showed that complements of so-called fuzzy circular interval graphs are a-perfect.

Fuzzy circular interval graphs are defined as follows: given a collection of intervals from a circle C, these are graphs whose nodes correspond to points xi on C and xixj is an edge if there is an interval in containing xi and xj, where nodes corresponding to different endpoints of one interval may or may not be adjacent. Circular interval graphs are special cases without fuzziness (as also nodes corresponding to different endpoints of one interval are always adjacent).

A further subclass of fuzzy circular interval graphs are so-called concave-round graphs whose nodes can be circularly enumerated such that the closed neighborhood of each node forms an interval in the enumeration. Moreover, the class of concave-round graphs contains the class of proper circular-arc graphs.

Hence, a-perfect graphs constitute an interesting subclass of rank-perfect graphs G for which we improved the previously known bound on crE(G) of order O(|G|) from Chvátal et al. (1989) to an upper bound of order .