1. Critical graphs and facets of the
stable set polytope
László Lipták
The Fields Institute
University of Toronto
Toronto, Ontario, Canada
László Lovász
Microsoft Research
One Microsoft Way,
Redmond, WA

2. Independent (stable) sets
maximum size of an independent
set of nodes
Difficult (NP-hard)!

3. α-critical graphs
[no isolated nodes; connected]

4. Erdős - Gallai
Hajnal
: defect of G
Gallai
odd cycles
Hajnal
even subdivisions of K4
Andrásfai

5. criticality is preserved by even subdivision
d is preserved
enough to find graphs w/o

6. + = + = + = Composing α–critical graphs Gallai, Plummer, Wessel
basis graphs
Enough to find graphs with deg >2

7. Andrásfai
Surányi, L
there is a finite number of basis graphs L

8. The stable set polytopec a b c b G a

9. Facets of the stable set polytopeG b a b a c

10. (1) and (2) suffice iff G is bipartite
(nonnegativity constraints)
(edge constraints)
(1) and (2) suffice iff G is bipartite b b a c a

11.  (clique constraints) (1) and (3) suffice iff G is perfect
Fulkerson-Chvátal
(odd hole constraints)
(rank constraints)
G -critical 
rank constraint for U=V is a facet
View facets
as generalizations
of -critical graphs
Chvátal

12. nontrivial facet of STAB(G):
vertex of this facet: stable set
integrality gap of this facet:
defect of facet
non-trivial
L-Schrijver

13. critical facet of STAB(G): deleting any edge, does not
remain valid 1 2 2 1 1 2 1 1 a a a a
irreducible facet of STAB(G): no

14. In every critical facet aid
Structural significance of d
In every critical facet aid
Sewell
In every critical facet:
● deg(i)  ai+d  2 d
● for d>1, deg(i)  2 d+1
LL-LL

15. Finite basis theorem for facets
For d=2, all critical facets are
subdivisions of K4
Sewell 2 2 2 2 2 2 2 2 2 2
For every d, there is a finite number of
irreducible critical facets with defect d.
LL-LL

16. Deriving facets
trivial constraints
(edge constraints)

17. 1 5 2 4 3
(odd hole constraints)

18. 1 5 2 4 3
(clique constraints)

19. Every facet can be obtained by iterating this process at most n times
L-Schrijver
Even without
squaring
Linear objective functions
subject to all constraints derived in c iterations
can be optimized in polynomial time
L-Schrijver

20. rank of facet: smallest c
Which facets can be derived in c iterations?
rank of facet: smallest c
r: rank of
d : defect of
L-Schrijver

21. The average degree of a critical facet
odd hole contraints ?
The average degree of a critical facet
with rank r is <2r.
Lipták