Discrete Mathematics 252 (2002) 123–140
www.elsevier.com/locate/disc
The stable set polytope and some operations on graphs
JeanFonlupt
a
, Ahmed Hadjar
b; ∗
a
Equipe Combinatoire, Universit
Ã
e Pierre et Marie Curie, case 189, 4 place Jussieu,
75252 Paris Cedex 05 France
b
GERAD—
Ã
Ecole des Hautes
Ã
Etudes Commerciales, 3000 chemin de la C
ˆ
ote-Sainte-Catherine,
Montr
Ã
eal, Que., Canada H3T 2A7
Received 1 June 1998; revised 20 March 2001; accepted 2 April 2001
Abstract
We study some operations ongraphs inrelationto the stable set polytope, for instance,
identiÿcation of two nodes, linking a pair of nodes by an edge and composition of graphs by
subgraph identiÿcation. We show that, with appropriate conditions, the descriptions of the stable
set polytopes associated with the resulting graphs can be derived from those related to the initial
graphs by adding eventual clique inequalities. Thus, perfection and h-perfectionof graphs are
preserved.
c
2002 Elsevier Science B.V. All rights reserved.
Keywords: Stable set polytope; Perfect graphs; h-Perfect graphs; Compositionof graphs;
Compositionof polyhedra
1. Introduction
Let G =(V; E) be a simple undirected graph. A stable set S ⊆ V is a set of pairwise
non-adjacent nodes in G. The incidence vector of a stable set S is a vector x ∈ R
V
such that x(v)=1 if v ∈ S and x(v)=0 if v ∈ S. The stable set polytope STB(G)is
the convex hull of incidence vectors of all stable sets of G. The stable set problem,
knownto be NP-hard, canbe formulated as max{cx : x ∈ STB(G)}, where c ∈ R
V
is
the cost vector. The polytope STB(G) is full-dimensional, thus there is a unique (up
to multiplication by a positive constant) non-redundant inequality system describing
STB(G).
The usual inequalities that are valid for STB(G) and that will be considered later
are the non-negativity constraints
x(v) ¿ 0;v∈ V (1)
∗
Corresponding author.
E-mail address: hadjar@crt.umontreal.ca (A. Hadjar).
0012-365X/02/$ - see front matter
c
2002 Elsevier Science B.V. All rights reserved.
PII: S0012-365X(01)00273-4