Jean Fonlupt

Chromatic characterization of biclique covers

A biclique B of a simple graph G is the edge-set of a complete bipartite subgraph of G. A biclique cover of G is a collection of bicliques covering the edge-set of G. Given a graph G, we will study the following problem: find the minimum number of bicliques which cover the edge-set of G. This problem will be called the minimum biclique cover problem (MBC). First, we will define the families of independent and dependent sets of the edge-set E(G) of G: [emailxa0protected]?E(G) will be called independent if there exists a biclique [emailxa0protected]?E(G) such that [emailxa0protected]?B, and will be called dependent otherwise. From our study of minimal dependent sets we will derive a 0-1 linear programming formulation of the following problem: find the maximum weighted biclique in a graph. This formulation may have an exponential number of constraints with respect to the number of nodes of G but we will prove that the continuous relaxation of this integer program can be solved in polynomial time. Finally we will also study continuous relaxation methods for the problem (MBC). This research was motivated by an open problem of Fishburn and Hammer.

Compositions of Graphs and Polyhedra IV: Acyclic Spanning Subgraphs

Given a directed graph

Critical extreme points of the 2-edge connected spanning subgraph polytope

D

Recognizing Dart-Free Perfect Graphs

that has a two-vertex cut, this paper describes a technique to derive a linear system that defines the acyclic subgraph polytope of

Strongly Polynomial Algorithm for the Intersection of a Line with a Polymatroid

D

Critical Extreme Points of the 2-Edge Connected Spanning Subgraph Polytope

from systems related to the pieces. It also gives a technique to describe facets of this polytope by composition of facets for the pieces. The authors prove that, if the systems for the pieces are totally dual integral (TDI), then the system for

The stable set polytope and some operations on graphs

D

On the Clique-Rank and the Coloration of Perfect Graphs

is also. The authors prove that the cycle inequalities form a TDI system for any orientation of

Optimisation combinatoire : théorie et algorithmes

K_5

Recognizing dart-free perfect graphs

. These results are combined with Lucchesi- Younger theorem and a theorem of Wagner to prove that, for graphs with no