Ali Ridha Mahjoub

On the cut polytope

The cut polytopePC(G) of a graphG=(V, E) is the convex hull of the incidence vectors of all edge sets of cuts ofG. We show some classes of facet-defining inequalities ofPC(G). We describe three methods with which new facet-defining inequalities ofPC(G) can be constructed from known ones. In particular, we show that inequalities associated with chordless cycles define facets of this polytope; moreover, for these inequalities a polynomial algorithm to solve the separation problem is presented. We characterize the facet defining inequalities ofPC(G) ifG is not contractible toK5. We give a simple characterization of adjacency inPC(G) and prove that for complete graphs this polytope has diameter one and thatPC(G) has the Hirsch property. A relationship betweenPC(G) and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied.

Facets of the Bipartite Subgraph Polytope

The bipartite subgraph polytope PBG of a graph G = [V, E] is the convex hull of the incidence vectors of all edge sets of bipartite subgraphs of G. We show that all complete subgraphs of G of odd order and all so-called odd bicycle wheels contained in G induce facets of PBG. Moreover, we describe several methods with which new facet defining inequalities of PBG can be constructed from known ones. Examples of these methods are contraction of node sets in odd complete subgraphs, odd subdivision of edges, certain splittings of nodes, and subdivision of all edges of a cut. Using these methods we can construct facet defining inequalities of PBG having coefficients of order |V|2.

Two-edge connected spanning subgraphs and polyhedra

This paper studies the problem of finding a two-edge connected spanning subgraph of minimum weight. This problem is closely related to the widely studied traveling salesman problem and has applications to the design of reliable communication and transportation networks. We discuss the polytope associated with the solutions to this problem. We show that when the graph is series-parallel, the polytope is completely described by the trivial constraints and the so-called cut constraints. We also give some classes of facet defining inequalities of this polytope when the graph is general.

Compositions of Graphs and Polyhedra II: Stable Sets

A graph

On two-connected subgraph polytopes

G

Design of survivable IP-over-optical networks

with a two-node cutset decomposes into two pieces. A technique to describe the stable set polytope for

Two Edge-Disjoint Hop-Constrained Paths and Polyhedra

G

k-edge connected polyhedra on series-parallel graphs

based on stable set polytopes associated with the pieces is studied. This gives a way to characterize this polytope for classes of graphs that can be recursively decomposed. This also gives a procedure to describe new facets of this polytope. A compact system for the stable set problem in series-parallel graphs is derived. This technique is also applied to characterize facet-defining inequalities for graphs with no

On the stable set polytope of a series-parallel graph

K_{5}\e

Separation of Partition Inequalities

minor. The stable set problem is polynomially solvable for this class of graphs. Compositions of