Local complementation and interlacement graphs

AbstractLet M be a binary matroid on a set E. We show that by performing a sequence of local complementations at ei (i=1,…,n) on the principal interlacement graph of M, for any ordering of E = {e1,…,en}, we obtain a bipartite graph whose two sets of vertices define a partition of E into a base B and a cobase B⊥ of the matroid M. We then give a characterisation of bipartite chordable graphs and, as an application, we give a short proof of Pierre Rosenstiehl's characterization of planar graphs: a graph with a trivial bicycle space is planar if and only if its principal interlacement graph is chordable

Barycentric systems and stretchability

AbstractUsing a general resolution of barycentric systems we give a generalization of Tutte's theorem on convex drawing of planar graphs. We deduce a characterization of the edge coverings into pairwise non-crossing paths which are stretchable: such a system is stretchable if and only if each subsystem of at least two paths has at least three free vertices (vertices of the outer face of the induced subgraph which are internal to none of the paths of the subsystem). We also deduce that a contact system of pseudo-segments is stretchable if and only if it is extendible

Tr\'{e}maux trees and planarity

We present a simplified version of the DFS-based Left-Right planarity testing and embedding algorithm implemented in Pigale which has been considered as the fastest implemented one [J.M. Boyer, P.F. Cortese, M. Patrignani, and G. Di Battista. Stop minding your P's and Q's: implementing fast and simple DFS-based planarity and embedding algorithm. In Graph Drawing, volume 2912 of Lecture Notes in Computer Science, pages 25-36. Springer, 2004.]. We give here a simple full justification of the algorithm, based on a preliminary extended study of topological properties of DFS trees.Comment: Special Issue on Graph Drawin

A Short Proof of a Gauss Problem

A traversal of a self crossing closed plane curve, with points of multiplicity at most two, defines a double occurrence sequence. C.F. Gauss conjectured [2] that such sequences could be characterized by their interlacement properties. This conjecture was proved by P. Rosenstiehl in 1976 [15]. We shall give here a simple self-contained proof of his characterization. This new proof relies on the D-switch operation

On a Characterization of Gauss Codes

The traversal of a self crossing closed plane curve, with points of multiplicity at most two, defines a double occurrence sequence, the Gauss code of the curve. Using the D-switch operation, we give a new simple characterization of these sequences and deduce a simple self-contained proof of Rosenstiehl's characterization

Stretching of Jordan arc contact systems

Using a general resolution of barycentric systems we give a generalization of Tutte's theorem on convex drawing of planar graphs. We deduce a characterization of the edge coverings into pairwise non-crossing paths which are stretchable: such a system is stretchable if and only if each subsystem of at least two paths has at least 3 free vertices (vertices of the outer face of the induced subgraph which are internal to none of the paths of the subsystem). We also deduce that a contact system of pseudo-segments is stretchable if and only if it is extendible

Stretching of Jordan Arc Contact Systems

We prove that a contact system of Jordan arcs is stretchable if and only if it is extendable into a weak arrangement of pseudo-lines