Rhombic embeddings of planar quad-graphs

Rhombic Embeddings of Planar Quad-graphs

Given a finite or infinite planar graph all of whose faces have degree 4, we study embeddings in the plane in which all edges have length 1, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such embeddings. Résumé. Etant donné un graphe planaire, fini ou infini, dont toutes ...

Chordal embeddings of planar graphs

Robertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual differ by at most one. Lapoire solved the conjecture in the affirmative, using algebraic techniques. We give here a much shorter proof of this result.

Geodesic Embeddings and Planar Graphs

Geodesi Embeddings and Planar Graphs

Maintaining alterable planar embeddings of dynamic graphs

We present a data structure for maintaining a planar embedding of a dynamic plane graph. The graph may be updated with edge deletions and insertions of edges when it does not violate the embedding. Queries of vertex pairs return the list of possible places for inserting an edge. Furthermore, the embedding may be updated with flips that alter the embedding by swapping the orientation of a subgra...