Patrice Ossona de Mendez

Tree-depth, subgraph coloring and homomorphism bounds

We define the notions tree-depth and upper chromatic number of a graph and show their relevance to local-global problems for graph partitions. In particular we show that the upper chromatic number coincides with the maximal function which can be locally demanded in a bounded coloring of any proper minor closed class of graphs. The rich interplay of these notions is applied to a solution of bounds of proper minor closed classes satisfying local conditions. In particular, we prove the following result:For every graph M and a finite set F of connected graphs there exists a (universal) graph U = U(M, F) ∈ Forbh(F) such that any graph G ∈ Forbh(F) which does not have M as a minor satisfies G → U (i.e. is homomorphic to U).This solves the main open problem of restricted dualities for minor closed classes and as an application it yields the bounded chromatic number of exact odd powers of any graph in an arbitrary proper minor closed class. We also generalize the decomposition theorem of DeVos et al. [M. DeVos, G. Ding, B. Oporowski, D.P. Sanders, B. Reed, P. Seymour, D. Vertigan, Excluding any graph as a minor allows a low tree-width 2-coloring, J. Combin. Theory Ser. B 91 (2004) 25-41].

On Triangle Contact Graphs

It is proved that any plane graph may be represented by a triangle contact system, that is a collection of triangular disks which are disjoint except at contact points, each contact point being a node of exactly one triangle. Representations using contacts of T-or Y-shaped objects follow. Moreover, there is a one-to-one mapping between all the triangular contact representations of a maximal plane graph and all its partitions into three Schnyder trees.

Colorings and Homomorphisms of Minor Closed Classes

We relate acyclic (and star) chromatic number of a graph to the chromatic number of its minors and as a consequence we show that the set of all triangle free planar graphs is homomorphism bounded by a triangle free graph. This solves a problem posed in [[15]]. It also improves the best known bound for the star chromatic number of planar graphs from 80 to 30. Our method generalizes to all minor closed classes and puts Hadwiger conjecture in yet another context.

Bipolar orientations revisited

Abstract Acyclic orientations with exactly one source and one sink — the so-called bipolar orientations-arise in many graph algorithms and specially in graph drawing. The fundamental properties of these orientations are explored in terms of circuits, cocircuits and also in terms of “angles” in the planar case. Classical results get here new simple proofs; new results concern the extension of partial orientations, exhaustive enumerations, the existence of deletable and contractable edges, and continuous transitions between bipolar orientations.

On nowhere dense graphs

In this paper, we define and analyze the nowhere dense classes of graphs. This notion is a common generalization of proper minor closed classes, classes of graphs with bounded degree, locally planar graphs, classes with bounded expansion, to name just a few classes which are studied extensively in combinatorial and computer science contexts. In this paper, we show that this concept leads to a classification of general classes of graphs and to the dichotomy between nowhere dense and somewhere dense classes. This classification is surprisingly stable as it can be expressed in terms of the most commonly used basic combinatorial parameters, such as the independence number @a, the clique number @w, and the chromatic number @g. The remarkable stability of this notion and its robustness has a number of applications to mathematical logic, complexity of algorithms, and combinatorics. We also express the nowhere dense versus somewhere dense dichotomy in terms of edge densities as a trichotomy theorem.

A left-first search algorithm for planar graphs

We give anO(|V(G)|)-time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graphG so that (i) no two segments have an interior point in common, and (ii) two segments touch each other if and only if the corresponding vertices are adjacent. As a corollary, we obtain a strengthening of the following theorem of Ringel and Petrovič. The edges of any maximal bipartite plane graphG with outer facebwb′w′ can be colored by two colors such that the color classes form spanning trees ofG−b andG−b′, respectively. Furthermore, such a coloring can be found in linear time. Our method is based on a new linear-time algorithm for constructing bipolar orientations of 2-connected plane graphs.

A Formal Model for Topic Maps

Topic maps have been developed in order to represent the structures of relationships between subjects, independently of resources documenting them, and to allow standard representation and interoperability of such structures. The ISO 13250 XTMsp ecification [2] have provided a robust syntactic XML representation allowing processing and interchange of topic maps. But topic maps have so far suffered from a lack of formal description, or conceptual model. We propose here such a model, based on the mathematical notions of hypergraph and connexity. This model addresses the critical issue of topic map organization in semantic layers, and provides ways to check semantic consistency of topic maps. Moreover, it seems generic enough to be used as a foundation for other semantic standards, like RDF [3].

Structural Properties of Sparse Graphs

In this chapter we briefly outline the main motivation of our work and we relate it to other research. We do not include any definition here.

TRÉMAUX TREES AND PLANARITY

We present a simplified version of the DFS-based Left-Right planarity testing and embedding algorithm implemented in Pigale [1, 2], which has been considered as the fastest implemented one [3]. We give here a full justification of the algorithm, based on a topological properties of Tremaux trees.

Representations by Contact and Intersection of Segments

A necessary and sufficient condition is given for a connected bipartite graph to be the incidence graph of a contact family of segments and points. We deduce that any four-connected three-colorable plane graph is the contact graph of a family of segments and that any four-colored planar graph without an induced C4 using four colors is the intersection graph of a family of straight line segments.