We prove a decomposition theorem for graphs that do not contain a subdivision of K4 as an induced subgraph where K4 is the complete graph on four vertices. We obtain also a structure theorem for the… (More)

We consider the class A of graphs that contain no odd hole, no antihole of length at least 5, and no “prism” (a graph consisting of two disjoint triangles with three disjoint paths between them) and… (More)

Observation 2. Let F be a clique of a graph G, and let B* be the union of some connected components of G F. Then any two-pair {x, y} of G B* is a two-pair of G. We prove the Theorem by induction on… (More)

An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation of an s-graph B is any graph obtained by subdividing subdivisible edges of B into paths of arbitrary… (More)

A b-coloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The b-chromatic number of a graph G,… (More)

We consider the class A of graphs that contain no odd hole, no antihole, and no “prism” (a graph consisting of two disjoint triangles with three disjoint paths between them). We prove that every… (More)

A kernel of a directed graph D is a set of vertices which is both independent and absorbant. In 1983, Berge and Duchet conjectured that an undirected graph G is perfect if and only if the following… (More)