Jean-Marie Vanherpe

BIPARTITE GRAPHS TOTALLY DECOMPOSABLE BY CANONICAL DECOMPOSITION

We describe here a technique of decomposition of bipartite graphs which seems to be as interesting within this context as the well known modular and split techniques for the decomposition of general graphs. In particular, we characterize by forbidden subgraphs the family of bipartite graphs which are totally decomposable (i.e. reducible to single vertices) with respect to our decomposition. This family contains previously known families of graphs such as bicographs and P6-free bipartite graphs. As an application we provide polynomial solutions of optimization problems, some of them being NP-complete for general bipartite graphs.

On extended P 4 -reducible and extended P 4 -sparse graphs

Abstract A graph G was defined in [16] as P 4 -reducible , if no vertex in G belongs to more than one chordless path on four vertices or P 4 . A graph G is defined in [15] as P 4 -sparse if no set of five vertices induces more than one P 4 , in G . P 4 -sparse graphs generalize both P 4 -reducible and the well known class of p 4 -free graphs or cographs . In an extended abstract in [11] the first author introduced a method using the modular decomposition tree of a graph as the framework for the resolution of algorithmic problems. This method was applied to the study of P 4 -sparse and extended P 4 -sparse graphs. In this paper, we begin by presenting the complete information about the method used in [11]. We propose a unique tree representation of P 4 -sparse and a unique tree representation of P 4 -reducible graphs leading to a simple linear recognition algorithm for both classes of graphs. In this way we simplify and unify the solutions for these problems, presented in [16–19]. The tree representation of an n -vertex P 4 -sparse or a P 4 -reducible graph is the key for obtaining O ( n ) time algorithms for the weighted version of classical optimization problems solved in [20]. These problems are NP-complete on general graphs. Finally, by relaxing the restriction concerning the exclusion of the C 5 cycles from P 4 -sparse and P 4 -reducible graphs, we introduce the class of the extended P 4-sparse and the class of the extended P 4 -reducible graphs. We then show that a minimal amount of additional work suffices for extending most of our algorithms to these new classes of graphs.

LINEAR TIME RECOGNITION AND OPTIMIZATIONS FOR WEAK-BISPLIT GRAPHS, BI-COGRAPHS AND BIPARTITE P6-FREE GRAPHS

In [7] was introduced a new decomposition scheme for bipartite graphs that was called canonical decomposition. Weak-bisplit graphs are totally decomposable following this decomposition. We give here linear time algorithms for the recognition of weak-bisplit graphs as well as for two subclasses of this class, the P6-free bipartite graphs and the bi-cographs. Our algorithms extends the technics developped in [2] for cographss recognition. We conclude by presenting efficient solutions for some optimization problems when dealing with weak-bisplit graphs.

On Fulkerson conjecture

If

Bimodular Decomposition of Bipartite Graphs

G

On normal partitions in cubic graphs

is a bridgeless cubic graph, Fulkerson conjectured that we can find

On Compatible Normal Odd Partitions in Cubic Graphs

6

Linear time recognition of Weak bisplit graphs

perfect matchings (a {\em Fulkerson covering}) with the property that every edge of

On canonical decomposition of bipartite graphs

G

Mácajová and Škoviera conjecture on cubic graphs

is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has