Pierre Duchet

Recent problems and results about kernels in directed graphs

Abstract In Section 1, we survey the existence theorems for a kernel; in Section 2, we discuss a new conjecture which could constitute a bridge between the kernel problems and the perfect graph conjecture. In fact, we believe that a graph is ‘quasi-perfect’ if and only if it is perfect.

Graphs whose neighborhoods have no special cycles

To a graph G is canonically associated its neighborhood-hypergraph, N(G), formed by the closed neighborhoods of the vertices of G. We characterize the graphs G such that (i) N(G) has no induced cycle, or (ii) N(G) is a balanced hypergraph or (iii) N(G) is triangle free. (i) is another short proof of a result by Farber; (ii) answers a problem asked by C. Berge. The case of strict neighborhoods is also solved.

A sufficient condition for a digraph to be kernel-perfect

Galeana-Sanchez and Neumann-Lara proved that a sufficient condition for a digraph to have a kernel (i.e., an absorbent independent set) is the following: (P) every odd directed cycle possesses at least two directed chords whose terminal endpoints are consecutive on the cycle. Here it is proved that (P) is satisfied by those digraphs having these two properties: (i) the reversal of every 3-circuit is present, and (ii) every odd directed cycle v1… v2n+1V1 has two chords of the form (vi, vi+2). This is stronger than a result of Galeana-Sanchez.

Kernels in directed graphs: a poison game

Abstract We consider a two player game on a progressively and locally finite directed graph and we prove that the first player wins if and only if the graph has a local kernel. The result is sharp. From it, we derive a short proof of a general version of the Galeana-Sanchez & Neuman-Lara Theorem that give a sufficient condition for a digraph to be kernel-perfect.

Some operations preserving the existence of kernels

Abstract We consider some classical constructions of graphs: join of two graphs, duplication of a vertex, and show that they behave nicely from the point of view of the existence of kernels.

On kernels in perfect graphs

A kernel of a digraphD is a set of vertices which is both independent and absorbant. In 1983, C. Berge and P. Duchet conjectured that an undirected graphG is perfect if and only if the following condition is fulfilled: ifD is an orientation ofG (where pairs of opposite arcs are allowed) and if every clique ofD has a kernel thenD has a kernel. We prove here the conjecture for the complements of strongly perfect graphs and establish that a minimal counterexample to the conjecture is not a complete join of an independent set with another graph.

On sign-invariance graphs of uniform oriented matroids

Two elements of an oriented matroid constitute an invariant pair if all signed circuits containing them have the same sign (resp. different signs). The invariance graph of an oriented matroid M(E) is the graph with vertex set E and where edges are the invariant pairs. We prove that invariance graphs of uniform oriented matroids have maximum degree at most 2 (except in trivial cases) and that the alternating matroid is determined, up to reorientation, by its invariance graph. Representable uniform oriented matroids with empty invariance graphs are constructed.

More characterizations of triangulated graphs

New characterizations of triangulated and cotriangulated graphs are pre- sented. Cotriangulated graphs form a natural subclass of the class of strongly perfect graphs, and they are also characterized in terms of the shellability of some associated collection of sets. Finally, the notion of stability function of a graph is introduced, and it is proved that a graph is triangulated if and only if the polynomial representing its stability func- tion has all its coefficients equal to 0, +1 or -1.

On the orientation of Meyniel graphs

A kernel of a directed graph is a set of vertices K that is both absorbant and independent (i.e., every vertex not in K is the origin of an arc whose extremity is in K, and no arc of the graph has both endpoints in K). In 1983, Meyniel conjectured that any perfect graph, directed in such a way that every circuit of length three uses two reversible arcs, must have a kernel. This conjecture was proved for parity graphs. In this paper, we extend that result and prove that Meyniels conjecture holds for all graphs in which every odd cycle has two chords.

On quasi-kernels of a graphs

Abstract We define three classes of quasi-kernels for a directed graph. As a consequence, we show the existence of quasi-kernels in every progressively finite graph and in every locally finite graph, generalizing the result of Chvatal and Lovasz which deals with the finite case. Our method shows that the problem of finding a quasi-kernel in a finite digraph and the problem of finding the unique kernel of an acircuitic finite digraph have the same algorithmic complexity.We define three classes of quasi-kernels for a directed graph. As a consequence, we show the existence of quasi-kernels in every progressively finite graph and in every locally finite graph, generalizing the result of Chv~ital and Lov~tsz which deals with the finite case. Our method shows that the problem of finding a quasi-kernel in a finite digraph and the problem of finding the unique kernel of an acircuitic finite digraph have the same algorithmic complexity.