Florian Roussel

About Skew Partitions in Minimal Imperfect Graphs

V. Chvatal conjectured in 1985 that a minimal imperfect graph G cannot have a skew cutset (i.e., a cutset S decomposable into disjoint sets A and B joined by all possible edges). We prove here the conjecture in the particular case where at least one of A and B is a stable set.

HOLES AND DOMINOES IN MEYNIEL GRAPHS

Many practical problems (frequency assignement, register allocation, timetables) may be formulated as graph (vertex-)coloring problems, but finding solutions for them is difficult as long as they are treated in the most general case (where the graph is arbitrary), since vertex coloring has been proved to be NP-complete. The problem becomes much easier to solve if the graph resulting from the modelisation of the practical application belongs to some particular class of graphs, for which solutions to the problem are known. Meyniel graphs form such a class (a fast coloring algorithm can be found in [9]), for which an efficient recognizing algorithm is therefore needed. A graph G=(V,E) is said to be a Meyniel graph if every odd cycle of G on at least five vertices contains at least two chords. Meyniel graphs generalize both i-triangulated and parity graphs, two well known classes of perfect graphs that will be present in our paper in Section 7. In [2], Burlet and Fonlupt propose a characterization of Meyniel graphs which relies on the following property: the class of Meyniel graphs may be obtained from some basic Meyniel graphs using a binary operation called amalgam. Besides the theoretical interest of this result, a practical interest arises because of the polynomial recognition algorithm which can be obtained. Unfortunately, it is quite expensive to verify if a given graph is the amalgam of two graphs (therefore the complexity of the whole algorithm is in O(n7)), and this supports the idea that a new point of view is needed to find a more efficient algorithm. Our approach of Meyniel graphs will be directed through the search of a general structure. Intuitively, a Meyniel graph either will be simple (i.e. with no hole or domino), or will have a skeleton around which the rest of the graph will be regularly organized. As suggested, the first type of Meyniel graphs is simple to identify. For the second type, a deeper analysis is necessary; it yields a characterization theorem, which is used to deduce the O(m2+mn) recognition algorithm.

An O( n 2 ) algorithm to color Meyneil graphs

Abstract Meyniel graphs are the graphs in which every odd cycle with five vertices or more has at least two chords. In 1990, Hertz gave an O(mn) algorithm to color Meyniel graphs based on successive contractions of even pairs. We give here another algorithm which consists in simultaneously ordering (in a Lex-BFS way) and coloring (with the greedy algorithm) the vertices of the graph and we show that it needs only O(n2) operations.

ON GRAPHS WITH LIMITED NUMBER OF P4-PARTNERS

The study of graphs containing few P4s generated an important number of results related to perfection, recognition, optimization problems (see [12], [15], [8]). We define here a new, larger class of graphs and show that the indicated problems may be efficiently solved on this class too (thus generalizing some of the previous results). Namely, we give a linear time recognition algorithm for this class and we note that the optimization problems concerning the clique number, stability number, chromatic number and clique cover number are solvable in linear time.

A linear algorithm to color i -triangulated graphs

We show that i-triangulated graphs can be colored in linear time by applying lexicographic breadth-first search (abbreviated LexBFS) and the greedy coloring algorithm.

Scattering number and modular decomposition

Abstract The scattering number of a graph G equals max { c ( G ⧹ S ) − | S | S is a cutset of G } where c ( G ⧹ S ) denotes the number of connected components in G ⧹ S . Jung (1978) has given for any graph having no induced path on four vertices ( P 4 -free graph) a correspondence between the value of its scattering number and the existence of Hamiltonian paths or Hamiltonian cycles. Hochstattler and Tinhofer (to appear) studied the Hamiltonicity of P 4 -sparse graphs introduced by Hoang (1985). In this paper, using modular decomposition, we show that the results of Jung and Hochstattler and Tinhofer can be generalized to a subclass of the family of semi- P 4 -sparse graphs introduced in Fouquet and Giakoumakis (to appear).

Triangulated Neighbourhoods in C4-Free Berge Graphs

We call a T-vertex of a graph G = (V, E) a vertex z whose neighbourhood N(z) in G induces a triangulated graph, and we show that every C4-free Berge graph either is a clique or contains at least two non-adjacent T-vertices. An easy consequence of this result is that every C4-free Berge graph admits a T-elimination scheme, i.e. an ordering [v1, v2, . . . , vn] of its vertices such that vi is a T-vertex in the subgraph induced by vi, . . . , vn in G.

Recognizing i -triangulated graphs in O9 mn ) time

We use breadth-first search and contractions of subgraphs to give a new and faster algorithm to recognize i-triangulated graphs.

Loose vertices in C4-free Berge graphs

Abstract Following Maire (Graphs Combin. 10 (3) (1994) 263) we call a loose vertex a vertex whose neighbourhood induces a P 4 -free graph, and we show that every C 4 -free Berge graph G which is not a clique either is breakable (i.e. G or G has a star-cutset) or contains at least two non-adjacent loose vertices. Consequently, every minimal imperfect C 4 -free graph has loose vertices.

New classes of perfectly orderable graphs

This paper generalizes previous works on perfectly orderable graphs by Olariu (Discrete Math. 113 (1992) 143) and by Hoang et al. (Discrete Math. 102 (1992) 67). Chvatal defined a graph to be perfectly orderable (V. Chvatal, in: C. Berge, V. Chvatal (Eds.), Topics on Perfect Graphs, Annals of Discrete Mathematics, Vol. 21, North-Holland, Amsterdam, 1984, pp. 63–65) if there exists a linear order < on its set of vertices such that no induced path abcd with edges ab,bc,cd has both a<b and d<c. Given a graph G and a vertex v in G such that G−v is perfectly orderable, we set some conditions on v for which we deduce that G is perfectly orderable. Our method allows to construct a new class of such graphs, recognizable in polynomial time, containing quasi-brittle graphs, charming graphs and some other classes of perfectly orderable graphs.