Irena Rusu

Weighted parameters in ( P 5 , P 5 )-free graphs

Abstract We use the modular decomposition to give O(n(m + n) algorithms for finding a maximum weighted clique (respectively stable set) and an approximate weighted colouring (respectively partition into cliques) in a (P 5 , P 5 )- free graph. As a by-product, we obtain an O(m + n) algorithm for finding a minimum weighted transversal of the C5 in a (P 5 , P 5 )- free graph.

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.

Dirac-type characterizations of graphs without long chordless cycles

We call a chordless path υ1υ2...υi simplicial if it does not extend into any chordless path υ0υ1υ2.., υiυi+1. Trivially, for every positive integer k, a graph contains no chordless cycle of length k + 3 or more if each of its nonempty induced subgraphs contains a simplicial path with at most k vertices; we prove the converse. The case of k = 1 is a classic result of Dirac.

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.

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.

Berge graphs with chordless cycles of bounded length

The problem is considered under which conditions a 4-connected planar or projective planar graph has a Hamiltonian cycle containing certain prescribed edges and missing certain forbidden edges. The results are applied to obtain novel lower bounds on the number of distinct Hamiltonian cycles that must be present in a 5-connected graph that is embedded into the plane or into the projective plane with face-width at least five. Especially, we show that every 5-connected plane or projective plane triangulation on n vertices with no non-contractible cyles of length less than five contains at least

Domination graphs: examples and counterexamples

2^{{\cal O}(n^{1/4})}

Recognizing i -triangulated graphs in O9 mn ) time

distinct Hamiltonian cycles.