ResumoNós caracterizamos os grafos cliquc de grafos cordais e de caminhos. É apresentada ainda uma classe de grafos chamada árvores expandidas.Elas formam uma subclasse dos grafos disk-l1elly. l\'[ostra-se que o grafo clique de todo grafo cordal (portanto dos de caminho) é uma árvore expandida. Mais ainda, que toda árvore expandida é o grafo clique de algum grafo de caminho (portanto cordal). Diferentes caracterizações de árvores expandidas são descritas, conduzindo a um algoritmo de tempo polinornial para o reconhecimento de grafos nesta classe. AbstractWe characterize clique graphs of chordal and path graphs. A special class of graphs called expanded trees is introduced. They form a subclass of disk-Helly graphs. It is shown that the clique graph of every chordal (hence path) graph is an expanded tree. In addition, every expanded trce is the clique graph of some path (hence chordal) graph. Different characterizations of expanded trees are described, leading to a polynomial time algorithm for rccognizing clique graphs of chordal and path graphs.. 0-

We introduce ow metrics as a relaxation of path metrics (i.e. linear orderings). They are deÿned by polynomial-sized linear programs and have interesting properties including spreading. We use them to obtain relaxations for several NP-hard linear ordering problems such as minimum linear arrangement and minimum pathwidth. Our approach has the advantage of achieving the best-known approximation guarantees for these problems using the same relaxation and essentially the same rounding algorithm for all the problems while varying only the objective function from problem to problem. This is in contrast to the current state of the literature where each problem either has a new relaxation or a new rounding or both. We also characterize a natural projection of the ow polyhedron.

A graph G is convergent when there is some finite integer n > 0, such that the n-th iterated clique graph K"(G) has only one vertex. The smallest such n is the index of G. The Helly defect of a convergent graph is the smallest n such that K"(G) is clique Helly, that is, its maximal cliques satisfy the Helly property. Bandelt and Prisner proved that the Helly defect of a chordal graph is at most one and asked whether there is a graph whose Helly defect exceeds the difference of its index and diameter by more than one. In the present paper an affirmative answer to the question is given. For any arbitrary finite integer n, a graph is exhibited, the Helly defect of which exceeds by n the difference of its index and diameter.

No abstract

A graph G has p-intersection number at most d if it is possible to assign to every vertex u of G, a subset S(u) of some ground set U with |U | = d in such a way that distinct vertices u and v of G are adjacent in G if and only if |S(u) ∩ S(v)| ≥ p. We show that every minimal forbidden induced subgraph for the hereditary class G(d, p) of graphs whose p-intersection number is at most d, has order at most 3 · 2 d+1 + 1, and that the exponential dependence on d in this upper bound is necessary. For p ∈ {d − 1, d − 2}, we provide more explicit results characterizing the graphs in G(d, p) without isolated/universal vertices using forbidden induced subgraphs.