1,099 research outputs found

    Vertex elimination orderings for hereditary graph classes

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    We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vu\vskovi\'c. We use this method to prove the existence of elimination orderings in several classes of graphs, and to compute them in linear time. Some of the classes have already been studied, namely even-hole-free graphs, square-theta-free Berge graphs, universally signable graphs and wheel-free graphs. Some other classes are new. It turns out that all the classes that we study in this paper can be defined by excluding some of the so-called Truemper configurations. For several classes of graphs, we obtain directly bounds on the chromatic number, or fast algorithms for the maximum clique problem or the coloring problem

    Wada Dessins associated with Finite Projective Spaces and Frobenius Compatibility

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    \textit{Dessins d'enfants} (hypermaps) are useful to describe algebraic properties of the Riemann surfaces they are embedded in. In general, it is not easy to describe algebraic properties of the surface of the embedding starting from the combinatorial properties of an embedded dessin. However, this task becomes easier if the dessin has a large automorphism group. In this paper we consider a special type of dessins, so-called \textit{Wada dessins}. Their underlying graph illustrates the incidence structure of finite projective spaces \PR{m}{n}. Usually, the automorphism group of these dessins is a cyclic \textit{Singer group} Σℓ\Sigma_\ell permuting transitively the vertices. However, in some cases, a second group of automorphisms Φf\Phi_f exists. It is a cyclic group generated by the \textit{Frobenius automorphism}. We show under what conditions Φf\Phi_f is a group of automorphisms acting freely on the edges of the considered dessins.Comment: 23 page
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