1,099 research outputs found
Vertex elimination orderings for hereditary graph classes
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
\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} permuting transitively the
vertices. However, in some cases, a second group of automorphisms
exists. It is a cyclic group generated by the \textit{Frobenius automorphism}.
We show under what conditions is a group of automorphisms acting
freely on the edges of the considered dessins.Comment: 23 page
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