174 research outputs found
The PC-Tree algorithm, Kuratowski subdivisions, and the torus.
The PC-Tree algorithm of Shih and Hsu (1999) is a practical linear-time planarity algorithm that provides a plane embedding of the given graph if it is planar and a Kuratowski subdivision otherwise. Remarkably, there is no known linear-time algorithm for embedding graphs on the torus. We extend the PC-Tree algorithm to a practical, linear-time toroidality test for K3;3-free graphs called the PCK-Tree algorithm. We also prove that it is NP-complete to decide whether the edges of a graph can be covered with two Kuratowski subdivisions. This greatly reduces the possibility of a polynomial-time toroidality testing algorithm based solely on edge-coverings by subdivisions of Kuratowski subgraphs
Minor-Obstructions for Apex-Pseudoforests
A graph is called a pseudoforest if none of its connected components contains
more than one cycle. A graph is an apex-pseudoforest if it can become a
pseudoforest by removing one of its vertices. We identify 33 graphs that form
the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of
all minor-minimal graphs that are not apex-pseudoforests
Moduli spaces of rational weighted stable curves and tropical geometry
We study moduli spaces of rational weighted stable tropical curves, and their
connections with the classical Hassett spaces. Given a vector w of weights, the
moduli space of tropical w-stable curves can be given the structure of a
balanced fan if and only if w has only heavy and light entries. In this case,
we can express the moduli space as the Bergman fan of a graphic matroid.
Furthermore, we realize the tropical moduli space as a geometric
tropicalization, and as a Berkovich skeleton, of the classical moduli space.
This builds on previous work of Tevelev, Gibney--Maclagan, and
Abramovich--Caporaso--Payne. Finally, we construct the moduli spaces of
heavy/light weighted tropical curves as fiber products of unweighted spaces,
and explore parallels with the algebraic world.Comment: 26 pages, 8 TikZ figures. v3: Minor changes and corrections. Final
version to appear in Forum of Mathematics, Sigm
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
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