3,442 research outputs found
The structure of graphs not admitting a fixed immersion
We present an easy structure theorem for graphs which do not admit an immersion of the complete graph. The theorem motivates the definition of a variation of tree decompositions based on edge cuts instead of vertex cuts which we call tree-cut decompositions. We give a definition for the width of tree-cut decompositions, and using this definition along with the structure theorem for excluded clique immersions, we prove that every graph either has bounded tree-cut width or admits an immersion of a large wall
The structure of graphs not admitting a fixed immersion
We present an easy structure theorem for graphs which do not admit an
immersion of the complete graph. The theorem motivates the definition of a
variation of tree decompositions based on edge cuts instead of vertex cuts
which we call tree-cut decompositions. We give a definition for the width of
tree-cut decompositions, and using this definition along with the structure
theorem for excluded clique immersions, we prove that every graph either has
bounded tree-cut width or admits an immersion of a large wall
Lean Tree-Cut Decompositions: Obstructions and Algorithms
The notion of tree-cut width has been introduced by Wollan in [The structure of graphs not admitting a fixed immersion, Journal of Combinatorial Theory, Series B, 110:47 - 66, 2015]. It is defined via tree-cut decompositions, which are tree-like decompositions that highlight small (edge) cuts in a graph. In that sense, tree-cut decompositions can be seen as an edge-version of tree-decompositions and have algorithmic applications on problems that remain intractable on graphs of bounded treewidth. In this paper, we prove that every graph admits an optimal tree-cut decomposition that satisfies a certain Menger-like condition similar to that of the lean tree decompositions of Thomas [A Menger-like property of tree-width: The finite case, Journal of Combinatorial Theory, Series B, 48(1):67 - 76, 1990]. This allows us to give, for every k in N, an upper-bound on the number immersion-minimal graphs of tree-cut width k. Our results imply the constructive existence of a linear FPT-algorithm for tree-cut width
Deformations of constant mean curvature 1/2 surfaces in H2xR with vertical ends at infinity
We study constant mean curvature 1/2 surfaces in H2xR that admit a
compactification of the mean curvature operator. We show that a particular
family of complete entire graphs over H2 admits a structure of infinite
dimensional manifold with local control on the behaviors at infinity. These
graphs also appear to have a half-space property and we deduce a uniqueness
result at infinity. Deforming non degenerate constant mean curvature 1/2
annuli, we provide a large class of (non rotational) examples and construct
(possibly embedded) annuli without axis, i.e. with two vertical, asymptotically
rotational, non aligned ends.Comment: 35 pages. Addition of a half-space theore
A note on forbidding clique immersions
Robertson and Seymour proved that the relation of graph immersion is
well-quasi-ordered for finite graphs. Their proof uses the results of graph
minors theory. Surprisingly, there is a very short proof of the corresponding
rough structure theorem for graphs without -immersions; it is based on the
Gomory-Hu theorem. The same proof also works to establish a rough structure
theorem for Eulerian digraphs without -immersions, where
denotes the bidirected complete digraph of order
The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space \l^3
We show that a complete embedded maximal surface in the 3-dimensional
Lorentz-Minkowski space with a finite number of singularities is, up to a
Lorentzian isometry, an entire graph over any spacelike plane asymptotic to a
vertical half catenoid or a horizontal plane and with conelike singular points.
We study the space of entire maximal graphs over in
with conelike singularities and vertical limit normal vector at
infinity. We show that is a real analytic manifold of dimension
and the coordinates are given by the position of the singular points in
and the logarithmic growth at the end. We also introduce the moduli space
of {\em marked} graphs with singular points (a mark in a graph is an
ordering of its singularities), which is a -sheeted covering of
We prove that identifying marked graphs differing by translations, rotations
about a vertical axis, homotheties or symmetries about a horizontal plane, the
corresponding quotient space is an analytic manifold of dimension Comment: 32 pages, 4 figures, corrected typos, former Theorem 3.3 (now Theorem
2.2) modifie
Cutwidth: obstructions and algorithmic aspects
Cutwidth is one of the classic layout parameters for graphs. It measures how
well one can order the vertices of a graph in a linear manner, so that the
maximum number of edges between any prefix and its complement suffix is
minimized. As graphs of cutwidth at most are closed under taking
immersions, the results of Robertson and Seymour imply that there is a finite
list of minimal immersion obstructions for admitting a cut layout of width at
most . We prove that every minimal immersion obstruction for cutwidth at
most has size at most .
As an interesting algorithmic byproduct, we design a new fixed-parameter
algorithm for computing the cutwidth of a graph that runs in time , where is the optimum width and is the number of vertices.
While being slower by a -factor in the exponent than the fastest known
algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear
time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth
II: Algorithms for partial -trees of bounded degree, J. Algorithms,
56(1):25--49, 2005], our algorithm has the advantage of being simpler and
self-contained; arguably, it explains better the combinatorics of optimum-width
layouts
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