810 research outputs found
Edge-Orders
Canonical orderings and their relatives such as st-numberings have been used
as a key tool in algorithmic graph theory for the last decades. Recently, a
unifying concept behind all these orders has been shown: they can be described
by a graph decomposition into parts that have a prescribed vertex-connectivity.
Despite extensive interest in canonical orderings, no analogue of this
unifying concept is known for edge-connectivity. In this paper, we establish
such a concept named edge-orders and show how to compute (1,1)-edge-orders of
2-edge-connected graphs as well as (2,1)-edge-orders of 3-edge-connected graphs
in linear time, respectively. While the former can be seen as the edge-variants
of st-numberings, the latter are the edge-variants of Mondshein sequences and
non-separating ear decompositions. The methods that we use for obtaining such
edge-orders differ considerably in almost all details from the ones used for
their vertex-counterparts, as different graph-theoretic constructions are used
in the inductive proof and standard reductions from edge- to
vertex-connectivity are bound to fail.
As a first application, we consider the famous Edge-Independent Spanning Tree
Conjecture, which asserts that every k-edge-connected graph contains k rooted
spanning trees that are pairwise edge-independent. We illustrate the impact of
the above edge-orders by deducing algorithms that construct 2- and 3-edge
independent spanning trees of 2- and 3-edge-connected graphs, the latter of
which improves the best known running time from O(n^2) to linear time
Directed cycle double covers: structure and generation of hexagon graphs
Jaeger's directed cycle double cover conjecture can be formulated as a
problem of existence of special perfect matchings in a class of graphs that we
call hexagon graphs. In this work, we explore the structure of hexagon graphs.
We show that hexagon graphs are braces that can be generated from the ladder on
8 vertices using two types of McCuaig's augmentations.Comment: 20 page
Subdivisional spaces and graph braid groups
We study the problem of computing the homology of the configuration spaces of
a finite cell complex . We proceed by viewing , together with its
subdivisions, as a subdivisional space--a kind of diagram object in a category
of cell complexes. After developing a version of Morse theory for subdivisional
spaces, we decompose and show that the homology of the configuration spaces
of is computed by the derived tensor product of the Morse complexes of the
pieces of the decomposition, an analogue of the monoidal excision property of
factorization homology.
Applying this theory to the configuration spaces of a graph, we recover a
cellular chain model due to \'{S}wi\k{a}tkowski. Our method of deriving this
model enhances it with various convenient functorialities, exact sequences, and
module structures, which we exploit in numerous computations, old and new.Comment: 71 pages, 15 figures. Typo fixed. May differ slightly from version
published in Documenta Mathematic
O(1) Steiner Point Removal in Series-Parallel Graphs
We study how to vertex-sparsify a graph while preserving both the graph's
metric and structure. Specifically, we study the Steiner point removal (SPR)
problem where we are given a weighted graph and terminal set and must compute a weighted minor of which
approximates 's metric on . A major open question in the area of metric
embeddings is the existence of multiplicative distortion SPR solutions
for every (non-trivial) minor-closed family of graphs. To this end prior work
has studied SPR on trees, cactus and outerplanar graphs and showed that in
these graphs such a minor exists with distortion.
We give distortion SPR solutions for series-parallel graphs, extending
the frontier of this line of work. The main engine of our approach is a new
metric decomposition for series-parallel graphs which we call a hammock
decomposition. Roughly, a hammock decomposition is a forest-like structure that
preserves certain critical parts of the metric induced by a series-parallel
graph
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