1,530 research outputs found
Contractions, Removals and How to Certify 3-Connectivity in Linear Time
It is well-known as an existence result that every 3-connected graph G=(V,E)
on more than 4 vertices admits a sequence of contractions and a sequence of
removal operations to K_4 such that every intermediate graph is 3-connected. We
show that both sequences can be computed in optimal time, improving the
previously best known running times of O(|V|^2) to O(|V|+|E|). This settles
also the open question of finding a linear time 3-connectivity test that is
certifying and extends to a certifying 3-edge-connectivity test in the same
time. The certificates used are easy to verify in time O(|E|).Comment: preliminary versio
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
Contractions, removals and certifying 3-connectivity in linear time
As existence result, it is well known that every 3-connected graph G=(V,E) on
more than 4 vertices admits a sequence of contractions and a sequence of
removal operations to K_4 such that every intermediate graph in the sequences
is 3-connected. We show that both sequences can be computed in linear time,
improving the previous best known running time of O(|V|^2) to O(|V|+|E|). This
settles also the open question of finding a certifying 3-connectivity test in
linear time and extents to certify 3-edge-connectivity in linear time as well
Which point sets admit a k-angulation?
For k >= 3, a k-angulation is a 2-connected plane graph in which every
internal face is a k-gon. We say that a point set P admits a plane graph G if
there is a straight-line drawing of G that maps V(G) onto P and has the same
facial cycles and outer face as G. We investigate the conditions under which a
point set P admits a k-angulation and find that, for sets containing at least
2k^2 points, the only obstructions are those that follow from Euler's formula.Comment: 13 pages, 7 figure
Computing Vertex-Disjoint Paths in Large Graphs Using MAOs
We consider the problem of computing k in N internally vertex-disjoint paths between special vertex pairs of simple connected graphs. For general vertex pairs, the best deterministic time bound is, since 42 years, O(min{k,sqrt{n}}m) for each pair by using traditional flow-based methods.
The restriction of our vertex pairs comes from the machinery of maximal adjacency orderings (MAOs). Henzinger showed for every MAO and every 1 <= k <= delta (where delta is the minimum degree of the graph) the existence of k internally vertex-disjoint paths between every pair of the last delta-k+2 vertices of this MAO. Later, Nagamochi generalized this result by using the machinery of mixed connectivity. Both results are however inherently non-constructive.
We present the first algorithm that computes these k internally vertex-disjoint paths in linear time O(m), which improves the previously best time O(min{k,sqrt{n}}m). Due to the linear running time, this algorithm is suitable for large graphs. The algorithm is simple, works directly on the MAO structure, and completes a long history of purely existential proofs with a constructive method. We extend our algorithm to compute several other path systems and discuss its impact for certifying algorithms
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