2,669 research outputs found
Nonbipartite Dulmage-Mendelsohn Decomposition for Berge Duality
The Dulmage-Mendelsohn decomposition is a classical canonical decomposition
in matching theory applicable for bipartite graphs, and is famous not only for
its application in the field of matrix computation, but also for providing a
prototypal structure in matroidal optimization theory. The Dulmage-Mendelsohn
decomposition is stated and proved using the two color classes, and therefore
generalizing this decomposition for nonbipartite graphs has been a difficult
task. In this paper, we obtain a new canonical decomposition that is a
generalization of the Dulmage-Mendelsohn decomposition for arbitrary graphs,
using a recently introduced tool in matching theory, the basilica
decomposition. Our result enables us to understand all known canonical
decompositions in a unified way. Furthermore, we apply our result to derive a
new theorem regarding barriers. The duality theorem for the maximum matching
problem is the celebrated Berge formula, in which dual optimizers are known as
barriers. Several results regarding maximal barriers have been derived by known
canonical decompositions, however no characterization has been known for
general graphs. In this paper, we provide a characterization of the family of
maximal barriers in general graphs, in which the known results are developed
and unified
Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm
Given an undirected graph and two disjoint vertex pairs and
, the Shortest two disjoint paths problem (S2DP) asks for the minimum
total length of two vertex disjoint paths connecting with , and
with , respectively.
We show that for cubic planar graphs there are NC algorithms, uniform
circuits of polynomial size and polylogarithmic depth, that compute the S2DP
and moreover also output the number of such minimum length path pairs.
Previously, to the best of our knowledge, no deterministic polynomial time
algorithm was known for S2DP in cubic planar graphs with arbitrary placement of
the terminals. In contrast, the randomized polynomial time algorithm by
Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is
serial in nature, and cannot count the solutions.
Our results are built on an approach by Hirai and Namba, Algorithmica 2017,
for a generalisation of S2DP, and fast algorithms for counting perfect
matchings in planar graphs
On disjoint matchings in cubic graphs
For and a cubic graph let denote the maximum number
of edges that can be covered by matchings. We show that and . Moreover, it turns out that
.Comment: 41 pages, 8 figures, minor chage
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