64,279 research outputs found
A superlinear bound on the number of perfect matchings in cubic bridgeless graphs
Lovasz and Plummer conjectured in the 1970's that cubic bridgeless graphs
have exponentially many perfect matchings. This conjecture has been verified
for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky
and Seymour in 2008, but in general only linear bounds are known. In this
paper, we provide the first superlinear bound in the general case.Comment: 54 pages v2: a short (missing) proof of Lemma 10 was adde
Bitangents of tropical plane quartic curves
We study smooth tropical plane quartic curves and show that they satisfy
certain properties analogous to (but also different from) smooth plane quartics
in algebraic geometry. For example, we show that every such curve admits either
infinitely many or exactly 7 bitangent lines. We also prove that a smooth
tropical plane quartic curve cannot be hyperelliptic.Comment: 13 pages, 9 figures. Minor revisions; accepted for publication in
Mathematische Zeitschrif
A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem
The clustered planarity problem (c-planarity) asks whether a hierarchically
clustered graph admits a planar drawing such that the clusters can be nicely
represented by regions. We introduce the cd-tree data structure and give a new
characterization of c-planarity. It leads to efficient algorithms for
c-planarity testing in the following cases. (i) Every cluster and every
co-cluster (complement of a cluster) has at most two connected components. (ii)
Every cluster has at most five outgoing edges.
Moreover, the cd-tree reveals interesting connections between c-planarity and
planarity with constraints on the order of edges around vertices. On one hand,
this gives rise to a bunch of new open problems related to c-planarity, on the
other hand it provides a new perspective on previous results.Comment: 17 pages, 2 figure
Decycling a graph by the removal of a matching: new algorithmic and structural aspects in some classes of graphs
A graph is {\em matching-decyclable} if it has a matching such that
is acyclic. Deciding whether is matching-decyclable is an NP-complete
problem even if is 2-connected, planar, and subcubic. In this work we
present results on matching-decyclability in the following classes: Hamiltonian
subcubic graphs, chordal graphs, and distance-hereditary graphs. In Hamiltonian
subcubic graphs we show that deciding matching-decyclability is NP-complete
even if there are exactly two vertices of degree two. For chordal and
distance-hereditary graphs, we present characterizations of
matching-decyclability that lead to -time recognition algorithms
Passages in Graphs
Directed graphs can be partitioned in so-called passages. A passage P is a
set of edges such that any two edges sharing the same initial vertex or sharing
the same terminal vertex are both inside or are both outside of P. Passages
were first identified in the context of process mining where they are used to
successfully decompose process discovery and conformance checking problems. In
this article, we examine the properties of passages. We will show that passages
are closed under set operators such as union, intersection and difference.
Moreover, any passage is composed of so-called minimal passages. These
properties can be exploited when decomposing graph-based analysis and
computation problems.Comment: 8 page
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