666 research outputs found
A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface
Given a graph cellularly embedded on a surface of genus , a
cut graph is a subgraph of such that cutting along yields a
topological disk. We provide a fixed parameter tractable approximation scheme
for the problem of computing the shortest cut graph, that is, for any
, we show how to compute a approximation of
the shortest cut graph in time .
Our techniques first rely on the computation of a spanner for the problem
using the technique of brick decompositions, to reduce the problem to the case
of bounded tree-width. Then, to solve the bounded tree-width case, we introduce
a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which
may be of independent interest
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
On the smallest snarks with oddness 4 and connectivity 2
A snark is a bridgeless cubic graph which is not 3-edge-colourable. The
oddness of a bridgeless cubic graph is the minimum number of odd components in
any 2-factor of the graph.
Lukot'ka, M\'acajov\'a, Maz\'ak and \v{S}koviera showed in [Electron. J.
Combin. 22 (2015)] that the smallest snark with oddness 4 has 28 vertices and
remarked that there are exactly two such graphs of that order. However, this
remark is incorrect as -- using an exhaustive computer search -- we show that
there are in fact three snarks with oddness 4 on 28 vertices. In this note we
present the missing snark and also determine all snarks with oddness 4 up to 34
vertices.Comment: 5 page
- …