35 research outputs found
Circuits in graphs embedded on the torus
AbstractWe give a survey of some recent results on circuits in graphs embedded on the torus. Especially we focus on methods relating graphs embedded on the torus to integer polygons in the Euclidean plane
On Essential and Inessential Polygons in Embedded Graphs
AbstractIn this article, we present a number of results of the following type: A given subgraph of an embedded graph either is embedded in a disc or it has a face chain containing a non-contractible closed path. Our main application is to prove that any two faces of a 4-representative embedding are simultaneously contained in a disc bounded by a polygon. This result is used to prove the existence of ⌊(r−1)/8⌋ pairwise disjoint, pairwise homotopic non-contractible separating polygons in an r -representative orientable embedding. Our proof of this latter result is simple and mechanical
Ideal triangulations of 3-manifolds up to decorated transit equivalences
We consider 3-dimensional pseudo-manifolds M with a given set of marked point
V such that M-V is the interior of a compact 3-manifold with boundary. An ideal
triangulation T of (M, V ) has V as its set of vertices. A branching (T, b)
enhances T to a Delta-complex. Branched triangulations of (M, V ) are
considered up to the b-transit equivalence generated by isotopy and ideal
branched moves which keep V pointwise fixed. We extend a well known
connectivity result for naked triangulations by showing that branched ideal
triangulations of (M, V) are equivalent to each other. A pre-branching is a
system of transverse orientations at the 2-facets of T verifying a certain
global constraint; pre-branchings are considered up to a natural pb-transit
equivalence. If M is oriented, every branching b induces a pre-branching w(b)
and every b-transit induces a pb-transit. The quotient set of pre-branchings up
to transit equivalence is far to be trivial; we get some information about it
and we characterize the pre-branchings of type w(b). Pre-branched and branched
moves are naturally organized in subfamilies which give rise to restricted
transit equivalences. In the branching setting we revisit early results about
the sliding transit equivalence and outline a conceptually different approach
to the branched connectivity and eventually also to the naked one. The basic
idea is to point out some structures of differential topological nature which
are carried by every branched ideal triangulation, are preserved by the sliding
transits and can be modified by the whole branched transits. The non ambiguous
transit equivalence already widely studied on pre-branchings lifts to a
specialization of the sliding equivalence on branchings; we point out a few
specific insights, again in terms of carried structures preserved by the non
ambiguous and which can be modified by the whole sliding transits.Comment: 29 pages, 22 figure