1,062 research outputs found
Types of triangle in plane Hamiltonian triangulations and applications to domination and k-walks
We investigate the minimum number t(0)(G) of faces in a Hamiltonian triangulation G so that any Hamiltonian cycle C of G has at least t(0)(G) faces that do not contain an edge of C. We prove upper and lower bounds on the maximum of these numbers for all triangulations with a fixed number of facial triangles. Such triangles play an important role when Hamiltonian cycles in triangulations with 3-cuts are constructed from smaller Hamiltonian cycles of 4-connected subgraphs. We also present results linking the number of these triangles to the length of 3-walks in a class of triangulation and to the domination number
Power domination in maximal planar graphs
Power domination in graphs emerged from the problem of monitoring an
electrical system by placing as few measurement devices in the system as
possible. It corresponds to a variant of domination that includes the
possibility of propagation. For measurement devices placed on a set S of
vertices of a graph G, the set of monitored vertices is initially the set S
together with all its neighbors. Then iteratively, whenever some monitored
vertex v has a single neighbor u not yet monitored, u gets monitored. A set S
is said to be a power dominating set of the graph G if all vertices of G
eventually are monitored. The power domination number of a graph is the minimum
size of a power dominating set. In this paper, we prove that any maximal planar
graph of order n 6 admits a power dominating set of size at most (n--2)/4
Planar graphs as L-intersection or L-contact graphs
The L-intersection graphs are the graphs that have a representation as
intersection graphs of axis parallel shapes in the plane. A subfamily of these
graphs are {L, |, --}-contact graphs which are the contact graphs of axis
parallel L, |, and -- shapes in the plane. We prove here two results that were
conjectured by Chaplick and Ueckerdt in 2013. We show that planar graphs are
L-intersection graphs, and that triangle-free planar graphs are {L, |,
--}-contact graphs. These results are obtained by a new and simple
decomposition technique for 4-connected triangulations. Our results also
provide a much simpler proof of the known fact that planar graphs are segment
intersection graphs
A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets
Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy
groups of two topological spaces and whenever a map with
strong connectivity conditions on the fibers is given. We apply similar
techniques in o-minimal expansions of fields to compare the o-minimal homotopy
of a definable set with the homotopy of some of its bounded hyperdefinable
quotients . Under suitable assumption, we show that and . As a special case,
given a definably compact group, we obtain a new proof of Pillay's group
conjecture ")" largely independent of the
group structure of . We also obtain different proofs of various comparison
results between classical and o-minimal homotopy.Comment: 24 page
Converting between quadrilateral and standard solution sets in normal surface theory
The enumeration of normal surfaces is a crucial but very slow operation in
algorithmic 3-manifold topology. At the heart of this operation is a polytope
vertex enumeration in a high-dimensional space (standard coordinates).
Tollefson's Q-theory speeds up this operation by using a much smaller space
(quadrilateral coordinates), at the cost of a reduced solution set that might
not always be sufficient for our needs. In this paper we present algorithms for
converting between solution sets in quadrilateral and standard coordinates. As
a consequence we obtain a new algorithm for enumerating all standard vertex
normal surfaces, yielding both the speed of quadrilateral coordinates and the
wider applicability of standard coordinates. Experimentation with the software
package Regina shows this new algorithm to be extremely fast in practice,
improving speed for large cases by factors from thousands up to millions.Comment: 55 pages, 10 figures; v2: minor fixes only, plus a reformat for the
journal styl
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