1,331 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
Construction of planar 4-connected triangulations
In this article we describe a recursive structure for the class of 4-connected triangulations or - equivalently - cyclically 4-connected plane cubic graphs
Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus
We extend the notion of canonical ordering (initially developed for planar
triangulations and 3-connected planar maps) to cylindric (essentially simple)
triangulations and more generally to cylindric (essentially internally)
-connected maps. This allows us to extend the incremental straight-line
drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case)
and of Kant (in the -connected case) to this setting. Precisely, for any
cylindric essentially internally -connected map with vertices, we
can obtain in linear time a periodic (in ) straight-line drawing of that
is crossing-free and internally (weakly) convex, on a regular grid
, with and ,
where is the face-distance between the two boundaries. This also yields an
efficient periodic drawing algorithm for graphs on the torus. Precisely, for
any essentially -connected map on the torus (i.e., -connected in the
periodic representation) with vertices, we can compute in linear time a
periodic straight-line drawing of that is crossing-free and (weakly)
convex, on a periodic regular grid
, with and
, where is the face-width of . Since ,
the grid area is .Comment: 37 page
Some Triangulated Surfaces without Balanced Splitting
Let G be the graph of a triangulated surface of genus . A
cycle of G is splitting if it cuts into two components, neither of
which is homeomorphic to a disk. A splitting cycle has type k if the
corresponding components have genera k and g-k. It was conjectured that G
contains a splitting cycle (Barnette '1982). We confirm this conjecture for an
infinite family of triangulations by complete graphs but give counter-examples
to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should
contain splitting cycles of every possible type.Comment: 15 pages, 7 figure
Irreducible triangulations of surfaces with boundary
A triangulation of a surface is irreducible if no edge can be contracted to
produce a triangulation of the same surface. In this paper, we investigate
irreducible triangulations of surfaces with boundary. We prove that the number
of vertices of an irreducible triangulation of a (possibly non-orientable)
surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was
known only for surfaces without boundary (b=0). While our technique yields a
worse constant in the O(.) notation, the present proof is elementary, and
simpler than the previous ones in the case of surfaces without boundary
Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs
We study Markov chains for -orientations of plane graphs, these are
orientations where the outdegree of each vertex is prescribed by the value of a
given function . The set of -orientations of a plane graph has
a natural distributive lattice structure. The moves of the up-down Markov chain
on this distributive lattice corresponds to reversals of directed facial cycles
in the -orientation. We have a positive and several negative results
regarding the mixing time of such Markov chains.
A 2-orientation of a plane quadrangulation is an orientation where every
inner vertex has outdegree 2. We show that there is a class of plane
quadrangulations such that the up-down Markov chain on the 2-orientations of
these quadrangulations is slowly mixing. On the other hand the chain is rapidly
mixing on 2-orientations of quadrangulations with maximum degree at most 4.
Regarding examples for slow mixing we also revisit the case of 3-orientations
of triangulations which has been studied before by Miracle et al.. Our examples
for slow mixing are simpler and have a smaller maximum degree, Finally we
present the first example of a function and a class of plane
triangulations of constant maximum degree such that the up-down Markov chain on
the -orientations of these graphs is slowly mixing
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
- …