31 research outputs found
Tutte Embeddings of Tetrahedral Meshes
Tutte's embedding theorem states that every 3-connected graph without a
or minor (i.e. a planar graph) is embedded in the plane if the outer
face is in convex position and the interior vertices are convex combinations of
their neighbors. We show that this result extends to simply connected
tetrahedral meshes in a natural way: for the tetrahedral mesh to be embedded if
the outer polyhedron is in convex position and the interior vertices are convex
combination of their neighbors it is sufficient (but not necessary) that the
graph of the tetrahedral mesh contains no and no , and all
triangles incident on three boundary vertices are boundary triangles
The Strong Arnold Property for 4-connected flat graphs
We show that if is a 4-connected flat graph, then any real
symmetric matrix with exactly one negative eigenvalue and
satisfying, for any two distinct vertices and , if and
are adjacent, and if and are nonadjacent, has the Strong
Arnold Property: there is no nonzero real symmetric matrix with
and whenever and are equal or adjacent. (A graph
is {\em flat} if it can be embedded injectively in -dimensional Euclidean
space such that the image of any circuit is the boundary of some disk disjoint
from the image of the remainder of the graph.)
This applies to the Colin de Verdi\`ere graph parameter, and extends similar
results for 2-connected outerplanar graphs and 3-connected planar graphs
Disjoint Essential Cycles
AbstractGraphs that have two disjoint noncontractible cycles in every possible embedding in surfaces are characterized. Similar characterization is given for the class of graphs whose orientable embeddings (embeddings in surfaces different from the projective plane, respectively) always have two disjoint noncontractible cycles. For graphs which admit embeddings in closed surfaces without having two disjoint noncontractible cycles, such embeddings are structurally characterized
Nullspace embeddings for outerplanar graphs
We study relations between geometric embeddings of graphs and the spectrum of associated matrices, focusing on outerplanar embeddings of graphs. For a simple connected graph G=(V,E), we define a "good" G-matrix as a V×V matrix with negative entries corresponding to adjacent nodes, zero entries corresponding to distinct nonadjacent nodes, and exactly one negative eigenvalue. We give an algorithmic proof of the fact that it G is a 2-connected graph, then either the nullspace representation defined by any "good" G-matrix with corank 2 is an outerplanar embedding of G, or else there exists a "good" G-matrix with corank 3