165 research outputs found
Which point sets admit a k-angulation?
For k >= 3, a k-angulation is a 2-connected plane graph in which every
internal face is a k-gon. We say that a point set P admits a plane graph G if
there is a straight-line drawing of G that maps V(G) onto P and has the same
facial cycles and outer face as G. We investigate the conditions under which a
point set P admits a k-angulation and find that, for sets containing at least
2k^2 points, the only obstructions are those that follow from Euler's formula.Comment: 13 pages, 7 figure
Synchronized planarity with applications to constrained planarity problems
We introduce the problem Synchronized Planarity. Roughly speaking, its input is a loop-free multi-graph together with synchronization constraints that, e.g., match pairs of vertices of equal degree by providing a bijection between their edges. Synchronized Planarity then asks whether the graph admits a crossing-free embedding into the plane such that the orders of edges around synchronized vertices are consistent. We show, on the one hand, that Synchronized Planarity can be solved in quadratic time, and, on the other hand, that it serves as a powerful modeling language that lets us easily formulate several constrained planarity problems as instances of Synchronized Planarity. In particular, this lets us solve Clustered Planarity in quadratic time, where the most efficient previously known algorithm has an upper bound of O(nâž)
Planar Drawings of Fixed-Mobile Bigraphs
A fixed-mobile bigraph G is a bipartite graph such that the vertices of one
partition set are given with fixed positions in the plane and the mobile
vertices of the other part, together with the edges, must be added to the
drawing. We assume that G is planar and study the problem of finding, for a
given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In
the most general case, we show NP-hardness. For k=0 and under additional
constraints on the positions of the fixed or mobile vertices, we either prove
that the problem is polynomial-time solvable or prove that it belongs to NP.
Finally, we present a polynomial-time testing algorithm for a certain type of
"layered" 1-bend drawings
Optimally fast incremental Manhattan plane embedding and planar tight span construction
We describe a data structure, a rectangular complex, that can be used to
represent hyperconvex metric spaces that have the same topology (although not
necessarily the same distance function) as subsets of the plane. We show how to
use this data structure to construct the tight span of a metric space given as
an n x n distance matrix, when the tight span is homeomorphic to a subset of
the plane, in time O(n^2), and to add a single point to a planar tight span in
time O(n). As an application of this construction, we show how to test whether
a given finite metric space embeds isometrically into the Manhattan plane in
time O(n^2), and add a single point to the space and re-test whether it has
such an embedding in time O(n).Comment: 39 pages, 15 figure
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