COCOON 2012: Computing and Combinatorics pp 240-251

# Monotone Paths in Planar Convex Subdivisions

• Günter Rote
• Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

## Abstract

Consider a connected subdivision of the plane into n convex faces where every vertex is incident to at most Δ edges. Then, starting from every vertex there is a path with at least Ω(logΔ n) edges that is monotone in some direction. This bound is the best possible. Consider now a connected subdivision of the plane into n convex faces where exactly k faces are unbounded. Then, there is a path with at least Ω(log(n/k)/loglog(n/k)) edges that is monotone in some direction. This bound is also the best possible.

In 3-space, we show that for every n ≥ 4, there exists a polytope P with n vertices, bounded vertex degrees, and triangular faces such that every monotone path on the 1-skeleton of P has at most O(log2 n) edges. We also construct a polytope Q with n vertices, and triangular faces, (with unbounded degree however), such that every monotone path on the 1-skeleton of Q has at most O(logn) edges.

## Keywords

Simple Polygon Geometric Graph Interior Vertex Weakly Monotone Face Incident
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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