Monotone paths on polytopes

@article{Athanasiadis2000MonotonePO,
title={Monotone paths on polytopes},
author={Christos A. Athanasiadis and Paul H. Edelman and Victor Reiner},
journal={Mathematische Zeitschrift},
year={2000},
volume={235},
pages={315-334}
}
• Published 2000
• Mathematics
• Mathematische Zeitschrift
Abstract. We investigate the vertex-connectivity of the graph of f-monotone paths on a d-polytopeP with respect to a generic functionalf. The third author has conjectured that this graph is always (d $-1$)-connected. We resolve this conjecture positively for simple polytopes and show that the graph is 2-connected for any d-polytope with $d \geq 3$. However, we disprove the conjecture in general by exhibiting counterexamples for each $d \geq 4$ in which the graph has a vertex of degree two.We…

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References

SHOWING 1-10 OF 26 REFERENCES

Convex and Linear Orientations of Polytopal Graphs

• Mathematics
Discret. Comput. Geom.
• 2000
For each fixed d -polytope and any acyclic orientation of its graph, it is proved there exist both convex and concave functions that induce the given orientation.

The Generalized Baues Problem

We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a

Lectures on Polytopes

Based on a graduate course given at the Technische Universitat, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforward

Extension spaces of oriented matroids

• Mathematics
Discret. Comput. Geom.
• 1993
It is proved that the extension space is spherical for the class of strongly euclidean oriented matroids, and it is shown that the subspace of realizable extensions is always connected but not necessarily spherical.

Introduction to Graph Theory

1. Fundamental Concepts. What Is a Graph? Paths, Cycles, and Trails. Vertex Degrees and Counting. Directed Graphs. 2. Trees and Distance. Basic Properties. Spanning Trees and Enumeration.

The homotopy type of hyperplane posets

• Mathematics
• 1985
Previously, Edelman had defined a partial order on the regions of a euclidean space dissected by hyperplanes. The goal of this paper is to compute the homotopy type of open intervals in these posets.

Essential chains and homotopy type of posets

A short proof of a result of Billera, Kapranov, and Sturmfels (Cellular strings on polytopes, preprint, 1991), verifying a homotopy type conjecture of Baues (Geometry of loop spaces and the cobar

Cellular strings on polytopes

• Mathematics
• 1994
The complex of cellular strings with respect to a generic linear functional on a d-dimensional convex polytope has the homotopy type of the (d 2)-sphere. This result was conjectured in a special case