Decomposition of Geometric Graphs into Star Forests
J\'anos Pach, Morteza Saghafian, Patrick Schnider
TL;DR
This paper proves that a complete convex geometric graph on n vertices requires at least n-1 noncrossing star-forests for decomposition, establishing a tight bound and exploring similar questions for abstract graphs.
Contribution
It resolves a problem by establishing the minimum number of noncrossing star-forests needed to decompose a complete convex geometric graph, providing a tight bound.
Findings
Minimum of n-1 star-forests needed for decomposition
Bound is tight for complete convex geometric graphs
Discusses similar decomposition questions for abstract graphs
Abstract
We solve a problem of Dujmovi\'c and Wood (2007) by showing that a complete convex geometric graph on vertices cannot be decomposed into fewer than star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Graph Theory Research · Computational Geometry and Mesh Generation · Markov Chains and Monte Carlo Methods