IWOCA 2016: Combinatorial Algorithms pp 55-67

# Essential Constraints of Edge-Constrained Proximity Graphs

• Prosenjit Bose
• Jean-Lou De Carufel
• Alina Shaikhet
• Michiel Smid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)

## Abstract

Given a plane forest $$F = (V, E)$$ of $$|V| = n$$ points, we find the minimum set $$S \subseteq E$$ of edges such that the edge-constrained minimum spanning tree over the set V of vertices and the set S of constraints contains F. We present an $$O(n \log n)$$-time algorithm that solves this problem. We generalize this to other proximity graphs in the constraint setting, such as the relative neighbourhood graph, Gabriel graph, $$\beta$$-skeleton and Delaunay triangulation.

We present an algorithm that identifies the minimum set $$S\subseteq E$$ of edges of a given plane graph $$I=(V,E)$$ such that $$I \subseteq CG_\beta (V, S)$$ for $$1 \le \beta \le 2$$, where $$CG_\beta (V, S)$$ is the constraint $$\beta$$-skeleton over the set V of vertices and the set S of constraints. The running time of our algorithm is O(n), provided that the constrained Delaunay triangulation of I is given.

### Keywords

Proximity graphs Constraints Visibility MST Delaunay $$\beta$$-skeletons

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© Springer International Publishing Switzerland 2016

## Authors and Affiliations

• Prosenjit Bose
• 1
• Jean-Lou De Carufel
• 2
• Alina Shaikhet
• 1
• Michiel Smid
• 1
1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
2. 2.School of Electrical Engineering and Computer ScienceUniversity of OttawaOttawaCanada