Guarding Quadrangulations and Stacked Triangulations with Edges
Paul Jungeblut, Torsten Ueckerdt

TL;DR
This paper studies the minimal number of edges needed to guard all faces in specific classes of planar graphs, providing tight bounds for quadrangulations and stacked triangulations.
Contribution
It establishes tight bounds on the number of edges required to guard all faces in quadrangulations and stacked triangulations, improving previous results.
Findings
At most n/3 edges guard all faces in quadrangulations.
(n-2)/4 edges are necessary for some quadrangulations.
2n/7 edges suffice for stacked triangulations, and this bound is nearly optimal.
Abstract
Let be a plane graph. A face of is guarded by an edge if at least one vertex from is on the boundary of . For a planar graph class we ask for the minimal number of edges needed to guard all faces of any -vertex graph in . We prove that edges are always sufficient for quadrangulations and give a construction where edges are necessary. For -degenerate quadrangulations we improve this to a tight upper bound of edges. We further prove that edges are always sufficient for stacked triangulations (that are the -degenerate triangulations) and show that this is best possible up to a small additive constant.
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Taxonomy
TopicsAdvanced Graph Theory Research · Computational Geometry and Mesh Generation · Complexity and Algorithms in Graphs
