Guarding Quadrangulations and Stacked Triangulations with Edges

06/24/2020

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by   Paul Jungeblut, et al.

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Let G = (V,E) be a plane graph. A face f of G is guarded by an edge vw ∈ E if at least one vertex from {v,w} is on the boundary of f. For a planar graph class 𝒢 we ask for the minimal number of edges needed to guard all faces of any n-vertex graph in 𝒢. We prove that ⌊ n/3 ⌋ edges are always sufficient for quadrangulations and give a construction where ⌊ (n-2)/4 ⌋ edges are necessary. For 2-degenerate quadrangulations we improve this to a tight upper bound of ⌊ n/4 ⌋ edges. We further prove that ⌊ 2n/7 ⌋ edges are always sufficient for stacked triangulations (that are the 3-degenerate triangulations) and show that this is best possible up to a small additive constant.