Guarding Quadrangulations and Stacked Triangulations with Edges

Abstract

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.