Good edge of a dual line arrangement

A good dual-arrangement edge has degree-six endpoints and two triangular adjacent faces. Other edges are bad. The Euler defect identity for a projective line arrangement bounds the number of bad edges by $16t_2$: charge them to degree-four or higher-degree vertices and to nontriangular faces. This turns a count of ordinary lines into quantitative control of local triangular geometry.

A safe dual-arrangement edge is one for which every edge on a path of length at most two from either endpoint is a good dual-arrangement edge. This stronger local condition supplies a two-cell-thick triangular strip. Only a bounded multiple of the original bad edges can be unsafe: trace a shortest path to the first bad edge and reverse it through degree-six intermediate vertices.

In an embedded arrangement graph, an edge whose fixed-radius neighbourhood meets a bad edge can be charged to the first such edge along a shortest path. All intervening edges are good dual-arrangement edges, so their endpoints have bounded degree. The number of possible reversed paths is bounded in terms of the fixed radius alone. Therefore thickening a defect set by a fixed radius preserves an $O(t_2)$ edge count.