An Improved Bound for Orthogeodesic Point Set Embeddings of Trees

In an orthogeodesic embedding of a graph, each edge is embedded as an axis-parallel polyline that forms a shortest path in the `1 metric. In this paper we consider orthogeodesic plane embeddings of trees on grids. A grid is implicitly defined by a set P ⊂ R of points. Denote by ΓP the arrangement induced by all horizontal and vertical lines that pass through a point from P . When embedding a graph on the grid defined by P , vertices are mapped to points from P and edges are realized as polylines that bend at vertices of ΓP only. For integers n and ∆, denote by t∆(n) the minimum number such that for every set P of t∆(n) points in general position, every tree on n vertices with vertex degree at most ∆ admits an orthogeodesic plane embedding on the grid defined by P . We show t4(n) < 11n/8 and t3(n) < 9n/8, improving an earlier bound of 3n/2.