Rooted Uniform Monotone Minimum Spanning Trees

We study the construction of the minimum cost spanning geometric graph of a given rooted point set \(P\) where each point of \(P\) is connected to the root by a path that satisfies a given property. We focus on two properties, namely the monotonicity w.r.t. a single direction (\(y\)-monotonicity) and the monotonicity w.r.t. a single pair of orthogonal directions (\(xy\)-monotonicity). We propose algorithms that compute the rooted \(y\)-monotone (\(xy\)-monotone) minimum spanning tree of \(P\) in \(O(|P|\log^2 |P|)\) (resp. \(O(|P|\log^3 |P|)\)) time when the direction (resp. pair of orthogonal directions) of monotonicity is given, and in \(O(|P|^2\log|P|)\) time when the optimum direction (resp. pair of orthogonal directions) has to be determined. We also give simple algorithms which, given a rooted connected geometric graph, decide if the root is connected to every other vertex by paths that are all monotone w.r.t. the same direction (pair of orthogonal directions).