Crossing-Free Spanning Trees in Visibility Graphs of Points between Monotone Polygonal Obstacles
We consider the problem of deciding whether or not a geometric graph has a crossing-free spanning tree. This problem is known to be NP-hard even for very restricted types of geometric graphs. In this paper, we present an O(n 5) time algorithm to solve this problem for the special case of geometric graphs that arise as visibility graphs of a finite set of n points between two monotone polygonal obstacles. In addition, we give a combinatorial characterization of those visibility graphs induced by such obstacles that have a crossing-free spanning tree. As a byproduct, we obtain a family of counterexamples to the following conjecture by Rivera-Campo: A geometric graph has a crossing-free spanning tree if every subgraph obtained by removing a single vertex has a crossing-free spanning tree.
Keywordsgeometric graph crossing-free spanning tree polygonal obstacle
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