## Covering paths for planar point sets.(English)Zbl 1294.05102

Summary: Given $$n$$ points in the plane, a covering path is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $$n/2$$ segments, and $$n-1$$ straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of $$n$$ points in the plane admits a (possibly self-crossing) covering path consisting of $$n/2+O(n/\log n)$$ straight line segments. If the path is required to be noncrossing, we prove that $$(1-{\varepsilon})n$$ straight line segments suffice for a small constant $$\varepsilon > 0$$, and we exhibit $$n$$-element point sets that require at least $$5n/9-O(1)$$ segments in every such path. Further, the analogous question for noncrossing covering trees is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossing covering path for $$n$$ points in the plane requires $$\varOmega(n \log n)$$ time in the worst case.

### MSC:

 05C38 Paths and cycles 05C05 Trees

### Keywords:

covering path; covering tree; noncrossing graph
Full Text:

### References:

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