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LowDegree Minimum Spanning Trees
 Discrete Comput. Geom
, 1999
"... Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that under the Lp norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where ..."
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Cited by 20 (1 self)
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Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that under the Lp norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where the maximum degree is minimized. We give the bestknown bounds for the maximum MST degree for arbitrary Lp metrics in all dimensions, with a focus on the rectilinear metric in two and three dimensions. We show that for any finite set of points in the rectilinear plane there exists an MST with maximum degree of at most 4, and for threedimensional rectilinear space the maximum possible degree of a minimumdegree MST is either 13 or 14. 1 Introduction Minimum spanning tree (MST) construction is a classic optimization problem for which several efficient algorithms are known [9] [15] [19]. Solutions of many other problems hinge on the construction of an MST as an intermediary step [4], with th...
The Local Steiner Problem in Normed Planes
 Networks
, 2000
"... We present a geometric analysis of the local structure of vertices in a Steiner Minimum Tree in an arbitrary normed plane in terms of socalled absorbing and critical angles, thereby unifying various results known for specific norms. We find necessary and sufficient conditions for a set of segments ..."
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Cited by 7 (0 self)
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We present a geometric analysis of the local structure of vertices in a Steiner Minimum Tree in an arbitrary normed plane in terms of socalled absorbing and critical angles, thereby unifying various results known for specific norms. We find necessary and sufficient conditions for a set of segments emanating from a point to be the neighbourhood of a vertex in a Steiner Minimum Tree. As corollaries we show that the maximum possible degree of a Steiner point and of a given point are equal, and equal 3 or 4, except if the unit ball is an ane regular hexagon, where it is known that the maximum degree of a Steiner point is 4, and of a regular point is 6. We also characterize the planes where the maximum degree is 4, the socalled Xplanes, and present examples. In particular, if the unit ball is an ane regular 2ngon, Steiner points of degree 4 exist if and only if n = 2; 3; 4 or 6.