4 research outputs found
Packing and covering with balls on Busemann surfaces
In this note we prove that for any compact subset of a Busemann surface
(in particular, for any simple polygon with geodesic metric)
and any positive number , the minimum number of closed balls of radius
with centers at and covering the set is at most 19
times the maximum number of disjoint closed balls of radius centered
at points of : , where and
are the covering and the packing numbers of by -balls.Comment: 27 page
Packing and covering balls in graphs excluding a minor
We prove that for every integer there exists a constant such
that for every -minor-free graph , and every set of balls in ,
the minimum size of a set of vertices of intersecting all the balls of
is at most times the maximum number of vertex-disjoint balls in . This
was conjectured by Chepoi, Estellon, and Vax\`es in 2007 in the special case of
planar graphs and of balls having the same radius.Comment: v3: final versio
Packing and covering with balls on Busemann surfaces
In this note we prove that for any compact subset of a Busemann surface (in particular, for any simple polygon with geodesic metric) and any positive number , the minimum number of closed balls of radius with centers at and covering the set is at most 19 times the maximum number of disjoint closed balls of radius centered at points of : , where and are the covering and the packing numbers of by -balls