1,272 research outputs found
Steiner distance and convexity in graphs
We use the Steiner distance to define a convexity in the vertex set of a graph, which has a nice behavior in the well-known class of HHD-free graphs. For this graph class, we prove that any Steiner tree of a vertex set is included into the geodesical convex hull of the set, which extends the well-known fact that the Euclidean convex hull contains at least one Steiner tree for any planar point set. We also characterize the graph class where Steiner convexity becomes a convex geometry, and provide a vertex set that allows us to rebuild any convex set, using convex hull operation, in any graph
On geodesic and monophonic convexity
In this paper we deal with two types of graph convexities, which are the most natural path convexities in a graph and which are defined by a system P of paths in a connected graph G: the geodesic convexity (also called metric
convexity) which arises when we consider shortest paths, and the monophonic convexity (also called minimal path convexity) when we consider chordless paths. First, we present a realization theorem proving, that there is no general relationship between monophonic and geodetic hull sets. Second, we study the contour of a graph, showing that the contour must be monophonic. Finally, we consider the so-called edge Steiner sets. We prove that every edge Steiner set is edge monophonic.Ministerio de Ciencia y TecnologÃaFondo Europeo de Desarrollo RegionalGeneralitat de Cataluny
New bounds on the average distance from the Fermat-Weber center of a planar convex body
The Fermat-Weber center of a planar body is a point in the plane from
which the average distance to the points in is minimal. We first show that
for any convex body in the plane, the average distance from the
Fermat-Weber center of to the points of is larger than , where is the diameter of . This proves a conjecture
of Carmi, Har-Peled and Katz. From the other direction, we prove that the same
average distance is at most . The new bound substantially improves the previous bound of
due to
Abu-Affash and Katz, and brings us closer to the conjectured value of . We also confirm the upper bound conjecture for centrally
symmetric planar convex bodies.Comment: 13 pages, 2 figures. An earlier version (now obsolete): A. Dumitrescu
and Cs. D. T\'oth: New bounds on the average distance from the Fermat-Weber
center of a planar convex body, in Proceedings of the 20th International
Symposium on Algorithms and Computation (ISAAC 2009), 2009, LNCS 5878,
Springer, pp. 132-14
Optimal Point Placement for Mesh Smoothing
We study the problem of moving a vertex in an unstructured mesh of
triangular, quadrilateral, or tetrahedral elements to optimize the shapes of
adjacent elements. We show that many such problems can be solved in linear time
using generalized linear programming. We also give efficient algorithms for
some mesh smoothing problems that do not fit into the generalized linear
programming paradigm.Comment: 12 pages, 3 figures. A preliminary version of this paper was
presented at the 8th ACM/SIAM Symp. on Discrete Algorithms (SODA '97). This
is the final version, and will appear in a special issue of J. Algorithms for
papers from SODA '9
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