14,569 research outputs found
The Number of Triangles Needed to Span a Polygon Embedded in R^d
Given a closed polygon P having n edges, embedded in R^d, we give upper and
lower bounds for the minimal number of triangles t needed to form a
triangulated PL surface in R^d having P as its geometric boundary. The most
interesting case is dimension 3, where the polygon may be knotted. We use the
Seifert suface construction to show there always exists an embedded surface
requiring at most 7n^2 triangles. We complement this result by showing there
are polygons in R^3 for which any embedded surface requires at least 1/2n^2 -
O(n) triangles. In dimension 2 only n-2 triangles are needed, and in dimensions
5 or more there exists an embedded surface requiring at most n triangles. In
dimension 4 we obtain a partial answer, with an O(n^2) upper bound for embedded
surfaces, and a construction of an immersed disk requiring at most 3n
triangles. These results can be interpreted as giving qualitiative discrete
analogues of the isoperimetric inequality for piecewise linear manifolds.Comment: 16 pages, 4 figures. This paper is a retitled, revised version of
math.GT/020217
On a class of intersection graphs
Given a directed graph D = (V,A) we define its intersection graph I(D) =
(A,E) to be the graph having A as a node-set and two nodes of I(D) are adjacent
if their corresponding arcs share a common node that is the tail of at least
one of these arcs. We call these graphs facility location graphs since they
arise from the classical uncapacitated facility location problem. In this paper
we show that facility location graphs are hard to recognize and they are easy
to recognize when the graph is triangle-free. We also determine the complexity
of the vertex coloring, the stable set and the facility location problems on
that class
Rectangular Layouts and Contact Graphs
Contact graphs of isothetic rectangles unify many concepts from applications
including VLSI and architectural design, computational geometry, and GIS.
Minimizing the area of their corresponding {\em rectangular layouts} is a key
problem. We study the area-optimization problem and show that it is NP-hard to
find a minimum-area rectangular layout of a given contact graph. We present
O(n)-time algorithms that construct -area rectangular layouts for
general contact graphs and -area rectangular layouts for trees.
(For trees, this is an -approximation algorithm.) We also present an
infinite family of graphs (rsp., trees) that require (rsp.,
) area.
We derive these results by presenting a new characterization of graphs that
admit rectangular layouts using the related concept of {\em rectangular duals}.
A corollary to our results relates the class of graphs that admit rectangular
layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi
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