430,400 research outputs found
Metro-Line Crossing Minimization: Hardness, Approximations, and Tractable Cases
Crossing minimization is one of the central problems in graph drawing.
Recently, there has been an increased interest in the problem of minimizing
crossings between paths in drawings of graphs. This is the metro-line crossing
minimization problem (MLCM): Given an embedded graph and a set L of simple
paths, called lines, order the lines on each edge so that the total number of
crossings is minimized. So far, the complexity of MLCM has been an open
problem. In contrast, the problem variant in which line ends must be placed in
outermost position on their edges (MLCM-P) is known to be NP-hard. Our main
results answer two open questions: (i) We show that MLCM is NP-hard. (ii) We
give an -approximation algorithm for MLCM-P
Drawing Planar Graphs with Few Geometric Primitives
We define the \emph{visual complexity} of a plane graph drawing to be the
number of basic geometric objects needed to represent all its edges. In
particular, one object may represent multiple edges (e.g., one needs only one
line segment to draw a path with an arbitrary number of edges). Let denote
the number of vertices of a graph. We show that trees can be drawn with
straight-line segments on a polynomial grid, and with straight-line
segments on a quasi-polynomial grid. Further, we present an algorithm for
drawing planar 3-trees with segments on an
grid. This algorithm can also be used with a small modification to draw maximal
outerplanar graphs with edges on an grid. We also
study the problem of drawing maximal planar graphs with circular arcs and
provide an algorithm to draw such graphs using only arcs. This is
significantly smaller than the lower bound of for line segments for a
nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2017
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