84 research outputs found
On the size of planarly connected crossing graphs
We prove that if an -vertex graph can be drawn in the plane such that
each pair of crossing edges is independent and there is a crossing-free edge
that connects their endpoints, then has edges. Graphs that admit
such drawings are related to quasi-planar graphs and to maximal -planar and
fan-planar graphs.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Planar Drawings of Fixed-Mobile Bigraphs
A fixed-mobile bigraph G is a bipartite graph such that the vertices of one
partition set are given with fixed positions in the plane and the mobile
vertices of the other part, together with the edges, must be added to the
drawing. We assume that G is planar and study the problem of finding, for a
given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In
the most general case, we show NP-hardness. For k=0 and under additional
constraints on the positions of the fixed or mobile vertices, we either prove
that the problem is polynomial-time solvable or prove that it belongs to NP.
Finally, we present a polynomial-time testing algorithm for a certain type of
"layered" 1-bend drawings
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201
NodeTrix Planarity Testing with Small Clusters
We study the NodeTrix planarity testing problem for flat clustered graphs
when the maximum size of each cluster is bounded by a constant . We consider
both the case when the sides of the matrices to which the edges are incident
are fixed and the case when they can be chosen arbitrarily. We show that
NodeTrix planarity testing with fixed sides can be solved in
time for every flat clustered graph that can be
reduced to a partial 2-tree by collapsing its clusters into single vertices. In
the general case, NodeTrix planarity testing with fixed sides can be solved in
time for , but it is NP-complete for any . NodeTrix
planarity testing remains NP-complete also in the free sides model when .Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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