1,458 research outputs found
On the Parameterized Complexity of Bend-Minimum Orthogonal Planarity
Computing planar orthogonal drawings with the minimum number of bends is one
of the most relevant topics in Graph Drawing. The problem is known to be
NP-hard, even when we want to test the existence of a rectilinear planar
drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia, 2001).
From the parameterized complexity perspective, the problem is fixed-parameter
tractable when parameterized by the sum of three parameters: the number of
bends, the number of vertices of degree at most two, and the treewidth of the
input graph (Di Giacomo et al., 2022). We improve this last result by showing
that the problem remains fixed-parameter tractable when parameterized only by
the number of vertices of degree at most two plus the number of bends. As a
consequence, rectilinear planarity testing lies in \FPT~parameterized by the
number of vertices of degree at most two.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
Orthogonal Graph Drawing with Inflexible Edges
We consider the problem of creating plane orthogonal drawings of 4-planar
graphs (planar graphs with maximum degree 4) with constraints on the number of
bends per edge. More precisely, we have a flexibility function assigning to
each edge a natural number , its flexibility. The problem
FlexDraw asks whether there exists an orthogonal drawing such that each edge
has at most bends. It is known that FlexDraw is NP-hard
if for every edge . On the other hand, FlexDraw can
be solved efficiently if and is trivial if
for every edge .
To close the gap between the NP-hardness for and the
efficient algorithm for , we investigate the
computational complexity of FlexDraw in case only few edges are inflexible
(i.e., have flexibility~). We show that for any FlexDraw
is NP-complete for instances with inflexible edges with
pairwise distance (including the case where they
induce a matching). On the other hand, we give an FPT-algorithm with running
time , where
is the time necessary to compute a maximum flow in a planar flow network with
multiple sources and sinks, and is the number of inflexible edges having at
least one endpoint of degree 4.Comment: 23 pages, 5 figure
Transforming planar graph drawings while maintaining height
There are numerous styles of planar graph drawings, notably straight-line
drawings, poly-line drawings, orthogonal graph drawings and visibility
representations. In this note, we show that many of these drawings can be
transformed from one style to another without changing the height of the
drawing. We then give some applications of these transformations
Planar Octilinear Drawings with One Bend Per Edge
In octilinear drawings of planar graphs, every edge is drawn as an
alternating sequence of horizontal, vertical and diagonal ()
line-segments. In this paper, we study octilinear drawings of low edge
complexity, i.e., with few bends per edge. A -planar graph is a planar graph
in which each vertex has degree less or equal to . In particular, we prove
that every 4-planar graph admits a planar octilinear drawing with at most one
bend per edge on an integer grid of size . For 5-planar
graphs, we prove that one bend per edge still suffices in order to construct
planar octilinear drawings, but in super-polynomial area. However, for 6-planar
graphs we give a class of graphs whose planar octilinear drawings require at
least two bends per edge
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