37 research outputs found
2-Layer Graph Drawings with Bounded Pathwidth
We determine which properties of 2-layer drawings characterise bipartite
graphs of bounded pathwidth
Cubic Planar Graphs That Cannot Be Drawn On Few Lines
For every integer l, we construct a cubic 3-vertex-connected planar bipartite graph G with O(l^3) vertices such that there is no planar straight-line drawing of G whose vertices all lie on l lines. This strengthens previous results on graphs that cannot be drawn on few lines, which constructed significantly larger maximal planar graphs. We also find apex-trees and cubic bipartite series-parallel graphs that cannot be drawn on a bounded number of lines
Grid Recognition: Classical and Parameterized Computational Perspectives
Grid graphs, and, more generally, grid graphs, form one of the
most basic classes of geometric graphs. Over the past few decades, a large body
of works studied the (in)tractability of various computational problems on grid
graphs, which often yield substantially faster algorithms than general graphs.
Unfortunately, the recognition of a grid graph is particularly hard -- it was
shown to be NP-hard even on trees of pathwidth 3 already in 1987. Yet, in this
paper, we provide several positive results in this regard in the framework of
parameterized complexity (additionally, we present new and complementary
hardness results). Specifically, our contribution is threefold. First, we show
that the problem is fixed-parameter tractable (FPT) parameterized by where is the maximum size of a connected component of
. This also implies that the problem is FPT parameterized by
where is the treedepth of (to be compared with the hardness
for pathwidth 2 where ). Further, we derive as a corollary that strip
packing is FPT with respect to the height of the strip plus the maximum of the
dimensions of the packed rectangles, which was previously only known to be in
XP. Second, we present a new parameterization, denoted , relating graph
distance to geometric distance, which may be of independent interest. We show
that the problem is para-NP-hard parameterized by , but FPT parameterized
by on trees, as well as FPT parameterized by . Third, we show that
the recognition of grid graphs is NP-hard on graphs of pathwidth 2
where . Moreover, when and are unrestricted, we show that the
problem is NP-hard on trees of pathwidth 2, but trivially solvable in
polynomial time on graphs of pathwidth 1