3 research outputs found
Small feedback vertex sets in planar digraphs
Let be a directed planar graph on vertices, with no directed cycle of
length less than . We prove that contains a set of vertices
such that has no directed cycle, and if ,
if , and if . This
improves recent results of Golowich and Rolnick.Comment: 5 pages, 1 figure - v3 final versio
Feedback vertex sets in (directed) graphs of bounded degeneracy or treewidth
We study the minimum size of a feedback vertex set in directed and
undirected -vertex graphs of given degeneracy or treewidth. In the
undirected setting the bound is known to be tight for graphs
with bounded treewidth or bounded odd degeneracy . We show that neither
of the easy upper and lower bounds and can
be exact for the case of even degeneracy. More precisely, for even degeneracy
we prove that , there exists
a -degenerate graph for which .
For directed graphs of bounded degeneracy , we prove that
and that this inequality is strict when is odd. For
directed graphs of bounded treewidth , we show that and for every , there exists a -degenerate graph
for which . Further,
we provide several constructions of low degeneracy or treewidth and large .Comment: 19 pages, 7 figures, 2 table