9 research outputs found
Packing Directed Cycles Quarter- and Half-Integrally
The celebrated Erd\H{o}s-P\'osa theorem states that every undirected graph
that does not admit a family of vertex-disjoint cycles contains a feedback
vertex set (a set of vertices hitting all cycles in the graph) of size . After being known for long as Younger's conjecture, a similar
statement for directed graphs has been proven in 1996 by Reed, Robertson,
Seymour, and Thomas. However, in their proof, the dependency of the size of the
feedback vertex set on the size of vertex-disjoint cycle packing is not
elementary.
We show that if we compare the size of a minimum feedback vertex set in a
directed graph with the quarter-integral cycle packing number, we obtain a
polynomial bound. More precisely, we show that if in a directed graph there
is no family of cycles such that every vertex of is in at most four of
the cycles, then there exists a feedback vertex set in of size .
Furthermore, a variant of our proof shows that if in a directed graph there
is no family of cycles such that every vertex of is in at most two of
the cycles, then there exists a feedback vertex set in of size .
On the way there we prove a more general result about quarter-integral
packing of subgraphs of high directed treewidth: for every pair of positive
integers and , if a directed graph has directed treewidth
, then one can find in a family of
subgraphs, each of directed treewidth at least , such that every vertex of
is in at most four subgraphs.Comment: Accepted to European Symposium on Algorithms (ESA '19
Product structure of graph classes with bounded treewidth
We show that many graphs with bounded treewidth can be described as subgraphs
of the strong product of a graph with smaller treewidth and a bounded-size
complete graph. To this end, define the "underlying treewidth" of a graph class
to be the minimum non-negative integer such that, for some
function , for every graph there is a graph with
such that is isomorphic to a subgraph of . We introduce disjointed coverings of graphs
and show they determine the underlying treewidth of any graph class. Using this
result, we prove that the class of planar graphs has underlying treewidth 3;
the class of -minor-free graphs has underlying treewidth (for ); and the class of -minor-free graphs has underlying
treewidth . In general, we prove that a monotone class has bounded
underlying treewidth if and only if it excludes some fixed topological minor.
We also study the underlying treewidth of graph classes defined by an excluded
subgraph or excluded induced subgraph. We show that the class of graphs with no
subgraph has bounded underlying treewidth if and only if every component of
is a subdivided star, and that the class of graphs with no induced
subgraph has bounded underlying treewidth if and only if every component of
is a star
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum