8 research outputs found
Hitting minors, subdivisions, and immersions in tournaments
The Erd\H{o}s-P\'osa property relates parameters of covering and packing of
combinatorial structures and has been mostly studied in the setting of
undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim,
and Seymour to show that, for every directed graph (resp.
strongly-connected directed graph ), the class of directed graphs that
contain as a strong minor (resp. butterfly minor, topological minor) has
the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove
that if is a strongly-connected directed graph, the class of directed
graphs containing as an immersion has the edge-Erd\H{o}s-P\'osa property in
the class of tournaments.Comment: Accepted to Discrete Mathematics & Theoretical Computer Science.
Difference with the previous version: use of the DMTCS article class. For a
version with hyperlinks see the previous versio
On the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments
We consider the Erd\H{o}s-P\'osa property for immersions and topological
minors in tournaments. We prove that for every simple digraph , , and tournament , the following statements hold:
If in one cannot find arc-disjoint immersion copies of ,
then there exists a set of arcs that intersects all
immersion copies of in .
If in one cannot find vertex-disjoint topological minor
copies of , then there exists a set of vertices
that intersects all topological minor copies of in .
This improves the results of Raymond [DMTCS '18], who proved similar
statements under the assumption that is strongly connected.Comment: 15 pages, 1 figur
Polynomial Kernel for Immersion Hitting in Tournaments
For a fixed simple digraph H without isolated vertices, we consider the problem of deleting arcs from a given tournament to get a digraph which does not contain H as an immersion. We prove that for every H, this problem admits a polynomial kernel when parameterized by the number of deleted arcs. The degree of the bound on the kernel size depends on H
On digraphs without onion star immersions
The -onion star is the digraph obtained from a star with leaves by
replacing every edge by a triple of arcs, where in triples we orient two
arcs away from the center, and in the remaining triples we orient two arcs
towards the center. Note that the -onion star contains, as an immersion,
every digraph on vertices where each vertex has outdegree at most and
indegree at most , or vice versa. We investigate the structure in digraphs
that exclude a fixed onion star as an immersion. The main discovery is that in
such digraphs, for some duality statements true in the undirected setting we
can prove their directed analogues. More specifically, we show the next two
statements.
There is a function satisfying the
following: If a digraph contains a set of vertices such that for
any there are arc-disjoint paths from to , then
contains the -onion star as an immersion.
There is a function
satisfying the following: If and is a pair of vertices in a digraph
such that there are at least arc-disjoint paths from to and
there are at least arc-disjoint paths from to , then either
contains the -onion star as an immersion, or there is a family of
pairwise arc-disjoint paths with paths from to and paths from
to .Comment: 14 pages, 5 figure
Recommended from our members
Graph Theory
This was a workshop on graph theory, with a comprehensive approach. Highlights included the emerging theories of sparse graph limits and of infinite matroids, new techniques for colouring graphs on surfaces, and extensions of graph minor theory to directed graphs and to immersion
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
Hitting minors, subdivisions, and immersions in tournaments
The Erd\H{o}s-P\'osa property relates parameters of covering and packing of
combinatorial structures and has been mostly studied in the setting of
undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim,
and Seymour to show that, for every directed graph (resp.
strongly-connected directed graph ), the class of directed graphs that
contain as a strong minor (resp. butterfly minor, topological minor) has
the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove
that if is a strongly-connected directed graph, the class of directed
graphs containing as an immersion has the edge-Erd\H{o}s-P\'osa property in
the class of tournaments
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum