11 research outputs found
Excluding subdivisions of bounded degree graphs
Let be a fixed graph. What can be said about graphs that have no
subgraph isomorphic to a subdivision of ? Grohe and Marx proved that such
graphs satisfy a certain structure theorem that is not satisfied by graphs
that contain a subdivision of a (larger) graph . Dvo\v{r}\'ak found a
clever strengthening---his structure is not satisfied by graphs that contain a
subdivision of a graph , where has "similar embedding properties" as
. Building upon Dvo\v{r}\'ak's theorem, we prove that said graphs
satisfy a similar structure theorem. Our structure is not satisfied by graphs
that contain a subdivision of a graph that has similar embedding
properties as and has the same maximum degree as . This will be
important in a forthcoming application to well-quasi-ordering
Recommended from our members
Graph Theory
This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results
Packing Topological Minors Half-Integrally
The packing problem and the covering problem are two of the most general
questions in graph theory. The Erd\H{o}s-P\'{o}sa property characterizes the
cases when the optimal solutions of these two problems are bounded by functions
of each other. Robertson and Seymour proved that when packing and covering
-minors for any fixed graph , the planarity of is equivalent with the
Erd\H{o}s-P\'{o}sa property. Thomas conjectured that the planarity is no longer
required if the solution of the packing problem is allowed to be half-integral.
In this paper, we prove that this half-integral version of Erd\H{o}s-P\'{o}sa
property holds with respect to the topological minor containment, which easily
implies Thomas' conjecture. Indeed, we prove an even stronger statement in
which those subdivisions are rooted at any choice of prescribed subsets of
vertices. Precisely, we prove that for every graph , there exists a function
such that for every graph , every sequence of
subsets of and every integer , either there exist subgraphs
of such that every vertex of belongs to at most two
of and each is isomorphic to a subdivision of whose
branch vertex corresponding to belongs to for each , or
there exists a set with size at most intersecting all
subgraphs of isomorphic to a subdivision of whose branch vertex
corresponding to belongs to for each .
Applications of this theorem include generalizations of algorithmic
meta-theorems and structure theorems for -topological minor free (or
-minor free) graphs to graphs that do not half-integrally pack many
-topological minors (or -minors)
A global decomposition theorem for excluding immersions in graphs with no edge-cut of order three
A graph contains another graph as an immersion if can be obtained
from a subgraph of by splitting off edges and removing isolated vertices.
There is an obvious necessary degree condition for the immersion containment:
if contains as an immersion, then for every integer , the number of
vertices of degree at least in is at least the number of vertices of
degree at least in . In this paper, we prove that this obvious necessary
condition is "nearly" sufficient for graphs with no edge-cut of order 3: for
every graph , every -immersion free graph with no edge-cut of order 3 can
be obtained by an edge-sum of graphs, where each of the summands is obtained
from a graph violating the obvious degree condition by adding a bounded number
of edges. The condition for having no edge-cut of order 3 is necessary. A
simple application of this theorem shows that for every graph of maximum
degree , there exists an integer such that for every positive
integer , there are at most unlabelled -edge-connected
-immersion free -edge graphs with no isolated vertex, while there are
superexponentially many unlabelled -edge-connected -immersion free
-edge graphs with no isolated vertex. Our structure theorem will be applied
in a forthcoming paper about determining the clustered chromatic number of the
class of -immersion free graphs