14 research outputs found
Uniqueness and minimal obstructions for tree-depth
A k-ranking of a graph G is a labeling of the vertices of G with values from
{1,...,k} such that any path joining two vertices with the same label contains
a vertex having a higher label. The tree-depth of G is the smallest value of k
for which a k-ranking of G exists. The graph G is k-critical if it has
tree-depth k and every proper minor of G has smaller tree-depth.
We establish partial results in support of two conjectures about the order
and maximum degree of k-critical graphs. As part of these results, we define a
graph G to be 1-unique if for every vertex v in G, there exists an optimal
ranking of G in which v is the unique vertex with label 1. We show that several
classes of k-critical graphs are 1-unique, and we conjecture that the property
holds for all k-critical graphs. Generalizing a previously known construction
for trees, we exhibit an inductive construction that uses 1-unique k-critical
graphs to generate large classes of critical graphs having a given tree-depth.Comment: 14 pages, 4 figure
Minor-Obstructions for Apex-Pseudoforests
A graph is called a pseudoforest if none of its connected components contains
more than one cycle. A graph is an apex-pseudoforest if it can become a
pseudoforest by removing one of its vertices. We identify 33 graphs that form
the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of
all minor-minimal graphs that are not apex-pseudoforests
Forbidding Kuratowski Graphs as Immersions
The immersion relation is a partial ordering relation on graphs that is
weaker than the topological minor relation in the sense that if a graph
contains a graph as a topological minor, then it also contains it as an
immersion but not vice versa. Kuratowski graphs, namely and ,
give a precise characterization of planar graphs when excluded as topological
minors. In this note we give a structural characterization of the graphs that
exclude Kuratowski graphs as immersions. We prove that they can be constructed
by applying consecutive -edge-sums, for , starting from graphs that
are planar sub-cubic or of branch-width at most 10
On the tree-depth and tree-width in heterogeneous random graphs
In this note, we investigate the tree-depth and tree-width in a heterogeneous random graph obtained by including each edge eij (i≠j) of a complete graph Kn over n vertices independently with probability pn(eij). When the sequence of edge probabilities satisfies some density assumptions, we show both tree-depth and tree-width are of linear size with high probability. Moreover, we extend the method to random weighted graphs with non-identical edge weights and capture the conditions under which with high probability the weighted tree-depth is bounded by a constant
Forbidding Kuratowski graphs as immersions
The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph G contains a graph H as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely K 5 and K 3,3 , give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive i-edge-sums, for i ≤ 3, starting from graphs that are planar sub-cubic or of branchwidth at most 10
Efficient fully dynamic elimination forests with applications to detecting long paths and cycles
We present a data structure that in a dynamic graph of treedepth at most ,
which is modified over time by edge insertions and deletions, maintains an
optimum-height elimination forest. The data structure achieves worst-case
update time , which matches the best known parameter
dependency in the running time of a static fpt algorithm for computing the
treedepth of a graph. This improves a result of Dvo\v{r}\'ak et al. [ESA 2014],
who for the same problem achieved update time for some non-elementary
(i.e. tower-exponential) function . As a by-product, we improve known upper
bounds on the sizes of minimal obstructions for having treedepth from
doubly-exponential in to .
As applications, we design new fully dynamic parameterized data structures
for detecting long paths and cycles in general graphs. More precisely, for a
fixed parameter and a dynamic graph , modified over time by edge
insertions and deletions, our data structures maintain answers to the following
queries:
- Does contain a simple path on vertices?
- Does contain a simple cycle on at least vertices?
In the first case, the data structure achieves amortized update time
. In the second case, the amortized update time is . In both cases we assume access to a dictionary
on the edges of .Comment: 74 pages, 5 figure