16 research outputs found
Packing and covering immersion models of planar subcubic graphs
A graph is an immersion of a graph if can be obtained by some
sugraph after lifting incident edges. We prove that there is a polynomial
function , such that if is a
connected planar subcubic graph on edges, is a graph, and is a
non-negative integer, then either contains vertex/edge-disjoint
subgraphs, each containing as an immersion, or contains a set of
vertices/edges such that does not contain as an
immersion
Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes
Suppose F is a finite family of graphs. We consider the following meta-problem, called F-Immersion Deletion: given a graph G and an integer k, decide whether the deletion of at most k edges of G can result in a graph that does not contain any graph from F as an immersion. This problem is a close relative of the F-Minor Deletion problem studied by Fomin et al. [FOCS 2012], where one deletes vertices in order to remove all minor models of graphs from F.
We prove that whenever all graphs from F are connected and at least one graph of F is planar and subcubic, then the F-Immersion Deletion problem admits:
- a constant-factor approximation algorithm running in time O(m^3 n^3 log m)
- a linear kernel that can be computed in time O(m^4 n^3 log m) and
- a O(2^{O(k)} + m^4 n^3 log m)-time fixed-parameter algorithm,
where n,m count the vertices and edges of the input graph. Our findings mirror those of Fomin et al. [FOCS 2012], who obtained similar results for F-Minor Deletion, under the assumption that at least one graph from F is planar.
An important difference is that we are able to obtain a linear kernel for F-Immersion Deletion, while the exponent of the kernel of Fomin et al. depends heavily on the family F. In fact, this dependence is unavoidable under plausible complexity assumptions, as proven by Giannopoulou et al. [ICALP 2015]. This reveals that the kernelization complexity of F-Immersion Deletion is quite different than that of F-Minor Deletion
Tree-Partitions with Small Bounded Degree Trees
A "tree-partition" of a graph is a partition of such that
identifying the vertices in each part gives a tree. It is known that every
graph with treewidth and maximum degree has a tree-partition with
parts of size . We prove the same result with the extra property
that the underlying tree has maximum degree and
vertices
Cutwidth: obstructions and algorithmic aspects
Cutwidth is one of the classic layout parameters for graphs. It measures how
well one can order the vertices of a graph in a linear manner, so that the
maximum number of edges between any prefix and its complement suffix is
minimized. As graphs of cutwidth at most are closed under taking
immersions, the results of Robertson and Seymour imply that there is a finite
list of minimal immersion obstructions for admitting a cut layout of width at
most . We prove that every minimal immersion obstruction for cutwidth at
most has size at most .
As an interesting algorithmic byproduct, we design a new fixed-parameter
algorithm for computing the cutwidth of a graph that runs in time , where is the optimum width and is the number of vertices.
While being slower by a -factor in the exponent than the fastest known
algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear
time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth
II: Algorithms for partial -trees of bounded degree, J. Algorithms,
56(1):25--49, 2005], our algorithm has the advantage of being simpler and
self-contained; arguably, it explains better the combinatorics of optimum-width
layouts
Product structure of graph classes with strongly sublinear separators
We investigate the product structure of hereditary graph classes admitting
strongly sublinear separators. We characterise such classes as subgraphs of the
strong product of a star and a complete graph of strongly sublinear size. In a
more precise result, we show that if any hereditary graph class
admits separators, then for any fixed
every -vertex graph in is a subgraph
of the strong product of a graph with bounded tree-depth and a complete
graph of size . This result holds with if
we allow to have tree-depth . Moreover, using extensions of
classical isoperimetric inequalties for grids graphs, we show the dependence on
in our results and the above bound are
both best possible. We prove that -vertex graphs of bounded treewidth are
subgraphs of the product of a graph with tree-depth and a complete graph of
size , which is best possible. Finally, we investigate the
conjecture that for any hereditary graph class that admits
separators, every -vertex graph in is a
subgraph of the strong product of a graph with bounded tree-width and a
complete graph of size . We prove this for various classes
of interest.Comment: v2: added bad news subsection; v3: removed section "Polynomial
Expansion Classes" which had an error, added section "Lower Bounds", and
added a new author; v4: minor revisions and corrections
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 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum