12 research outputs found
Clustered Colouring in Minor-Closed Classes
The "clustered chromatic number" of a class of graphs is the minimum integer
such that for some integer every graph in the class is -colourable
with monochromatic components of size at most . We prove that for every
graph , the clustered chromatic number of the class of -minor-free graphs
is tied to the tree-depth of . In particular, if is connected with
tree-depth then every -minor-free graph is -colourable with
monochromatic components of size at most . This provides the first
evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about
defective colouring of -minor-free graphs. If then we prove that 4
colours suffice, which is best possible. We also determine those minor-closed
graph classes with clustered chromatic number 2. Finally, we develop a
conjecture for the clustered chromatic number of an arbitrary minor-closed
class
Contraction-Bidimensionality of Geometric Intersection Graphs
Given a graph G, we define bcg(G) as the minimum k for which G can be contracted to the uniformly triangulated grid Gamma_k. A graph class G has the SQGC property if every graph G in G has treewidth O(bcg(G)c) for some 1 <= c < 2. The SQGC property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a general family of graph classes that satisfy the SQGC property and includes bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for several intersection graph classes of 2-dimensional geometrical objects
Finding detours is fixed-parameter tractable
We consider the following natural "above guarantee" parameterization of the
classical Longest Path problem: For given vertices s and t of a graph G, and an
integer k, the problem Longest Detour asks for an (s,t)-path in G that is at
least k longer than a shortest (s,t)-path. Using insights into structural graph
theory, we prove that Longest Detour is fixed-parameter tractable (FPT) on
undirected graphs and actually even admits a single-exponential algorithm, that
is, one of running time exp(O(k)) poly(n). This matches (up to the base of the
exponential) the best algorithms for finding a path of length at least k.
Furthermore, we study the related problem Exact Detour that asks whether a
graph G contains an (s,t)-path that is exactly k longer than a shortest
(s,t)-path. For this problem, we obtain a randomized algorithm with running
time about 2.746^k, and a deterministic algorithm with running time about
6.745^k, showing that this problem is FPT as well. Our algorithms for Exact
Detour apply to both undirected and directed graphs.Comment: Extended abstract appears at ICALP 201
Linear bounds on treewidth in terms of excluded planar minors
One of the fundamental results in graph minor theory is that for every planar
graph , there is a minimum integer such that graphs with no minor
isomorphic to have treewidth at most . A lower bound for can
be obtained by considering the maximum integer such that contains
vertex-disjoint cycles. There exists a graph of treewidth
which does not contain vertex-disjoint cycles, from which it follows that
. In particular, if is linear in
for graphs from a subclass of planar graphs, it is
necessary that -vertex graphs from the class contain at most
vertex-disjoint cycles. We ask whether this is also a
sufficient condition, and demonstrate that this is true for classes of planar
graphs with bounded component size. For an -vertex graph which is a
disjoint union of cycles, we show that ,
and improve this to when . In particular
this bound is linear when . We present a linear bound
for when is a subdivision of an -edge planar graph for any
constant . We also improve the best known bounds for when is
the wheel graph or the grid, obtaining a bound of for the
latter.Comment: 18 page
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