779 research outputs found
Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth
We give a fixed-parameter tractable algorithm that, given a parameter and
two graphs , either concludes that one of these graphs has treewidth
at least , or determines whether and are isomorphic. The running
time of the algorithm on an -vertex graph is ,
and this is the first fixed-parameter algorithm for Graph Isomorphism
parameterized by treewidth.
Our algorithm in fact solves the more general canonization problem. We namely
design a procedure working in time that, for a
given graph on vertices, either concludes that the treewidth of is
at least , or: * finds in an isomorphic-invariant way a graph
that is isomorphic to ; * finds an isomorphism-invariant
construction term --- an algebraic expression that encodes together with a
tree decomposition of of width .
Hence, the isomorphism test reduces to verifying whether the computed
isomorphic copies or the construction terms for and are equal.Comment: Full version of a paper presented at FOCS 201
Minors for alternating dimaps
We develop a theory of minors for alternating dimaps --- orientably embedded
digraphs where, at each vertex, the incident edges (taken in the order given by
the embedding) are directed alternately into, and out of, the vertex. We show
that they are related by the triality relation of Tutte. They do not commute in
general, though do in many circumstances, and we characterise the situations
where they do. The relationship with triality is reminiscent of similar
relationships for binary functions, due to the author, so we characterise those
alternating dimaps which correspond to binary functions. We give a
characterisation of alternating dimaps of at most a given genus, using a finite
set of excluded minors. We also use the minor operations to define simple Tutte
invariants for alternating dimaps and characterise them. We establish a
connection with the Tutte polynomial, and pose the problem of characterising
universal Tutte-like invariants for alternating dimaps based on these minor
operations.Comment: 51 pages, 7 figure
Grad and Classes with Bounded Expansion II. Algorithmic Aspects
Classes of graphs with bounded expansion are a generalization of both proper
minor closed classes and degree bounded classes. Such classes are based on a
new invariant, the greatest reduced average density (grad) of G with rank r,
∇r(G). These classes are also characterized by the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. These results lead to several new linear time algorithms, such as an
algorithm for counting all the isomorphs of a fixed graph in an input graph or
an algorithm for checking whether there exists a subset of vertices of a priori
bounded size such that the subgraph induced by this subset satisfies some
arbirtrary but fixed first order sentence. We also show that for fixed p,
computing the distances between two vertices up to distance p may be performed
in constant time per query after a linear time preprocessing. We also show,
extending several earlier results, that a class of graphs has sublinear
separators if it has sub-exponential expansion. This result result is best
possible in general
Linear kernels for outbranching problems in sparse digraphs
In the -Leaf Out-Branching and -Internal Out-Branching problems we are
given a directed graph with a designated root and a nonnegative integer
. The question is to determine the existence of an outbranching rooted at
that has at least leaves, or at least internal vertices,
respectively. Both these problems were intensively studied from the points of
view of parameterized complexity and kernelization, and in particular for both
of them kernels with vertices are known on general graphs. In this
work we show that -Leaf Out-Branching admits a kernel with vertices
on -minor-free graphs, for any fixed family of graphs
, whereas -Internal Out-Branching admits a kernel with
vertices on any graph class of bounded expansion.Comment: Extended abstract accepted for IPEC'15, 27 page
- …