5,428 research outputs found
-Blocks: a connectivity invariant for graphs
A -block in a graph is a maximal set of at least vertices no two
of which can be separated in by fewer than other vertices. The block
number of is the largest integer such that has a
-block.
We investigate how interacts with density invariants of graphs, such
as their minimum or average degree. We further present algorithms that decide
whether a graph has a -block, or which find all its -blocks.
The connectivity invariant has a dual width invariant, the
block-width of . Our algorithms imply the duality theorem
: a graph has a block-decomposition of width and adhesion if and only if it contains no -block.Comment: 22 pages, 5 figures. This is an extended version the journal article,
which has by now appeared. The version here contains an improved version of
Theorem 5.3 (which is now best possible) and an additional section with
examples at the en
A Logic of Reachable Patterns in Linked Data-Structures
We define a new decidable logic for expressing and checking invariants of
programs that manipulate dynamically-allocated objects via pointers and
destructive pointer updates. The main feature of this logic is the ability to
limit the neighborhood of a node that is reachable via a regular expression
from a designated node. The logic is closed under boolean operations
(entailment, negation) and has a finite model property. The key technical
result is the proof of decidability. We show how to express precondition,
postconditions, and loop invariants for some interesting programs. It is also
possible to express properties such as disjointness of data-structures, and
low-level heap mutations. Moreover, our logic can express properties of
arbitrary data-structures and of an arbitrary number of pointer fields. The
latter provides a way to naturally specify postconditions that relate the
fields on entry to a procedure to the fields on exit. Therefore, it is possible
to use the logic to automatically prove partial correctness of programs
performing low-level heap mutations
On the Generalised Colouring Numbers of Graphs that Exclude a Fixed Minor
The generalised colouring numbers and
were introduced by Kierstead and Yang as a generalisation
of the usual colouring number, and have since then found important theoretical
and algorithmic applications. In this paper, we dramatically improve upon the
known upper bounds for generalised colouring numbers for graphs excluding a
fixed minor, from the exponential bounds of Grohe et al. to a linear bound for
the -colouring number and a polynomial bound for the weak
-colouring number . In particular, we show that if
excludes as a minor, for some fixed , then
and
.
In the case of graphs of bounded genus , we improve the bounds to
(and even if
, i.e. if is planar) and
.Comment: 21 pages, to appear in European Journal of Combinatoric
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
Turaev genus, knot signature, and the knot homology concordance invariants
We give bounds on knot signature, the Ozsvath-Szabo tau invariant, and the
Rasmussen s invariant in terms of the Turaev genus of the knot.Comment: 15 pages, 5 figure
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