74 research outputs found
Directed Width Parameters and Circumference of Digraphs
We prove that the directed treewidth, DAG-width and Kelly-width of a digraph
are bounded above by its circumference plus one
Connection Matrices and the Definability of Graph Parameters
In this paper we extend and prove in detail the Finite Rank Theorem for
connection matrices of graph parameters definable in Monadic Second Order Logic
with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and
J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying
known and new non-definability results of graph properties and finding new
non-definability results for graph parameters. We also prove a Feferman-Vaught
Theorem for the logic CFOL, First Order Logic with the modular counting
quantifiers
Connection Matrices and the Definability of Graph Parameters
In this paper we extend the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with modular counting CMSOL of B. Godlin, T. Kotek and J.A. Makowsky (2008 and 2009), and demonstrate its vast applicability in simplifying known and new non-definability results of graph properties and finding new non-definability results for graph parameters. We also prove a Feferman-Vaught Theorem for the logic CFOL, First Order Logic with the modular counting quantifiers
Turing Kernelization for Finding Long Paths in Graphs Excluding a Topological Minor
The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether k-PATH admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length k, using an oracle that answers queries of size k^{O(1)}?
We show this can be done when the input graph avoids a fixed graph H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and K_{3,t}-minor-free graphs. Moreover, we show that k-PATH even admits a polynomial Turing kernel when the input graph is not H-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H-topological-minor-free graph that does not contain a k-path has a separation that can safely be reduced after communication with the oracle
Parameterized Algorithms for Generalizations of Directed Feedback Vertex Set
The Directed Feedback Vertex Set (DFVS) problem takes as input a directed
graph~ and seeks a smallest vertex set~ that hits all cycles in . This
is one of Karp's 21 -complete problems. Resolving the
parameterized complexity status of DFVS was a long-standing open problem until
Chen et al. [STOC 2008, J. ACM 2008] showed its fixed-parameter tractability
via a -time algorithm, where .
Here we show fixed-parameter tractability of two generalizations of DFVS:
- Find a smallest vertex set such that every strong component of
has size at most~: we give an algorithm solving this problem in time
. This generalizes an algorithm by Xiao
[JCSS 2017] for the undirected version of the problem.
- Find a smallest vertex set such that every non-trivial strong component
of is 1-out-regular: we give an algorithm solving this problem in time
.
We also solve the corresponding arc versions of these problems by
fixed-parameter algorithms
Recommended from our members
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
- …