15 research outputs found
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Odd-Minors I: Excluding small parity breaks
Given a graph class~, the -blind-treewidth of a
graph~ is the smallest integer~ such that~ has a tree-decomposition
where every bag whose torso does not belong to~ has size at
most~. In this paper we focus on the class~ of bipartite graphs
and the class~ of planar graphs together with the odd-minor
relation. For each of the two parameters, -blind-treewidth and
-blind-treewidth, we prove an analogue of the
celebrated Grid Theorem under the odd-minor relation. As a consequence we
obtain FPT-approximation algorithms for both parameters. We then provide
FPT-algorithms for \textsc{Maximum Independent Set} on graphs of bounded
-blind-treewidth and \textsc{Maximum Cut} on graphs of bounded
-blind-treewidth
On digraphs without onion star immersions
The -onion star is the digraph obtained from a star with leaves by
replacing every edge by a triple of arcs, where in triples we orient two
arcs away from the center, and in the remaining triples we orient two arcs
towards the center. Note that the -onion star contains, as an immersion,
every digraph on vertices where each vertex has outdegree at most and
indegree at most , or vice versa. We investigate the structure in digraphs
that exclude a fixed onion star as an immersion. The main discovery is that in
such digraphs, for some duality statements true in the undirected setting we
can prove their directed analogues. More specifically, we show the next two
statements.
There is a function satisfying the
following: If a digraph contains a set of vertices such that for
any there are arc-disjoint paths from to , then
contains the -onion star as an immersion.
There is a function
satisfying the following: If and is a pair of vertices in a digraph
such that there are at least arc-disjoint paths from to and
there are at least arc-disjoint paths from to , then either
contains the -onion star as an immersion, or there is a family of
pairwise arc-disjoint paths with paths from to and paths from
to .Comment: 14 pages, 5 figure
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Chordless Cycle Packing Is Fixed-Parameter Tractable
A chordless cycle or hole in a graph G is an induced cycle of length at least 4. In the Hole Packing problem, a graph G and an integer k is given, and the task is to find (if exists) a set of k pairwise vertex-disjoint chordless cycles. Our main result is showing that Hole Packing is fixed-parameter tractable (FPT), that is, can be solved in time f(k)n^O(1) for some function f depending only on k
Structural and Topological Graph Theory and Well-Quasi-Ordering
Στη σειρά εργασιών Ελασσόνων Γραφημάτων, οι Neil Robertson και Paul Seymour
μεταξύ άλλων σπουδαίων αποτελεσμάτων, απέδειξαν την εικασία του Wagner που σήμερα
είναι γνωστή ως το Θεώρημα των Robertson και Seymour.
Σε κάθε τους βήμα προς την συναγωγή της τελικής απόδειξης
της εικασίας, κάθε ειδική περίπτωση αυτής που αποδείκνυαν ήταν συνέπεια ενός "δομικού θεωρήματος"
το οποίο σε γενικές γραμμές ισχυριζόταν ότι ικανοποιητικά γενικά γραφήματα περιέχουν ως ελάσσονα γραφήματα
ή άλλες δομές που είναι χρήσιμα για την απόδειξη, ή ισοδύναμα, ότι η δομή των
γραφημάτων τα οποία δεν περιέχουν ένα χρήσιμο για την απόδειξη γράφημα ως έλασσον
είναι κατά κάποιο τρόπο περιορισμένη συνάγοντας έτσι και πάλι μια χρήσιμη πληροφορία για την απόδειξη.
Στην παρούσα εργασία, παρουσιάζουμε -σχετικά μικρές- αποδείξεις διαφόρων ειδικών περιπτώσεων του Θεωρήματος των Robertson και Seymour,
αναδεικνύοντας με αυτό τον τρόπο την αλληλεπίδραση της δομικής θεωρίας γραφημάτων με την θεωρία των
καλών-σχεδόν-διατάξεων.
Παρουσιάζουμε ακόμα την ίσως πιο ενδιαφέρουσα ειδική περίπτωση του Θεωρήματος των Robertson και Seymour,
η οποία ισχυρίζεται ότι η εμβαπτισιμότητα
σε κάθε συγκεκριμένη επιφάνεια δύναται να χαρακτηριστεί μέσω της απαγόρευσης πεπερασμένων το πλήθος γραφημάτων
ως ελάσσονα. Το τελευταίο αποτέλεσμα συνάγεται ως ένα αποτέλεσμα της θεωρίας των καλών-σχεδόν-διατάξεων
αναδεικνύοντας με αυτό τον τρόπο την αλληλεπίδρασή της με την τοπολογική θεωρία γραφημάτων. Τέλος, σταχυολογούμε
αποτελέσματα αναφορικά με την καλή-σχεδόν-διάταξη κλάσεων γραφημάτων από άλλες -πέραν της
σχέσης έλασσον- σχέσεις γραφημάτων.In their Graph Minors series, Neil Robertson and Paul Seymour among other great results
proved Wagner's conjecture which is today known as the Robertson and Seymour's theorem.
In every step along their way to the final proof, each special case of the conjecture which they were proving
was a consequence of a "structure theorem", that sufficiently general graphs contain
minors or other sub-objects that are useful for the proof - or equivalently,
that graphs that do not contain a useful minor have a certain restricted structure, deducing that way also a useful information for the proof.
The main object of this thesis is the presentation of -relatively short-
proofs of several Robertson and Seymour's theorem's special cases, illustrating by this way the interplay between
structural graph theory and graphs' well-quasi-ordering.
We present also the proof of the perhaps most important special case of the Robertson and Seymour's theorem
which states that embeddability in any fixed surface can be characterized by forbidding finitely many minors.
The later result is deduced as a well-quasi-ordering result,
indicating by this way the interplay among topological graph theory and well-quasi-ordering theory.
Finally, we survey results regarding the well-quasi-ordering of graphs by other than the minor graphs' relations
Adapting the Directed Grid Theorem into an FPT Algorithm
The Grid Theorem of Robertson and Seymour [JCTB, 1986], is one of the most
important tools in the field of structural graph theory, finding numerous
applications in the design of algorithms for undirected graphs. An analogous
version of the Grid Theorem in digraphs was conjectured by Johnson et al.
[JCTB, 2001], and proved by Kawarabayashi and Kreutzer [STOC, 2015]. Namely,
they showed that there is a function such that every digraph of directed
tree-width at least contains a cylindrical grid of size as a
butterfly minor and stated that their proof can be turned into an XP algorithm,
with parameter , that either constructs a decomposition of the appropriate
width, or finds the claimed large cylindrical grid as a butterfly minor. In
this paper, we adapt some of the steps of the proof of Kawarabayashi and
Kreutzer to improve this XP algorithm into an FPT algorithm. Towards this, our
main technical contributions are two FPT algorithms with parameter . The
first one either produces an arboreal decomposition of width or finds a
haven of order in a digraph , improving on the original result for
arboreal decompositions by Johnson et al. The second algorithm finds a
well-linked set of order in a digraph of large directed tree-width. As
tools to prove these results, we show how to solve a generalized version of the
problem of finding balanced separators for a given set of vertices in FPT
time with parameter , a result that we consider to be of its own interest.Comment: 31 pages, 9 figure