15 research outputs found

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Odd-Minors I: Excluding small parity breaks

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    Given a graph class~C\mathcal{C}, the C\mathcal{C}-blind-treewidth of a graph~GG is the smallest integer~kk such that~GG has a tree-decomposition where every bag whose torso does not belong to~C\mathcal{C} has size at most~kk. In this paper we focus on the class~B\mathcal{B} of bipartite graphs and the class~P\mathcal{P} of planar graphs together with the odd-minor relation. For each of the two parameters, B\mathcal{B}-blind-treewidth and (BP){(\mathcal{B}\cup\mathcal{P})}-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 B\mathcal{B}-blind-treewidth and \textsc{Maximum Cut} on graphs of bounded (BP){(\mathcal{B}\cup\mathcal{P})}-blind-treewidth

    On digraphs without onion star immersions

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    The tt-onion star is the digraph obtained from a star with 2t2t leaves by replacing every edge by a triple of arcs, where in tt triples we orient two arcs away from the center, and in the remaining tt triples we orient two arcs towards the center. Note that the tt-onion star contains, as an immersion, every digraph on tt vertices where each vertex has outdegree at most 22 and indegree at most 11, 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 f ⁣:NNf\colon \mathbb{N}\to \mathbb{N} satisfying the following: If a digraph DD contains a set XX of 2t+12t+1 vertices such that for any x,yXx,y\in X there are f(t)f(t) arc-disjoint paths from xx to yy, then DD contains the tt-onion star as an immersion. There is a function g ⁣:N×NNg\colon \mathbb{N}\times \mathbb{N}\to \mathbb{N} satisfying the following: If xx and yy is a pair of vertices in a digraph DD such that there are at least g(t,k)g(t,k) arc-disjoint paths from xx to yy and there are at least g(t,k)g(t,k) arc-disjoint paths from yy to xx, then either DD contains the tt-onion star as an immersion, or there is a family of 2k2k pairwise arc-disjoint paths with kk paths from xx to yy and kk paths from yy to xx.Comment: 14 pages, 5 figure

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum

    LIPIcs, Volume 248, ISAAC 2022, Complete Volume

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volum

    Chordless Cycle Packing Is Fixed-Parameter Tractable

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    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

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    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

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    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 f(k)f(k) such that every digraph of directed tree-width at least f(k)f(k) contains a cylindrical grid of size kk as a butterfly minor and stated that their proof can be turned into an XP algorithm, with parameter kk, 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 kk. The first one either produces an arboreal decomposition of width 3k23k-2 or finds a haven of order kk in a digraph DD, improving on the original result for arboreal decompositions by Johnson et al. The second algorithm finds a well-linked set of order kk in a digraph DD 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 TT in FPT time with parameter T|T|, a result that we consider to be of its own interest.Comment: 31 pages, 9 figure

    27th Annual European Symposium on Algorithms: ESA 2019, September 9-11, 2019, Munich/Garching, Germany

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