Fork-decompositions of matroids

For the abstract of this paper, please see the PDF file

An upgraded Wheels-and-Whirls Theorem for 3-connected matroids

Let M be a 3-connected matroid that is not a wheel or a whirl. In this paper, we prove that M has an element e such that M\e or M/e is 3-connected and has no 3-separation that is not equivalent to one induced by M. © 2011 Elsevier Inc

Διδιαστατότητα: Θεωρία και Αλγοριθμικές Εφαρμογές

Πολλά συνδυαστικά υπολογιστικά προβλήματα είναι στη γενική μορφή τους δύσβατα, υπό την έννοια πως ακόμη και για εισόδους μέτριου μεγέθους, η εύρεση μιας ακριβούς και βέλτιστης λύσης είναι μάλλον ανέφικτη, δεδομένου ότι συνήθως απαιτεί την κλήση αλγορίθμων, των οποίων η χρονική πολυπλοκότητα είναι εκθετική ως προς το μέγεθος του προβλήματος. Συχνά τα προβλήματα αυτά μπορούν να οριστούν σε γραφήματα. Πρόσθετες δομικές ιδιότητες ενός γραφήματος, όπως η εμβαπτισιμότητα σε κάποια επιφάνεια, παρέχουν μια λαβή για το σχεδιασμό αποδοτικότερων αλγορίθμων. Η θεωρία της διδιαστατότητας αναπτύχθηκα στα πλαίσια της Παραμετρικής Πολυπλοκότητας και, βασιζόμενη στα αποτελέσματα της θεωρίας των Ελασσόνων Γραφημάτων, παρέχει ένας μετα-αλγοριθμικό πλαίσιο για την αντιμετώπιση ενός συνόλου προβλημάτων σε πλατύ φάσμα κλάσεων γραφημάτων, πιο συγκεκριμένα σε όλες τις γενικεύσεις γραφημάτων εμβαπτίσιμων σε κάποια επιφάνεια. Στη διδακτορική διατριβή αυτή θεωρούμε ζητήματα συνδυαστικής φύσης σχετικά με την εφαρμογή της θεωρίας της Διδιαστατότητας και τις δυνατότητες επέκτασης του πεδίου εφαρμογής της.Many combinatorial computational problems are considered in their general form intractable, in the sense that even for modest size problems, providing an exact optimal solution is practically infeasible, as it typically involves the use of algorithms whose running time is exponential in the size of the problem. Often these problems can be modeled by graphs. Then, additional structural properties of a graph, such as surface embeddability, can provide a handle for the design of more ecient algorithms. The theory of Bidimensionality, dened in the context of Parameterized Complexity, builds on the celebrated results of Graph Minor theory and establishes a meta algorithmic framework for addressing problems in a broad range of graph classes, namely all generalizations of graphs embeddable on some surface. In this doctoral thesis we explore topics of combinatorial nature related to the implementation of the theory of Bidimensionality and to the possibilities of the extension of its applicability range

FORK-DECOMPOSITIONS OF MATROIDS

One of the central problems in matroid theory is Rota’s conjecture that, for all prime powers q, the class of GF (q)–representable matroids has a finite set of excluded minors. This conjecture has been settled for q ≤ 4 but remains open otherwise. Further progress towards this conjecture has been hindered by the fact that, for all q> 5, there are 3–connected GF (q)–representable matroids having arbitrarily many inequivalent GF (q)–representations. This fact refutes a 1988 conjecture of Kahn that 3–connectivity would be strong enough to ensure an absolute bound on the number of such inequivalent representations. This paper introduces fork-connectivity, a new type of self-dual 4–connectivity, which we conjecture is strong enough to guarantee the existence of such a bound but weak enough to allow for an analogue of Seymour’s Splitter Theorem. We prove that every fork-connected matroid can be reduced to a vertically 4–connected matroid by a sequence of operations that generalize ∆ − Y and Y − ∆ exchanges. It follows from this that the analogue of Kahn’s Conjecture holds for fork-connected matroids if and only if it holds for vertically 4–connected matroids. The class of fork-connected matroids includes the class of 3–connected forked matroids. By taking direct sums and 2–sums of matroids in the latter class, we get the class M of forked matroids, which is closed under duality and minors. The class M is a natural subclass of the class of matroids of branch-width at most 3 and includes the matroids of path-width at most 3. We give a constructive characterization of the members of M and prove that M has finitely many excluded minors

FORK-DECOMPOSITIONS OF MATROIDS

Abstract. One of the central problems in matroid theory is Rota’s conjecture that, for all prime powers q, the class of GF(q)–representable matroids has a finite set of excluded minors. This conjecture has been settled for q ≤ 4 but remains open otherwise. Further progress towards this conjecture has been hindered by the fact that, for all q> 5, there are 3–connected GF(q)–representable matroids having arbitrarily many inequivalent GF(q)–representations. This fact refutes a 1988 conjecture of Kahn that 3–connectivity would be strong enough to ensure an absolute bound on the number of such inequivalent representations. This paper introduces fork-connectivity, a new type of self-dual 4–connectivity, which we conjecture is strong enough to guarantee the existence of such a bound but weak enough to allow for an analogue of Seymour’s Splitter Theorem. We prove that every fork-connected matroid can be reduced to a vertically 4–connected matroid by a sequence of operations that generalize ∆ − Y and Y − ∆ exchanges. It follows from this that the analogue of Kahn’s Conjecture holds for fork-connected matroids if and only if it holds for vertically 4–connected matroids. The class of fork-connected matroids includes the class of 3–connected forked matroids. By taking direct sums and 2–sums of matroids in the latter class, we get the class M of forked matroids, which is closed under duality and minors. The class M is a natural subclass of the class of matroids of branch-width at most 3 and includes the matroids of path-width at most 3. We give a constructive characterization of the members of M and prove that M has finitely many excluded minors. 1