Computing excluded minors for classes of matroids representable over partial fields

We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids.We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids

On the computability of obstruction sets for well-quasi-ordered graph classes

Στην παρούσα διπλωματική εργασία θα μελετήσουμε αλγόριθμους για τον υπολογισμό συνόλων παρεμπόδησης καλώς μερικώς διατεταγμένων κλάσεων γραφημάτων. Το Θεώρημα Ελασσόνων Γραφημάτων (ΘΕΓ), των Neil Robertson και Paul Seymour (Graph Minor Theorem) εγγυάται πως κάθε κλάση κλειστή ως προς τη σχέση των ελασσόνων έχει πεπερασμένο σύνολο παρεμόδησης. Αν η C είναι μια τέτοια κλάση, τότε το σύνολο παρεμπόδησης της C είναι το ελαχιστικό σύνολο γραφημάτων H έτσι ώστε, ένα γράφημα G ανήκει στην κλάση C αν και μόνο αν κανένα από τα γραφήματα στο σύνολο H δεν περιέχεται ως ελάσσον στο G. Το αντίστοιχο αποτέλεσμα για μια άλλη καλή μερική διάταξη, την σχέση της εμβύθισης, αποδείχθηκε στην ίδια σειρά εργασιών (Graph Minors). Όμως αυτά τα αποτελέσματα είναι μη-κατασκευαστικά: ξέρουμε πως κάθε κλάση κλειστή ως προς ελάσσονα ή εμβυθίσεις έχει πεπερασμένο σύνολο παρεμπόδησης αλλά από αυτά τα αποτελέσματα δεν υποδεικνύουν κάποιο αλγόριθμο για να το υπολογίσουμε. Οι K. Cattell, M. J. Dinneen, R. Downey, M. R. Fellows and M. Langston στην εργασία "On computing graph minor obstruction sets" και οι I. Adler, M. Grohe and S. Kreutzer στην εργασία "Computing Excluded Minors" παρουσιάζουν αλγόριθμους για να ξεπεράσουμε αυτό το πρόβλημα στις κλάσεις γραφημάτων κλειστές ως προς ελάσσονα, καθώς και εφαρμογές των μεθόδων τους, όπως το πρόβλημα της ένωσης. Προσαρμόζοντας τις μεθόδους της δεύτερης από τις προηγουμένες εργασίες σε εμβυθήσεις οι Α. Γιαννοπούλου, Δ. Ζώρος και ο συγγραφέας, ύπο την επίβλεψη του Δ. Μ. Θηλυκού αποδεικνύουν το αντίστοιχο αποτέλεσμα για κλάσεις γραφημάτων κλειστές ως προς εμβύθιση, καθώς και έναν αλγόριθμο για το πρόβλημα της ένωσης για εμβυθίσεις.In this MSc thesis we are going to present algorithms for computing obstruction sets of well--quasi--ordered graph classes. Neil Robertson and Paul Seymour's Graph Minor Theorem (GMT) guarantees that any minor-closed graph class has a finite obstruction set. If C is such a class, the obstruction set of C is the minimal set of graphs H such that G belongs to C if and only if none of the graphs in H is contained as a minor in G. The analogous result for another well-quasi-ordering, the immersion ordering, was shown in the same series of papers (Graph Minors). But these results are non-constructive; we know that a minor or immersion-closed graph class has a finite obstruction set but the GMT does not imply any algorithm for computing it. K. Cattell, M. J. Dinneen, R. Downey, M. R. Fellows and M. Langston in "On computing graph minor obstruction sets" and I. Adler, M. Grohe and S. Kreutzer in "Computing Excluded Minors" present algorithms to overcome this problem for minor-closed graph classes, as well as, applications of their methods proving that the obstruction sets of various graph classes are computable, such as the union problem. By adapting some of the methods of Adler, Grohe and Kreutzer to immersions, the analogue result for immersion obstruction sets and an algorithm for the union problem on immersion-closed graph classes are proven by A. Giannopoulou, D. Zoros and the author, under the supervision of D. M. Thilikos

Minor-Obstructions for Apex-Pseudoforests

A graph is called a pseudoforest if none of its connected components contains more than one cycle. A graph is an apex-pseudoforest if it can become a pseudoforest by removing one of its vertices. We identify 33 graphs that form the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of all minor-minimal graphs that are not apex-pseudoforests

On Supergraphs Satisfying CMSO Properties

Let CMSO denote the counting monadic second order logic of graphs. We give a constructive proof that for some computable function f, there is an algorithm A that takes as input a CMSO sentence F, a positive integer t, and a connected graph G of maximum degree at most D, and determines, in time f(|F|,t)*2^O(D*t)*|G|^O(t), whether G has a supergraph G\u27 of treewidth at most t such that G\u27 satisfies F. The algorithmic metatheorem described above sheds new light on certain unresolved questions within the framework of graph completion algorithms. In particular, using this metatheorem, we provide an explicit algorithm that determines, in time f(d)*2^O(D*d)*|G|^O(d), whether a connected graph of maximum degree D has a planar supergraph of diameter at most d. Additionally, we show that for each fixed k, the problem of determining whether G has a k-outerplanar supergraph of diameter at most d is strongly uniformly fixed parameter tractable with respect to the parameter d. This result can be generalized in two directions. First, the diameter parameter can be replaced by any contraction-closed effectively CMSO-definable parameter p. Examples of such parameters are vertex-cover number, dominating number, and many other contraction-bidimensional parameters. In the second direction, the planarity requirement can be relaxed to bounded genus, and more generally, to bounded local treewidth

Fixed-Parameter Tractable Distances to Sparse Graph Classes

We show that for various classes $\mathcal{C}$ of sparse graphs, and several measures of distance to such classes (such as edit distance and elimination distance), the problem of determining the distance of a given graph $\small{G}$ to $\mathcal{C}$ is fixed-parameter tractable. The results are based on two general techniques. The first of these, building on recent work of Grohe et al. establishes that any class of graphs that is slicewise nowhere dense and slicewise first-order definable is FPT. The second shows that determining the elimination distance of a graph $\small{G}$ to a minor-closed class $\mathcal{C}$ is FPT. We demonstrate that several prior results (of Golovach, Moser and Thilikos and Mathieson) on the fixed-parameter tractability of distance measures are special cases of our first method