Surface cubications mod flips

Let $\Sigma$ be a compact surface. We prove that the set of surface cubications modulo flips, up to isotopy, is in one-to-one correspondence with $\Z/2\Z\oplus H_1(\Sigma,\Z/2\Z)$.Comment: revised version, 18

Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes

Suppose F is a finite family of graphs. We consider the following meta-problem, called F-Immersion Deletion: given a graph G and an integer k, decide whether the deletion of at most k edges of G can result in a graph that does not contain any graph from F as an immersion. This problem is a close relative of the F-Minor Deletion problem studied by Fomin et al. [FOCS 2012], where one deletes vertices in order to remove all minor models of graphs from F. We prove that whenever all graphs from F are connected and at least one graph of F is planar and subcubic, then the F-Immersion Deletion problem admits: - a constant-factor approximation algorithm running in time O(m^3 n^3 log m) - a linear kernel that can be computed in time O(m^4 n^3 log m) and - a O(2^{O(k)} + m^4 n^3 log m)-time fixed-parameter algorithm, where n,m count the vertices and edges of the input graph. Our findings mirror those of Fomin et al. [FOCS 2012], who obtained similar results for F-Minor Deletion, under the assumption that at least one graph from F is planar. An important difference is that we are able to obtain a linear kernel for F-Immersion Deletion, while the exponent of the kernel of Fomin et al. depends heavily on the family F. In fact, this dependence is unavoidable under plausible complexity assumptions, as proven by Giannopoulou et al. [ICALP 2015]. This reveals that the kernelization complexity of F-Immersion Deletion is quite different than that of F-Minor Deletion

Forbidding Kuratowski Graphs as Immersions

The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph $G$ contains a graph $H$ as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely $K_{5}$ and $K_{3,3}$, give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive $i$-edge-sums, for $i\leq 3$, starting from graphs that are planar sub-cubic or of branch-width at most 10

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

Forbidding Kuratowski graphs as immersions

The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph G contains a graph H as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely K 5 and K 3,3 , give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive i-edge-sums, for i ≤ 3, starting from graphs that are planar sub-cubic or of branchwidth at most 10

On the cohomology of reciprocity sheaves

In this paper we show the existence of an action of Chow correspondences on the cohomology of reciprocity sheaves. In order to do so, we prove a number of structural results, such as a projective bundle formula, a blow-up formula, a Gysin sequence, and the existence of proper pushforward. In this way we recover and generalize analogous statements for the cohomology of Hodge sheaves and Hodge-Witt sheaves. Among the applications, we construct new birational invariants of smooth projective varieties and obstructions to the existence of zero-cycles of degree one from the cohomology of reciprocity sheaves.Comment: 116 pages. Theorem 0.3 reformulated with an easier condition to verify; Added new application Cor. 11.19. New reference

A survey of parameterized algorithms and the complexity of edge modification

The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio