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

Decompositions and Algorithms for the Disjoint Paths Problem in Planar Graphs

Στο πρόβλημα των Διακεκριμενων Μονοπατιων μας ζητείται να εξετάσουμε, δεδομένου ενός γραφήματος G και ενος συνόλου k ζευγών τερματικών,αν τα ζεύγη των τερματικών μπορούν να συνδεθούν με διακεκριμένα μονοπάτια. Στα "Graph Minors", μια σειρά 23 εργασιών μεταξύ 1984 και 2011, οι Neil Robertson και Paul D. Seymour, ανάμεσα σε άλλα σπουδαία αποτελέσματα που επηρέασαν βαθιά την Θεωρία Γραφημάτων, παρουσίασαν έναν f(k)*n^3 αλγόριθμο για το πρόβλημα των Διακεκριμενων Μονοπατιων. Για να το καταφέρουν αυτό, εισήγαγαν την "τεχνκή της άσχετης κορυφής" σύμφωνα με την οποία σε κάθε στιγμιότυπο δεντροπλάτους μεγαλύτερου του g(k) υπάρχει μια "άσχετη" κορυφή της οποίας η αφαίρεση δημιουργεί ένα ισοδύναμο στιγμιότυπο του προβλήματος. Εδώ μελετάμε το πρόβλημα σε επίπεδα γραφήματα και αποδεικνύουμε ότι για κάθε σταθερό k κάθε στιγμιότυπο του προβλήματος των Διακεκριμενων Μονοπατιων σε επιπεδα γραφηματα μπορεί να μετασχηματιστεί σε ένα ισοδύναμο που έχει φραγμένο δενδροπλάτος, αφαιρώντας ταυτόχρονα ένα σύνολο κορυφών από το δεδομένο επίπεδο γράφημα. Ως συνέπεια αυτού, το πρόβλημα των Διακεκριμένων Μονοπατιών σε επίπεδα γραφήματα μπορεί να λυθεί σε γραμμικό χρόνο για κάθε σταθερό πλήθος τερματικών.> In the Disjoint Paths Problem, given a graph G and a set of k pairs of terminals, we ask whether the pairs of terminals can be linked by pairwise disjoint paths. > In the Graph Minors series of 23 papers between 1984 and 2011, Neil Robertson and Paul D. Seymour, among other great results that heavily influenced Graph Theory, provided an f(k)\cdot n^{3} algorithm for the Disjoint Paths Problem. To achieve this, they introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k) there is an “irrelevant” vertex whose removal creates an equivalent instance of the problem. > > We study the problem in planar graphs and we prove that for every fixed k every instance of the Planar Disjoint Paths Problem can be transformed to an equivalent one that has bounded treewidth, by simultaneously discarding a set of vertices of the given planar graph. As a consequence the Planar Disjoint Paths Problem can be solved in linear time for every fixed number of terminals

A more accurate view of the Flat Wall Theorem

We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular, we suggest two variants of the theorem and we introduce a new, more versatile, concept of wall homogeneity as well as the notion of regularity in flat walls. All proposed concepts and results aim at facilitating the use of the irrelevant vertex technique in future algorithmic applications.Comment: arXiv admin note: text overlap with arXiv:2004.1269

An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL

In general, a graph modification problem is defined by a graph modification operation $\boxtimes$ and a target graph property ${\cal P}$. Typically, the modification operation $\boxtimes$ may be vertex removal}, edge removal}, edge contraction}, or edge addition and the question is, given a graph $G$ and an integer $k$, whether it is possible to transform $G$ to a graph in ${\cal P}$ after applying $k$ times the operation $\boxtimes$ on $G$. This problem has been extensively studied for particilar instantiations of $\boxtimes$ and ${\cal P}$. In this paper we consider the general property ${\cal P}_{{\phi}}$ of being planar and, moreover, being a model of some First-Order Logic sentence ${\phi}$ (an FOL-sentence). We call the corresponding meta-problem Graph $\boxtimes$-Modification to Planarity and ${\phi}$ and prove the following algorithmic meta-theorem: there exists a function $f:\Bbb{N}^{2}\to\Bbb{N}$ such that, for every $\boxtimes$ and every FOL sentence ${\phi}$, the Graph $\boxtimes$-Modification to Planarity and ${\phi}$ is solvable in $f(k,|{\phi}|)\cdot n^2$ time. The proof constitutes a hybrid of two different classic techniques in graph algorithms. The first is the irrelevant vertex technique that is typically used in the context of Graph Minors and deals with properties such as planarity or surface-embeddability (that are not FOL-expressible) and the second is the use of Gaifman's Locality Theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems

Branchwidth is (1,g)-self-dual

A graph parameter is self-dual in some class of graphs embeddable in some surface if its value does not change in the dual graph by more than a constant factor. We prove that the branchwidth of connected hypergraphs without bridges and loops that are embeddable in some surface of Euler genus at most g is an (1,g)-self-dual parameter. This is the first proof that branchwidth is an additively self-dual width parameter.Comment: 10 page