Contraction Obstructions for Connected Graph Searching

We consider the connected variant of the classic mixed search game where, in each search step, cleaned edges form a connected subgraph. We consider graph classes with bounded connected (and monotone) mixed search number and we deal with the question whether the obstruction set, with respect of the contraction partial ordering, for those classes is finite. In general, there is no guarantee that those sets are finite, as graphs are not well quasi ordered under the contraction partial ordering relation. In this paper we provide the obstruction set for $k=2$, where $k$ is the number of searchers we are allowed to use. This set is finite, it consists of 177 graphs and completely characterises the graphs with connected (and monotone) mixed search number at most 2. Our proof reveals that the "sense of direction" of an optimal search searching is important for connected search which is in contrast to the unconnected original case. We also give a double exponential lower bound on the size of the obstruction set for the classes where this set is finite

FPT Algorithms for Plane Completion Problems

The Plane Subgraph (resp. Topological Minor) Completion problem asks, given a (possibly disconnected) plane (multi)graph Gamma and a connected plane (multi)graph Delta, whether it is possible to add edges in Gamma without violating the planarity of its embedding so that it contains some subgraph (resp. topological minor) that is topologically isomorphic to Delta. We give FPT algorithms that solve both problems in f(|E(Delta)|)*|E(Gamma)|^{2} steps. Moreover, for the Plane Subgraph Completion problem we show that f(k)=2^{O(k*log(k))}

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

Obstructions and Algorithms for Graph Layouts Problems

Η σύγχρονη Θεωρία Γραφημάτων έχει επηρεαστεί σε μεγάλο βαθμό από την δουλειά των N. Robertson και P. Seymour. Μέσα από αυτήν, πληθώρα από δομικά, συνδυαστικά, καθώς και αλγοριθμικά αποτελέσματα εισήχθηκαν, σε μία σειρά από εργασίες που είχε απώτερο στόχο την απόδειξη της εικασίας του Wagner. Σε αυτή την διδακτορική διατριβή επικεντρωνόμαστε στην μελέτη των Συνόλων Παρεμπόδισης, μία από τις σημαντικότερες συνδυαστικές έννοιες αυτής της θεωρίας. Πιο συγκεκριμένα, μελετάμε την συνδυαστική και αλγοριθμική πτυχή, καθώς και την υπολογισιμότητα, των γραφημάτων παρεμπόδισης, σε σχέση με παραμέτρους γραφημάτων που πηγάζουν από Διατάξεις σε Γραφήματα, Προβλήματα Διαγραφής κορυφών και Προβλήματα Ανίχνευσης Γραφημάτων. Η μελέτη αυτή είναι βασισμένη σε σχέσεις μερικής διάταξης γραφημάτων, όπως τα ελάσσονα, οι εμβυθίσεις και οι συνθλίψεις. Η μελέτη μας περιλαμβάνει αποτελέσματα σχετικά με την ύπαρξη και υπολογισιμότητα των συνόλων παρεμπόδισης, συνδυαστικά φράγματα στο μέγεθος τους, καθώς και την αλληλεπίδρασή τους με την Παραμετρική Πολυπλοκότητα και την Πυρηνοποίηση.Modern Graph Theory is heavily influenced by the seminal work of N. Robertson and P. Seymour, known as Graph Minors. Through this work, a wealth of structural, combinatorial, as well as algorithmic, results has been introduced, in a series of papers having as ultimate goal to give an affirmative answer to Wagner’s Conjecture. In this doctoral thesis we focus on the study of Obstruction Sets, one of the most important combinatorial concepts of this theory. In particular, we study the combinatorics and the algorithmic and computability aspects of graph obstructions and their relation to graph parameters emerging from Graph Layouts, Vertex Deletion Problems, and Graph Searching Problems. We consider several partial ordering relations on graphs such as minors, immersions, and contractions. Our study includes results on the existence and the computability of obstruction sets, combinatorial bounds on their size, and their interplay with parameterized complexity and kernelization

Παρεμποδίσεις και αλγόριθμοι για προβλήματα διατάξεων σε γραφήματα

Modern Graph Theory is heavily influenced by the seminal work of N. Robertson and P. Seymour, known as Graph Minors. Through this work, a wealth of structural, combinatorial, as well as algorithmic, results has been introduced, in a series of papers having as ultimate goal to give an affirmative answer to Wagner’s Conjecture. In this doctoral thesis we focus on the study of Obstruction Sets, one of the most important combinatorial concepts of this theory. In particular, we study the combinatorics and the algorithmic and computability aspects of graph obstructions and their relation to graph parameters emerging from Graph Layouts, Vertex Deletion Problems, and Graph Searching Problems. We consider several partial ordering relations on graphs such as minors, immersions, and contractions. Our study includes results on the existence and the computability of obstruction sets, combinatorial bounds on their size, and their interplay with parameterized complexity and kernelization.Η σύγχρονη Θεωρία Γραφημάτων έχει επηρεαστεί σε μεγάλο βαθμό από την δουλειά των N. Robertson και P. Seymour. έσα από αυτήν, πληθώρα από δομικά, συνδυαστικά, καθώς και αλγοριθμικά αποτελέσματα εισήχθηκαν, σε μία σειρά από εργασίες που είχε απώτερο στόχο την απόδειξη της εικασίας του Wagner. Σε αυτή την διδακτορική διατριβή επικεντρωνόμαστε στην μελέτη των Συνόλων Παρεμπόδισης, μία από τις σημαντικότερες συνδυαστικές έννοιες αυτής της θεωρίας. Πιο συγκεκριμένα, μελετάμε την συνδυαστική και αλγοριθμική πτυχή, καθώς και την υπολογισιμότητα, των γραφημάτων παρεμπόδισης, σε σχέση με παραμέτρους γραφημάτων που πηγάζουν από Διατάξεις σε Γραφήματα, Προβλήματα Διαγραφής ορυφών και Προβλήματα Ανίχνευσης Γραφημάτων. Η μελέτη αυτή είναι βασισμένη σε σχέσεις μερικής διάταξης γραφημάτων, όπως τα ελάσσονα, οι εμβυθίσεις και οι συνθλίψεις. Η μελέτη μας περιλαμβάνει αποτελέσματα σχετικά με την ύπαρξη και υπολογισιμότητα των συνόλων παρεμπόδισης, συνδυαστικά φράγματα στο μέγεθος τους, καθώς και την αλληλεπίδρασή τους με την Παραμετρική Πολυπλοκότητα και την Πυρηνοποίηση

Contraction Obstructions for Connected Graph Searching

International audienceWe consider the connected variant of the classic mixed search game where, in each search step, cleaned edges form a connected subgraph. We consider graph classes with bounded connected monotone mixed search number and we deal with the the question weather the obstruction set, with respect of the contraction partial ordering, for those classes is finite. In general, there is no guarantee that those sets are finite, as graphs are not well quasi ordered under the contraction partial ordering relation. In this paper we provide the obstruction set for k = 2. This set is finite, it consists of 174 graphs and completely characterizes the graphs with connected monotone mixed search number at most 2. Our proof reveals that the "sense of direction" of an optimal search searching is important for connected search which is in contrast to the unconnected original case

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, in the class of connected hypergraphs without bridges and loops that are embeddable in some surface of Euler genus at most g, branchwidth (1,g) -is a -self-dual parameter, i.e., for every hypergraph g in the class, the branchwidth of its dual is at most g the branchwidth of G plus g. This is the first proof that branchwidth is an additively self-dual width parameter