52 research outputs found

    Connectivity and Cycles in Graphs

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    https://digitalcommons.memphis.edu/speccoll-faudreerj/1199/thumbnail.jp

    Connectivity and Cycles

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    https://digitalcommons.memphis.edu/speccoll-faudreerj/1191/thumbnail.jp

    Cyclability: Combinatorial Properties, Algorithms and Complexity

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    Ένα γράφημα G καλείται k-κυκλώσιμο, αν για κάθε k από τις κορυφές του υπάρχει ένας κύκλος στο G που τις περιέχει. Η κυκλωσιμότητα ενός γραφήματος G είναι ο μέγιστος ακέραιος k για τον οποίο το G είναι k-κυκλώσιμο και είναι μία παράμετρος που σχετίζεται με τη συνεκτικότητα. Σε αυτή τη διδακτορική διατριβή μελετάμε, κυρίως από τη σκοπιά της Παραμετρικής Πολυπλοκότητας, το πρόβλημα ΚΥΚΛΩΣΙΜΟΤΗΤΑ: Δεδομένου ενός γραφήματος G = (V,E) και ενός μη αρνητικού ακεραίου k (η παράμετρος), να αποφασιστεί αν η κυκλωσιμότητα του G είναι ίση με k. Το πρώτο μας αποτέλεσμα είναι αρνητικό και δείχνει ότι η ύπαρξη ενός FPT-αλγορίθμου για την επίλυση του προβλήματος ΚΥΚΛΩΣΙΜΟΤΗΤΑ είναι απίθανη (εκτός αν FPT = co- W[1], το οποίο θεωρείται απίθανο). Πιο συγκεκριμένα, αποδεικνύουμε ότι το πρόβλημα ΚΥΚΛΩΣΙΜΟΤΗΤΑ είναι co-W[1]-δύσκολο, ακόμα και αν περιορίσουμε την είσοδο στο να είναι χωριζόμενο γράφημα. Από την άλλη, δίνουμε έναν FPT-αλγόριθμο για το ίδιο πρόβλημα περιορισμένο στην κλάση των επίπεδων γραφημάτων. Για να το πετύχουμε αυτό αποδεικνύουμε μια σειρά από συνδυαστικά αποτελέσματα σχετικά με την κυκλωσιμότητα και εφαρμόζουμε μια εκδοχή δύο βημάτων της περίφημης τεχνικής της άσχετης κορυφής, που εισήχθη από τους Robertson και Seymour στη σειρά εργασιών τους για Ελλάσονα Γραφήματα, ως ένα κρίσιμο συστατικό του αλγορίθμου τους για την επίλυση του προβλήματος των ΔΙΑΚΕΚΡΙΜΕΝΩΝ ΜΟΝΟΠΑΤΙΩΝ. Για να αποδείξουμε την ορθότητα του αλγορίθμου μας εισάγουμε έννοιες, όπως αυτή των ζωτικών κυκλικών συνδέσμων, και αποδεικνύουμε αποτελέσματα με ανεξάρτητου γραφοθεωρητικού ενδιαφέροντος. Κλείνουμε τη μελέτη μας με ένα δεύτερο αρνητικό αποτέλεσμα: Αποδεικνύουμε ότι για το πρόβλημα της ΚΥΚΛΩΣΙΜΟΤΗΤΑΣ δεν υπάρχουν πολυωνυμικοί πυρήνες, ακόμα και αν περιοριστούμε σε κυβικά επίπεδα γραφήματα, εκτός και αν δεν ισχύει μια υπόθεση της κλασσικής Θεωρίας Πολυπλοκότητας (ότι NP υποσύνολο του co-NP/poly).A graph G is called k-cyclable, if for every k of its vertices there exists a cycle in G that contains them. The cyclability of G is the maximum integer k for which G is k-cyclable and it is a connectivity related graph parameter. In this doctoral thesis we study, mainly from the Parameterized Complexity point of view, the Cyclability problem: Given a graph G = (V,E) and an integer k (the parameter), decide whether the cyclability of G is equal to k. Our first result is a negative one and shows that the existence of an FPT-algorithm for solving Cyclability is unlikely (unless FPT = co-W[1], which is considered unlikely). More specifically, we prove that Cyclability is co-W[1]-hard, even if we restrict the input to be a split graph. On the other hand, we give an FPT-algorithm for the same problem when restricted to the class of planar graphs. To do this, we prove a series of combinatorial results regarding cyclability and apply a two-step version of the so called irrelevant vertex technique, which was introduced by Robertson and Seymour in their Graph Minors series (Irrelevant vertices in linkage problems) as a crucial ingredient for their algorithm solving the Disjoint Paths problem. To prove the correctness of our algorithm, we introduce notions, like the one of vital cyclic linkages, and give results of independent graph-theoretic interest. We conclude our study with a negative result: We prove that Cyclability admits no polynomial kernel, even when restricted to cubic planar graphs, unless a classical complexity theoretic assumption (that NP is a subset of co-NP/poly) fails

    A look at cycles containing specified elements of a graph

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    AbstractThis article is intended as a brief survey of problems and results dealing with cycles containing specified elements of a graph. It is hoped that this will help researchers in the area to identify problems and areas of concentration

    Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs

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    AbstractThe circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n0.694), and the circumference of a 3-connected claw-free graph is Ω(n0.121). We generalize and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m0.753) edges. We use this result together with the Ryjáček closure operation to improve the lower bound on the circumference of a 3-connected claw-free graph to Ω(n0.753). Our proofs imply polynomial time algorithms for finding large Eulerian subgraphs of 3-edge-connected graphs and long cycles in 3-connected claw-free graphs

    Modification to planarity is fixed parameter tractable

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    A replacement action is a function L that maps each k-vertex labeled graph to another k-vertex graph. We consider a general family of graph modification problems, called L-Replacement to C, where the input is a graph G and the question is whether it is possible to replace in G some k-vertex subgraph H of it by L(H) so that the new graph belongs to the graph class C. L-Replacement to C can simulate several modification operations such as edge addition, edge removal, edge editing, and diverse completion and superposition operations. In this paper, we prove that for any action L, if C is the class of planar graphs, there is an algorithm that solves L-Replacement to C in O(|G| 2 ) steps. We also present several applications of our approach to related problems.publishedVersio

    Modification to Planarity is Fixed Parameter Tractable

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    A replacement action is a function L that maps each k-vertex labeled graph to another k-vertex graph. We consider a general family of graph modification problems, called L-Replacement to C, where the input is a graph G and the question is whether it is possible to replace in G some k-vertex subgraph H of it by L(H) so that the new graph belongs to the graph class C. L-Replacement to C can simulate several modification operations such as edge addition, edge removal, edge editing, and diverse completion and superposition operations. In this paper, we prove that for any action L, if C is the class of planar graphs, there is an algorithm that solves L-Replacement to C in O(|G|^{2}) steps. We also present several applications of our approach to related problems

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

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    In general, a graph modification problem is defined by a graph modification operation \boxtimes and a target graph property P{\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 GG and an integer kk, whether it is possible to transform GG to a graph in P{\cal P} after applying kk times the operation \boxtimes on GG. This problem has been extensively studied for particilar instantiations of \boxtimes and P{\cal P}. In this paper we consider the general property Pϕ{\cal P}_{{\phi}} of being planar and, moreover, being a model of some First-Order Logic sentence ϕ{\phi} (an FOL-sentence). We call the corresponding meta-problem Graph \boxtimes-Modification to Planarity and ϕ{\phi} and prove the following algorithmic meta-theorem: there exists a function f:N2Nf:\Bbb{N}^{2}\to\Bbb{N} such that, for every \boxtimes and every FOL sentence ϕ{\phi}, the Graph \boxtimes-Modification to Planarity and ϕ{\phi} is solvable in f(k,ϕ)n2f(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
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