103 research outputs found
Towards an Isomorphism Dichotomy for Hereditary Graph Classes
In this paper we resolve the complexity of the isomorphism problem on all but
finitely many of the graph classes characterized by two forbidden induced
subgraphs. To this end we develop new techniques applicable for the structural
and algorithmic analysis of graphs. First, we develop a methodology to show
isomorphism completeness of the isomorphism problem on graph classes by
providing a general framework unifying various reduction techniques. Second, we
generalize the concept of the modular decomposition to colored graphs, allowing
for non-standard decompositions. We show that, given a suitable decomposition
functor, the graph isomorphism problem reduces to checking isomorphism of
colored prime graphs. Third, we extend the techniques of bounded color valence
and hypergraph isomorphism on hypergraphs of bounded color size as follows. We
say a colored graph has generalized color valence at most k if, after removing
all vertices in color classes of size at most k, for each color class C every
vertex has at most k neighbors in C or at most k non-neighbors in C. We show
that isomorphism of graphs of bounded generalized color valence can be solved
in polynomial time.Comment: 37 pages, 4 figure
Odd-Minors I: Excluding small parity breaks
Given a graph class~, the -blind-treewidth of a
graph~ is the smallest integer~ such that~ has a tree-decomposition
where every bag whose torso does not belong to~ has size at
most~. In this paper we focus on the class~ of bipartite graphs
and the class~ of planar graphs together with the odd-minor
relation. For each of the two parameters, -blind-treewidth and
-blind-treewidth, we prove an analogue of the
celebrated Grid Theorem under the odd-minor relation. As a consequence we
obtain FPT-approximation algorithms for both parameters. We then provide
FPT-algorithms for \textsc{Maximum Independent Set} on graphs of bounded
-blind-treewidth and \textsc{Maximum Cut} on graphs of bounded
-blind-treewidth
Dynamic programming on bipartite tree decompositions
We revisit a graph width parameter that we dub bipartite treewidth, along
with its associated graph decomposition that we call bipartite tree
decomposition. Bipartite treewidth can be seen as a common generalization of
treewidth and the odd cycle transversal number. Intuitively, a bipartite tree
decomposition is a tree decomposition whose bags induce almost bipartite graphs
and whose adhesions contain at most one vertex from the bipartite part of any
other bag, while the width of such decomposition measures how far the bags are
from being bipartite. Adapted from a tree decomposition originally defined by
Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by
Tazari [Th. Comp. Sci. 2012], bipartite treewidth appears to play a crucial
role for solving problems related to odd-minors, which have recently attracted
considerable attention. As a first step toward a theory for solving these
problems efficiently, the main goal of this paper is to develop dynamic
programming techniques to solve problems on graphs of small bipartite
treewidth. For such graphs, we provide a number of para-NP-completeness
results, FPT-algorithms, and XP-algorithms, as well as several open problems.
In particular, we show that -Subgraph-Cover, Weighted Vertex
Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are
parameterized by bipartite treewidth. We provide the following complexity
dichotomy when is a 2-connected graph, for each of -Subgraph-Packing,
-Induced-Packing, -Scattered-Packing, and -Odd-Minor-Packing problem:
if is bipartite, then the problem is para-NP-complete parameterized by
bipartite treewidth while, if is non-bipartite, then it is solvable in
XP-time. We define 1--treewidth by replacing the bipartite graph
class by any class . Most of the technology developed here works for
this more general parameter.Comment: Presented in IPEC 202
Graph Algorithms and Complexity Aspects on Special Graph Classes
Graphs are a very flexible tool within mathematics, as such, numerous problems can be solved by formulating them as an instance of a graph. As a result, however, some of the structures found in real world problems may be lost in a more general graph. An example of this is the 4-Colouring problem which, as a graph problem, is NP-complete. However, when a map is converted into a graph, we observe that this graph has structural properties, namely being (K_5, K_{3,3})-minor-free which can be exploited and as such there exist algorithms which can find 4-colourings of maps in polynomial time.
This thesis looks at problems which are NP-complete in general and determines the complexity of the problem when various restrictions are placed on the input, both for the purpose of finding tractable solutions for inputs which have certain structures, and to increase our understanding of the point at which a problem becomes NP-complete.
This thesis looks at four problems over four chapters, the first being Parallel Knock-Out. This chapter will show that Parallel Knock-Out can be solved in O(n+m) time on P_4-free graphs, also known as cographs, however, remains hard on split graphs, a subclass of P_5-free graphs. From this a dichotomy is shown on -free graphs for any fixed integer .
The second chapter looks at Minimal Disconnected Cut. Along with some smaller results, the main result in this chapter is another dichotomy theorem which states that Minimal Disconnected Cut is polynomial time solvable for 3-connected planar graphs but NP-hard for 2-connected planar graphs.
The third chapter looks at Square Root. Whilst a number of results were found, the work in this thesis focuses on the Square Root problem when restricted to some classes of graphs with low clique number.
The final chapter looks at Surjective H-Colouring. This chapter shows that Surjective H-Colouring is NP-complete, for any fixed, non-loop connected graph H with two reflexive vertices and for any fixed graph H’ which can be obtained from H by replacing vertices with true twins. This result enabled us to determine the complexity of Surjective H-Colouring on all fixed graphs H of size at most 4
A priori bounds for GIETs, affine shadows and rigidity of foliations in genus two
We prove a rigidity result for foliations on surfaces of genus two, which can be seen as a generalization to higher genus of Herman’s theorem on circle diffeomorphisms and, correspondingly, flows on the torus. We prove in particular that, if a smooth, orientable foliation with non-degenerate (Morse) singularities on a closed surface of genus two is minimal, then, under a full measure condition for the rotation number, it is differentiably conjugate to a linear foliation. The corresponding result at the level of Poincaré sections is that, for a full measure set of (standard) interval exchange transformations (IETs for short) with d= 4 or d= 5 continuity intervals and irreducible combinatorics, any generalized interval exchange transformation (GIET for short) which is topologically conjugate to a standard IET from this set and satisfies an obstruction expressed in terms of boundary operator (which is automatically satisfied when the GIET arises as a Poincaré map of a smooth foliation) is C1 -conjugate to it. This in particular settles a conjecture by Marmi, Moussa and Yoccoz in genus two. Our results also show that this conjecture on the rigidity of GIETs can be reduced to the study of affine IETs, or more precisely of Birkhoff sums of piecewise constant observables over standard IETs, in genus g≥ 3 . Our approach is via renormalization, namely we exploit a suitable acceleration of the Rauzy-Veech induction (an acceleration which makes Oseledets generic effective) on the space of GIETs. For in ly renormalizable, irrational GIETs of any number of intervals d≥ 2 we prove a dynamical dichotomy on the behaviour of the orbits under renormalization, by proving that either an orbit is recurrent to certain bounded sets in the space of GIETs, or it diverges and it is approximated (up to lower order terms) by the orbit of an affine IET (a case that we refer to as affine shadowing). This result can in particular be used, in conjunction with previous work by Marmi-Moussa and Yoccoz on the existence of wandering intervals for affine IETs, to prove, a priori bounds in genus two and is therefore at the base of the rigidity result
- …