89 research outputs found
Cliques, colouring and satisfiability : from structure to algorithms
We examine the implications of various structural restrictions on the computational
complexity of three central problems of theoretical computer science
(colourability, independent set and satisfiability), and their relatives. All problems
we study are generally NP-hard and they remain NP-hard under various restrictions.
Finding the greatest possible restrictions under which a problem is computationally
difficult is important for a number of reasons. Firstly, this can make it easier to
establish the NP-hardness of new problems by allowing easier transformations. Secondly,
this can help clarify the boundary between tractable and intractable instances
of the problem.
Typically an NP-hard graph problem admits an infinite sequence of narrowing
families of graphs for which the problem remains NP-hard. We obtain a number
of such results; each of these implies necessary conditions for polynomial-time
solvability of the respective problem in restricted graph classes. We also identify
a number of classes for which these conditions are sufficient and describe explicit
algorithms that solve the problem in polynomial time in those classes. For the
satisfiability problem we use the language of graph theory to discover the very first
boundary property, i.e. a property that separates tractable and intractable instances
of the problem. Whether this property is unique remains a big open problem
Perfect Graphs
This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement
On the trace of random walks on random graphs
We study graph-theoretic properties of the trace of a random walk on a random
graph. We show that for any there exists such that the
trace of the simple random walk of length on the
random graph for is, with high probability,
Hamiltonian and -connected. In the special case (i.e.
when ), we show a hitting time result according to which, with high
probability, exactly one step after the last vertex has been visited, the trace
becomes Hamiltonian, and one step after the last vertex has been visited for
the 'th time, the trace becomes -connected.Comment: 32 pages, revised versio
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
Graph parameters and the speed of hereditary properties
In this thesis we study the speed of hereditary properties of graphs and how this defines some of the structure of the properties. We start by characterizing several graph parameters by means of minimal hereditary classes. We then give a global characterization of properties of low speed, before looking at properties with higher speeds starting at the Bell number. We then introduce a new parameter, clique-width, and show that there are an infinite amount of minimal hereditary properties with unbounded clique-width. We then look at the factorial layer in more detail and focus on P7-free bipartite graphs. Finally we discuss word-representable graphs
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