1 We study efficient exact algorithms for several problems including VERTEX BIPARTIZATION, FEEDBACK VERTEX SET, 4-HITTING SET, MAX CUT in graphs with maximum degree at most 4. Our main results include: • an O ∗ (1.9526 n) 2 algorithm for VERTEX BIPARTIZATION problem in undirected graphs; • an O

, and c is a constant. We extend this to an algorithm enumerating all solutions in O(d k ·m) time for a (larger) constant d. As a further result, we present a fixed-parameter algorithm with runtime O(2 k · m 2) for the NP-complete Edge Bipartization problem, which asks for at most k edges to remove from a

A fullerene graph is a cubic bridgeless planar graph with twelve 5-faces such that all other faces are 6-faces. We show that any fullerene graph on n vertices can be bipartized by removing O (√n) edges. This bound is asymptotically optimal.

A fullerene graph is a cubic bridgeless planar graph with twelve 5-faces such that all other faces are 6-faces. We show that any fullerene graph on n vertices can be bipartized by removing O( n) edges. This bound is asymptotically optimal.

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present a fixed-parameter algorithm for the NPcomplete Edge Bipartization problem with runtime O(2 k · m 2). 1

This thesis studies exponential time algorithms, more precisely, algorithms exactly solving problems for which no polynomial time algorithm is known and likely to exist. Interested in worst–case upper bounds on the running times, several known techniques to design and analyze such algorithms

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We describe an easy way to check whether a hamiltonian graph of order n with a given hamiltonian cycle is planar. This is done by solving the bipartation problem for the corresponding circle graph. If the graph is planar an embedding is constructed. The algorithm runs in O(n) time and space. 1

Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics and function of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of measurements capable of expressing the most relevant topological features. This article presents a survey of such measurements. It includes general considerations about complex network characterization, a brief review of the principal models, and the presentation of the main existing measurements organized into classes. Special attention is given to relating complex network analysis with the areas of pattern recognition and feature selection, as well as on surveying some concepts and measurements from traditional graph theory which are potentially useful for complex network research. Depending on the network and the analysis task one has in mind, a specific set of features may be chosen. It is hoped that the present survey will help the