The parallel solution of domination problems on chordal and strongly chordal graphs

AbstractWe present efficient parallel algorithms for the domination problem on strongly chordal graphs and related problems, such as the set cover problem for α-acyclic hypergraphs and the dominating clique problem for strongly chordal graphs. Moreover, we present an efficient parallel algorithm which checks, for any chordal graph, whether it has a dominating clique

Unique Perfect Phylogeny Characterizations via Uniquely Representable Chordal Graphs

The perfect phylogeny problem is a classic problem in computational biology, where we seek an unrooted phylogeny that is compatible with a set of qualitative characters. Such a tree exists precisely when an intersection graph associated with the character set, called the partition intersection graph, can be triangulated using a restricted set of fill edges. Semple and Steel used the partition intersection graph to characterize when a character set has a unique perfect phylogeny. Bordewich, Huber, and Semple showed how to use the partition intersection graph to find a maximum compatible set of characters. In this paper, we build on these results, characterizing when a unique perfect phylogeny exists for a subset of partial characters. Our characterization is stated in terms of minimal triangulations of the partition intersection graph that are uniquely representable, also known as ur-chordal graphs. Our characterization is motivated by the structure of ur-chordal graphs, and the fact that the block structure of minimal triangulations is mirrored in the graph that has been triangulated

On coding labeled trees

Trees are probably the most studied class of graphs in Computer Science. In this thesis we study bijective codes that represent labeled trees by means of string of node labels. We contribute to the understanding of their algorithmic tractability, their properties, and their applications. The thesis is divided into two parts. In the first part we focus on two types of tree codes, namely Prufer-like codes and Transformation codes. We study optimal encoding and decoding algorithms, both in a sequential and in a parallel setting. We propose a unified approach that works for all Prufer-like codes and a more generic scheme based on the transformation of a tree into a functional digraph suitable for all bijective codes. Our results in this area close a variety of open problems. We also consider possible applications of tree encodings, discussing how to exploit these codes in Genetic Algorithms and in the generation of random trees. Moreover, we introduce a modified version of a known code that, in Genetic Algorithms, outperform all the other known codes. In the second part of the thesis we focus on two possible generalizations of our work. We first take into account the classes of k-trees and k-arch graphs (both superclasses of trees): we study bijective codes for this classes of graphs and their algorithmic feasibility. Then, we shift our attention to Informative Labeling Schemes. In this context labels are no longer considered as simple unique node identifiers, they rather convey information useful to achieve efficient computations on the tree. We exploit this idea to design a concurrent data structure for the lowest common ancestor problem on dynamic trees. We also present an experimental comparison between our labeling scheme and the one proposed by Peleg for static trees

A note on the independence number, domination number and related parameters of random binary search trees and random recursive trees

We identify the mean growth of the independence number of random binary search trees and random recursive trees and show normal fluctuations around their means. Similarly we also show normal limit laws for the domination number and variations of it for these two cases of random tree models. Our results are an application of a recent general theorem of Holmgren and Janson on fringe trees in these two random tree models

Clique trees of infinite locally finite chordal graphs

We investigate clique trees of infinite locally finite chordal graphs. Our main contribution is a bijection between the set of clique trees and the product of local finite families of finite trees. Even more, the edges of a clique tree are in bijection with the edges of the corresponding collection of finite trees. This allows us to enumerate the clique trees of a chordal graph and extend various classic characterisations of clique trees to the infinite setting

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions