<p>In phylogenetic analysis one study the relationship between different species. By comparing DNA from two different species it is possible to get a numerical value representing the difference between the species. For a set of species, all pair-wise comparisons result in a dissimilarity matrix <em>d</em>.</p><p>In this thesis I present a few methods for constructing a phylogenetic tree from <em>d</em>. The common denominator for these methods is that they do not generate a tree, but instead give a connected graph. The resulting graph will be a tree, in areas where the data perfectly matches a tree. When <em>d</em> does not perfectly match a tree, the resulting graph will instead show the different possible topologies, and how strong support they have from the data.</p><p>Finally I have tested the methods both on real measured data and constructed test cases.</p>

In phylogenetic analysis one study the relationship between different species. By comparing DNA from two different species it is possible to get a numerical value representing the difference between the species. For a set of species, all pair-wise comparisons result in a dissimilarity matrix d. In this thesis I present a few methods for constructing a phylogenetic tree from d. The common denominator for these methods is that they do not generate a tree, but instead give a connected graph. The resulting graph will be a tree, in areas where the data perfectly matches a tree. When d does not perfectly match a tree, the resulting graph will instead show the different possible topologies, and how strong support they have from the data. Finally I have tested the methods both on real measured data and constructed test cases.

<p>This PhD thesis, consisting of an introduction, four papers, and some supplementary results, studies the problem of finding an <i>optimal realization</i> of a given finite metric space: a weighted graph which preserves the metric's distances and has minimal total edge weight. This problem is known to be NP-hard, and solutions are not necessarily unique.</p><p>It has been conjectured that <i>extremally weighted</i> optimal realizations may be found as subgraphs of the <i>hereditarily optimal realization</i> Γ_d, a graph which in general has a higher total edge weight than the optimal realization but has the advantages of being unique, and possible to construct explicitly via the <i>tight span</i> of the metric.</p><p>In Paper I, we prove that the graph Γ_d is equivalent to the 1-skeleton of the tight span precisely when the metric considered is <i>totally split-decomposable</i>. For the subset of totally split-decomposable metrics known as <i>consistent</i> metrics this implies that Γ_d is isomorphic to the easily constructed <i>Buneman graph</i>.</p><p>In Paper II, we show that for any metric on at most five points, any optimal realization can be found as a subgraph of Γ_d.</p><p>In Paper III we provide a series of counterexamples; metrics for which there exist extremally weighted optimal realizations which are not subgraphs of Γ_d. However, for these examples there also exists at least one optimal realization which is a subgraph.</p><p>Finally, Paper IV examines a weakened conjecture suggested by the above counterexamples: can we always find some optimal realization as a subgraph in Γ_d? Defining <i>extremal</i> optimal realizations as those having the maximum possible number of shortest paths, we prove that any embedding of the vertices of an extremal optimal realization into Γ_d is injective. Moreover, we prove that this weakened conjecture holds for the subset of consistent metrics which have a 2-dimensional tight span</p>

This PhD thesis, consisting of an introduction, four papers, and some supplementary results, studies the problem of finding an optimal realization of a given finite metric space: a weighted graph which preserves the metric's distances and has minimal total edge weight. This problem is known to be NP-hard, and solutions are not necessarily unique. It has been conjectured that extremally weighted optimal realizations may be found as subgraphs of the hereditarily optimal realization Γd, a graph which in general has a higher total edge weight than the optimal realization but has the advantages of being unique, and possible to construct explicitly via the tight span of the metric. In Paper I, we prove that the graph Γd is equivalent to the 1-skeleton of the tight span precisely when the metric considered is totally split-decomposable. For the subset of totally split-decomposable metrics known as consistent metrics this implies that Γd is isomorphic to the easily constructed Buneman graph. In Paper II, we show that for any metric on at most five points, any optimal realization can be found as a subgraph of Γd. In Paper III we provide a series of counterexamples; metrics for which there exist extremally weighted optimal realizations which are not subgraphs of Γd. However, for these examples there also exists at least one optimal realization which is a subgraph. Finally, Paper IV examines a weakened conjecture suggested by the above counterexamples: can we always find some optimal realization as a subgraph in Γd? Defining extremal optimal realizations as those having the maximum possible number of shortest paths, we prove that any embedding of the vertices of an extremal optimal realization into Γd is injective. Moreover, we prove that this weakened conjecture holds for the subset of consistent metrics which have a 2-dimensional tight span

This thesis presents the first sub-quadratic circle graph recognition algorithm, and develops improved algorithms for two important hierarchical decomposition schemes: modular decomposition and split decomposition. The modular decomposition algorithm results from unifying two different approaches previously employed to solve the problem: divide-and-conquer and factorizing permutations. It runs in linear-time, and is straightforward in its understanding, correctness, and implementation. It merely requires a collection of trees and simple traversals of these trees. The split-decomposition algorithm is similar in being straightforward in its understanding and correctness. An efficient implementation of the algorithm is described that uses the union-find data-structure. A novel charging argument is used to prove the running-time. The algorithm is the first to use the recent reformulation of split decomposition in terms of graph-labelled trees. This facilitates its extension to circle graph recognition. In particular, it allows us to efficiently apply a new lexicographic breadth-first search characterization of circle graphs developed in the thesis. Lexicographic breadth-first search is additionally responsible for the efficiency of the split decomposition algorithm, and contributes to the simplicity of the modular decomposition algorithm.

This thesis presents the first sub-quadratic circle graph recognition algorithm, and develops improved algorithms for two important hierarchical decomposition schemes: modular decomposition and split decomposition. The modular decomposition algorithm results from unifying two different approaches previously employed to solve the problem: divide-and-conquer and factorizing permutations. It runs in linear-time, and is straightforward in its understanding, correctness, and implementation. It merely requires a collection of trees and simple traversals of these trees. The split-decomposition algorithm is similar in being straightforward in its understanding and correctness. An efficient implementation of the algorithm is described that uses the union-find data-structure. A novel charging argument is used to prove the running-time. The algorithm is the first to use the recent reformulation of split decomposition in terms of graph-labelled trees. This facilitates its extension to circle graph recognition. In particular, it allows us to efficiently apply a new lexicographic breadth-first search characterization of circle graphs developed in the thesis. Lexicographic breadth-first search is additionally responsible for the efficiency of the split decomposition algorithm, and contributes to the simplicity of the modular decomposition algorithm.