arXiv subjects

Marc Hellmuth

Publications and source records attributed to Marc Hellmuth.

86 records · Page 5Linked to original sources

arXiv2016

Construction of Gene and Species Trees from Sequence Data incl. Orthologs, Paralogs, and Xenologs

Phylogenetic reconstruction aims at finding plausible hypotheses of the evolutionary history of genes or species based on genomic sequence information. The distinction of orthologous genes (genes that having a common ancestry and diverged after a speciation) is crucial and lies at the heart of many genomic studies. However, existing methods that rely only on 1:1 orthologs to infer species trees are strongly restricted to a small set of allowed genes that provide information about the species tree. The use of larger gene sets that consist in addition of non-orthologous genes (e.g. so-called paralogous or xenologous genes) considerably increases the information about the evolutionary history of the respective species. In this work, we introduce a novel method to compute species phylogenies based on sequence data including orthologs, paralogs or even xenologs.

q-bio.PE↗

arXiv2015

Associativity and non-associativity of some hypergraph products

Several variants of hypergraph products have been introduced as generalizations of the strong and direct products of graphs. Here we show that only some of them are associative. In addition to the Cartesian product, these are the minimal rank preserving direct product, and the normal product. Counter-examples are given for the strong product as well as the non-rank-preserving and the maximal rank preserving direct product.

math.CO↗

arXiv2015

Techniques for the Cograph Editing Problem: Module Merge is equivalent to Editing P4s

Cographs are graphs in which no four vertices induce a simple connected path $P_4$. Cograph editing is to find for a given graph $G = (V,E)$ a set of at most $k$ edge additions and deletions that transform $G$ into a cograph. This combinatorial optimization problem is NP-hard. It has, recently found applications in the context of phylogenetics, hence good heuristics are of practical importance. It is well-known that the cograph editing problem can be solved independently on the so-called strong prime modules of the modular decomposition of $G$. We show here that editing the induced $P_4$'s of a given graph is equivalent to resolving strong prime modules by means of a newly defined merge operation on the submodules. This observation leads to a new exact algorithm for the cograph editing problem that can be used as a starting point for the construction of novel heuristics.

cs.DM↗

arXiv2015

On Tree Representations of Relations and Graphs: Symbolic Ultrametrics and Cograph Edge Decompositions

Tree representations of (sets of) symmetric binary relations, or equivalently edge-colored undirected graphs, are of central interest, e.g.\ in phylogenomics. In this context symbolic ultrametrics play a crucial role. Symbolic ultrametrics define an edge-colored complete graph that allows to represent the topology of this graph as a vertex-colored tree. Here, we are interested in the structure and the complexity of certain combinatorial problems resulting from considerations based on symbolic ultrametrics, and on algorithms to solve them. This includes, the characterization of symbolic ultrametrics that additionally distinguishes between edges and non-edges of \emph{arbitrary} edge-colored graphs $G$ and thus, yielding a tree representation of $G$, by means of so-called cographs. Moreover, we address the problem of finding "closest" symbolic ultrametrics and show the NP-completeness of the three problems: symbolic ultrametric editing, completion and deletion. Finally, as not all graphs are cographs, and hence, don't have a tree representation, we ask, furthermore, what is the minimum number of cotrees needed to represent the topology of an arbitrary non-cograph $G$. This is equivalent to find an optimal cograph edge $k$-decomposition $\{E_1,\dots,E_k\}$ of $E$ so that each subgraph $(V,E_i)$ of $G$ is a cograph. We investigate this problem in full detail, resulting in several new open problems, and NP-hardness results. For all optimization problems proven to be NP-hard we will provide integer linear program (ILP) formulations to efficiently solve them.

cs.DM↗

arXiv2015

Fast Factorization of Cartesian products of Hypergraphs

Cartesian products of graphs and hypergraphs have been studied since the 1960s. For (un)directed hypergraphs, unique \emph{prime factor decomposition (PFD)} results with respect to the Cartesian product are known. However, there is still a lack of algorithms, that compute the PFD of directed hypergraphs with respect to the Cartesian product. In this contribution, we focus on the algorithmic aspects for determining the Cartesian prime factors of a finite, connected, directed hypergraph and present a first polynomial time algorithm to compute its PFD. In particular, the algorithm has time complexity $O(|E||V|r^2)$ for hypergraphs $H=(V,E)$, where the rank $r$ is the maximum number of vertices contained in an hyperedge of $H$. If $r$ is bounded, then this algorithm performs even in $O(|E|\log^2(|V|))$ time. Thus, our method additionally improves also the time complexity of PFD-algorithms designed for undirected hypergraphs that have time complexity $O(|E||V|r^6Δ^6)$, where $Δ$ is the maximum number of hyperedges a vertex is contained in.

cs.DM↗

arXiv2015

On Symbolic Ultrametrics, Cotree Representations, and Cograph Edge Decompositions and Partitions

Symbolic ultrametrics define edge-colored complete graphs K_n and yield a simple tree representation of K_n. We discuss, under which conditions this idea can be generalized to find a symbolic ultrametric that, in addition, distinguishes between edges and non-edges of arbitrary graphs G=(V,E) and thus, yielding a simple tree representation of G. We prove that such a symbolic ultrametric can only be defined for G if and only if G is a so-called cograph. A cograph is uniquely determined by a so-called cotree. As not all graphs are cographs, we ask, furthermore, what is the minimum number of cotrees needed to represent the topology of G. The latter problem is equivalent to find an optimal cograph edge k-decomposition {E_1,...,E_k} of E so that each subgraph (V,E_i) of G is a cograph. An upper bound for the integer k is derived and it is shown that determining whether a graph has a cograph 2-decomposition, resp., 2-partition is NP-complete.

cs.DM↗

arXiv2014

Spiders can be recognized by counting their legs

Spiders are arthropods that can be distinguished from their closest relatives, the insects, by counting their legs. Spiders have 8, insects just 6. Spider graphs are a very restricted class of graphs that naturally appear in the context of cograph editing. The vertex set of a spider (or its complement) is naturally partitioned into a clique (the body), an independent set (the legs), and a rest (serving as the head). Here we show that spiders can be recognized directly from their degree sequences through the number of their legs (vertices with degree 1). Furthermore, we completely characterize the degree sequences of spiders.

cs.DM↗

arXiv2014

The Relaxed Square Property

Graph products are characterized by the existence of non-trivial equivalence relations on the edge set of a graph that satisfy a so-called square property. We investigate here a generalization, termed RSP-relations. The class of graphs with non-trivial RSP-relations in particular includes graph bundles. Furthermore, RSP-relations are intimately related with covering graph constructions. For K_23-free graphs finest RSP-relations can be computed in polynomial-time. In general, however, they are not unique and their number may even grow exponentially. They behave well for graph products, however, in sense that a finest RSP-relations can be obtained easily from finest RSP-relations on the prime factors.

cs.DM↗

arXiv2014

On the Cartesian Skeleton and the Factorization of the Strong Product of Digraphs

The three standard products (the Cartesian, the direct and the strong product) of undirected graphs have been wellinvestigated, unique prime factor decomposition (PFD) are known and polynomial time algorithms have been established for determining the prime factors. For directed graphs, unique PFD results with respect to the standard products are known. However, there is still a lack of algorithms, that computes the PFD of directed graphs with respect to the direct and the strong product in general. In this contribution, we focus on the algorithmic aspects for determining the PFD of directed graphs with respect to the strong product. Essential for computing the prime factors is the construction of a so-called Cartesian skeleton. This article introduces the notion of the Cartesian skeleton of directed graphs as a generalization of the Cartesian skeleton of undirected graphs. We provide new, fast and transparent algorithms for its construction. Moreover, we present a first polynomial time algorithm for determining the PFD with respect to the strong product of arbitrary connected digraphs.

cs.DM↗

arXiv2013

Square Property, Equitable Partitions, and Product-like Graphs

Equivalence relations on the edge set of a graph $G$ that satisfy restrictive conditions on chordless squares play a crucial role in the theory of Cartesian graph products and graph bundles. We show here that such relations in a natural way induce equitable partitions on the vertex set of $G$, which in turn give rise to quotient graphs that can have a rich product structure even if $G$ itself is prime.

cs.DM↗

arXiv2013

Fast Recognition of Partial Star Products and Quasi Cartesian Products

This paper is concerned with the fast computation of a relation $\R$ on the edge set of connected graphs that plays a decisive role in the recognition of approximate Cartesian products, the weak reconstruction of Cartesian products, and the recognition of Cartesian graph bundles with a triangle free basis. A special case of $\R$ is the relation $δ^\ast$, whose convex closure yields the product relation $σ$ that induces the prime factor decomposition of connected graphs with respect to the Cartesian product. For the construction of $\R$ so-called Partial Star Products are of particular interest. Several special data structures are used that allow to compute Partial Star Products in constant time. These computations are tuned to the recognition of approximate graph products, but also lead to a linear time algorithm for the computation of $δ^\ast$ for graphs with maximum bounded degree. Furthermore, we define \emph{quasi Cartesian products} as graphs with non-trivial $δ^\ast$. We provide several examples, and show that quasi Cartesian products can be recognized in linear time for graphs with bounded maximum degree. Finally, we note that quasi products can be recognized in sublinear time with a parallelized algorithm.

cs.DM↗

arXiv2013

Partial Star Products: A Local Covering Approach for the Recognition of Approximate Cartesian Product Graphs

This paper is concerned with the recognition of approximate graph products with respect to the Cartesian product. Most graphs are prime, although they can have a rich product-like structure. The proposed algorithms are based on a local approach that covers a graph by small subgraphs, so-called partial star products, and then utilizes this information to derive the global factors and an embedding of the graph under investigation into Cartesian product graphs.

cs.DM↗

arXiv2013

Strong Products of Hypergraphs: Unique Prime Factorization Theorems and Algorithms

It is well-known that all finite connected graphs have a unique prime factor decomposition (PFD) with respect to the strong graph product which can be computed in polynomial time. Essential for the PFD computation is the construction of the so-called Cartesian skeleton of the graphs under investigation. In this contribution, we show that every connected thin hypergraph H has a unique prime factorization with respect to the normal and strong (hypergraph) product. Both products coincide with the usual strong graph product whenever H is a graph. We introduce the notion of the Cartesian skeleton of hypergraphs as a natural generalization of the Cartesian skeleton of graphs and prove that it is uniquely defined for thin hypergraphs. Moreover, we show that the Cartesian skeleton of hypergraphs can be determined in O(|E|^2) time and that the PFD can be computed in O(|V|^2|E|) time, for hypergraphs H = (V,E) with bounded degree and bounded rank.

cs.DM↗

arXiv2008

Largest Laplacian Eigenvalue and Degree Sequences of Trees

We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization.

math.CO↗