arXiv subjects

Marc Hellmuth

Publications and source records attributed to Marc Hellmuth.

At least 37 records · Page 2Linked to original sources

arXiv2023

Unique Least Common Ancestors and Clusters in Directed Acyclic Graphs

We investigate the connections between clusters and least common ancestors (LCAs) in directed acyclic graphs (DAGs). We focus on the class of DAGs having unique least common ancestors for certain subsets of their minimal elements since these are of interest, particularly as models of phylogenetic networks. Here, we use the close connection between the canonical k-ary transit function and the closure function on a set system to show that pre-k-ary clustering systems are exactly those that derive from a class of DAGs with unique LCAs. Moreover, we show that k-ary T-systems and k-weak hierarchies are associated with DAGs that satisfy stronger conditions on the existence of unique LCAs for sets of size at most k.

cs.DM↗

arXiv2023

Relative Timing Information and Orthology in Evolutionary Scenarios

Evolutionary scenarios describing the evolution of a family of genes within a collection of species comprise the mapping of the vertices of a gene tree $T$ to vertices and edges of a species tree $S$. The relative timing of the last common ancestors of two extant genes (leaves of $T$) and the last common ancestors of the two species (leaves of $S$) in which they reside is indicative of horizontal gene transfers (HGT) and ancient duplications. Orthologous gene pairs, on the other hand, require that their last common ancestors coincides with a corresponding speciation event. The relative timing information of gene and species divergences is captured by three colored graphs that have the extant genes as vertices and the species in which the genes are found as vertex colors: the equal-divergence-time (EDT) graph, the later-divergence-time (LDT) graph and the prior-divergence-time (PDT) graph, which together form an edge partition of the complete graph. Here we give a complete characterization in terms of informative and forbidden triples that can be read off the three graphs and provide a polynomial time algorithm for constructing an evolutionary scenario that explains the graphs, provided such a scenario exists. We show that every EDT graph is perfect. While the information about LDT and PDT graphs is necessary to recognize EDT graphs in polynomial-time for general scenarios, this extra information can be dropped in the HGT-free case. However, recognition of EDT graphs without knowledge of putative LDT and PDT graphs is NP-complete for general scenarios. In contrast, PDT graphs can be recognized in polynomial-time. We finally connect the EDT graph to the alternative definitions of orthology that have been proposed for scenarios with horizontal gene transfer. With one exception, the corresponding graphs are shown to be colored cographs.

q-bio.PE↗

arXiv2023

Fitch Graph Completion

Horizontal gene transfer is an important contributor to evolution. According to Walter M.\ Fitch, two genes are xenologs if they are separated by at least one HGT. More formally, the directed Fitch graph has a set of genes is its vertices, and directed edges $(x,y)$ for all pairs of genes $x$ and $y$ for which $y$ has been horizontally transferred at least once since it diverged from the last common ancestor of $x$ and $y$. Subgraphs of Fitch graphs can be inferred by comparative sequence analysis. In many cases, however, only partial knowledge about the ``full'' Fitch graph can be obtained. Here, we characterize Fitch-satisfiable graphs that can be extended to a biologically feasible ``full'' Fitch graph and derive a simple polynomial-time recognition algorithm. We then proceed to showing that finding the Fitch graphs with total maximum (confidence) edge-weights is an NP-hard problem.

cs.DS↗

arXiv2023

Resolving Prime Modules: The Structure of Pseudo-cographs and Galled-Tree Explainable Graphs

The modular decomposition of a graph $G$ is a natural construction to capture key features of $G$ in terms of a labeled tree $(T,t)$ whose vertices are labeled as "series" ($1$), "parallel" ($0$) or "prime". However, full information of $G$ is provided by its modular decomposition tree $(T,t)$ only, if $G$ is a cograph, i.e., $G$ does not contain prime modules. In this case, $(T,t)$ explains $G$, i.e., $\{x,y\}\in E(G)$ if and only if the lowest common ancestor $\mathrm{lca}_T(x,y)$ of $x$ and $y$ has label "$1$". Pseudo-cographs, or, more general, GaTEx graphs $G$ are graphs that can be explained by labeled galled-trees, i.e., labeled networks $(N,t)$ that are obtained from the modular decomposition tree $(T,t)$ of $G$ by replacing the prime vertices in $T$ by simple labeled cycles. GaTEx graphs can be recognized and labeled galled-trees that explain these graphs can be constructed in linear time. In this contribution, we provide a novel characterization of GaTEx graphs in terms of a set $\mathfrak{F}_{\mathrm{GT}}$ of 25 forbidden induced subgraphs. This characterization, in turn, allows us to show that GaTEx graphs are closely related to many other well-known graph classes such as $P_4$-sparse and $P_4$-reducible graphs, weakly-chordal graphs, perfect graphs with perfect order, comparability and permutation graphs, murky graphs as well as interval graphs, Meyniel graphs or very strongly-perfect and brittle graphs. Moreover, we show that every GaTEx graph as twin-width at most 1 and and provide linear-time algorithms to solve several NP-hard problems (clique, coloring, independent set) on GaTEx graphs by utilizing the structure of the underlying galled-trees they explain.

math.CO↗

arXiv2023

The Theory of Gene Family Histories

Most genes are part of larger families of evolutionary related genes. The history of gene families typically involves duplications and losses of genes as well as horizontal transfers into other organisms. The reconstruction of detailed gene family histories, i.e., the precise dating of evolutionary events relative to phylogenetic tree of the underlying species has remained a challenging topic despite their importance as a basis for detailed investigations into adaptation and functional evolution of individual members of the gene family. The identification of orthologs, moreover, is a particularly important subproblem of the more general setting considered here. In the last few years, an extensive body of mathematical results has appeared that tightly links orthology, a formal notion of best matches among genes, and horizontal gene transfer. The purpose of this chapter is the broadly outline some of the key mathematical insights and to discuss their implication for practical applications. In particular, we focus on tree-free methods, i.e., methods to infer orthology or horizontal gene transfer as well as gene trees, species trees and reconciliations between them without using \emph{a priori} knowledge of the underlying trees or statistical models for the inference of phylogenetic trees. Instead, the initial step aims to extract binary relations among genes.

q-bio.PE↗

arXiv2023

On a generalization of median graphs: $k$-median graphs

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. To be more formal, a graph $G$ is a median graph if, for all $μ, u,v\in V(G)$, it holds that $|I(μ,u)\cap I(μ,v)\cap I(u,v)|=1$ where $I(x,y)$ denotes the set of all vertices that lie on shortest paths connecting $x$ and $y$. In this paper we are interested in a natural generalization of median graphs, called $k$-median graphs. A graph $G$ is a $k$-median graph, if there are $k$ vertices $μ_1,\dots,μ_k\in V(G)$ such that, for all $u,v\in V(G)$, it holds that $|I(μ_i,u)\cap I(μ_i,v)\cap I(u,v)|=1$, $1\leq i\leq k$. By definition, every median graph with $n$ vertices is an $n$-median graph. We provide several characterizations of $k$-median graphs that, in turn, are used to provide many novel characterizations of median graphs.

math.CO↗

arXiv2023

Quasi-Best Match Graphs

Quasi-best match graphs (qBMGs) are a hereditary class of directed, properly vertex-colored graphs. They arise naturally in mathematical phylogenetics as a generalization of best match graphs, which formalize the notion of evolutionary closest relatedness of genes (vertices) in multiple species (vertex colors). They are explained by rooted trees whose leaves correspond to vertices. In contrast to BMGs, qBMGs represent only best matches at a restricted phylogenetic distance. We provide characterizations of qBMGs that give rise to polynomial-time recognition algorithms and identify the BMGs as the qBMGs that are color-sink-free. Furthermore, two-colored qBMGs are characterized as directed graphs satisfying three simple local conditions, two of which have appeared previously, namely bi-transitivity in the sense of Das et al. (2021) and a hierarchy-like structure of out-neighborhoods, i.e., $N(x)\cap N(y)\in\{N(x),N(y),\emptyset\}$ for any two vertices $x$ and $y$. Further results characterize qBMGs that can be explained by binary phylogenetic trees.

math.CO↗

arXiv2022

From Modular Decomposition Trees to Level-1 Networks: Pseudo-Cographs, Polar-Cats and Prime Polar-Cats

The modular decomposition of a graph $G$ is a natural construction to capture key features of $G$ in terms of a labeled tree $(T,t)$ whose vertices are labeled as "series" ($1$), "parallel" ($0$) or "prime". However, full information of $G$ is provided by its modular decomposition tree $(T,t)$ only, if $G$ does not contain prime modules. In this case, $(T,t)$ explains $G$, i.e., $\{x,y\}\in E(G)$ if and only if the lowest common ancestor $\mathrm{lca}_T(x,y)$ of $x$ and $y$ has label "$1$". This information, however, gets lost whenever $(T,t)$ contains vertices with label "prime". In this contribution, we aim at replacing "prime" vertices in $(T,t)$ by simple 0/1-labeled cycles, which leads to the concept of rooted labeled level-1 networks $(N,t)$. We characterize graphs that can be explained by such level-1 networks $(N,t)$, which generalizes the concept of graphs that can be explained by labeled trees, that is, cographs. We provide three novel graph classes: \emph{polar-cats} are a proper subclass of \emph{pseudo-cographs} which forms a proper subclass of \emph{prime polar-cats}. In particular, every cograph is a pseudo-cograph and prime polar-cats are precisely those graphs that can be explained by a labeled level-1 network. The class of prime polar-cats is defined in terms of the modular decomposition of graphs and the property that all prime modules "induce" polar-cats. We provide a plethora of structural results and characterizations for graphs of these new classes. In addition, we show under which conditions there is a unique least-resolved labeled level-1 network that explains a given graph and provide linear-time algorithms to recognize all these types of graphs and to construct level-1 networks to explain them.

math.CO↗

arXiv2022

Clustering Systems of Phylogenetic Networks

Rooted acyclic graphs appear naturally when the phylogenetic relationship of a set $X$ of taxa involves not only speciations but also recombination, horizontal transfer, or hybridization, that cannot be captured by trees. A variety of classes of such networks have been discussed in the literature, including phylogenetic, level-1, tree-child, tree-based, galled tree, regular, or normal networks as models of different types of evolutionary processes. Clusters arise in models of phylogeny as the sets $\mathtt{C}(v)$ of descendant taxa of a vertex $v$. The clustering system $\mathscr{C}_N$ comprising the clusters of a network $N$ conveys key information on $N$ itself. In the special case of rooted phylogenetic trees, $T$ is uniquely determined by its clustering system $\mathscr{C}_T$. Although this is no longer true for networks in general, it is of interest to relate properties of $N$ and $\mathscr{C}_N$. Here, we systematically investigate the relationships of several well-studied classes of networks and their clustering systems. The main results are correspondences of classes of networks and clustering system of the following form: If $N$ is a network of type $\mathbb{X}$, then $\mathcal{C}_N$ satisfies $\mathbb{Y}$, and conversely if $\mathscr{C}$ is a clustering system satisfying $\mathbb{Y}$ then there is network $N$ of type $\mathbb{X}$ such that $\mathscr{C}\subseteq\mathscr{C}_N$.This, in turn, allows us to investigate the mutual dependencies between the distinct types of networks in much detail.

q-bio.PE↗

arXiv2021

Orientation of Fitch Graphs and Detection of Horizontal Gene Transfer in Gene Trees

Horizontal gene transfer events partition a gene tree $T$ and thus, its leaf set into subsets of genes whose evolutionary history is described by speciation and duplication events alone. Indirect phylogenetic methods can be used to infer such partitions $\mathcal{P}$ from sequence similarity or evolutionary distances without any a priory knowledge about the underlying tree $T$. In this contribution, we assume that such a partition $\mathcal{P}$ of a set of genes $X$ is given and that, independently, an estimate $T$ of the original gene tree on $X$ has been derived. We then ask to what extent $T$ and the xenology information, i.e., $\mathcal{P}$ can be combined to determine the horizontal transfer edges in $T$. We show that for each pair of genes $x$ and $y$ with $x,y$ being in different parts of $\mathcal{P}$, it can be decided whether there always exists or never exists a horizontal gene transfer in $T$ along the path connecting $y$ and the most recent common ancestor of $x$ and $y$. This problem is equivalent to determining the presence or absence of the directed edge $(x,y)$ in so-called Fitch graphs; a more fine-grained version of graphs that represent the dependencies between the sets in $\mathcal{P}$. We then consider the generalization to insufficiently resolved gene trees and show that analogous results can be obtained. We show that the classification of $(x,y)$ can be computed in constant time after linear-time preprocessing. Using simulated gene family histories, we observe empirically that the vast majority of horizontal transfer edges in the gene tree $T$ can be recovered unambiguously.

cs.DM↗

arXiv2021

Compatibility of Partitions with Trees, Hierarchies, and Split Systems

The question whether a partition $\mathcal{P}$ and a hierarchy $\mathcal{H}$ or a tree-like split system $\mathfrak{S}$ are compatible naturally arises in a wide range of classification problems. In the setting of phylogenetic trees, one asks whether the sets of $\mathcal{P}$coincide with leaf sets of connected components obtained by deleting some edges from the tree $T$ that represents $\mathcal{H}$ or $\mathfrak{S}$, respectively. More generally, we ask whether a refinement $T^*$ of $T$ exists such that $T^*$ and $\mathcal{P}$ are compatible in this sense. The latter is closely related to the question as to whether there exists a tree at all that is compatible with $\mathcal{P}$. We report several characterizations for (refinements of) hierarchies and split systems that are compatible with (systems of) partitions. In addition, we provide a linear-time algorithm to check whether refinements of trees and a given partition are compatible. The latter problem becomes NP-complete but fixed-parameter tractable if a system of partitions is considered instead of a single partition. In this context, we also explore the close relationship of the concept of compatibility and so-called Fitch maps.

cs.DM↗

arXiv2021

Planar Median Graphs and Cubesquare-Graphs

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median graphs. More specifically, we characterize when a planar graph $G$ is a median graph in terms of forbidden subgraphs and the structure of isometric cycles in $G$, and also in terms of subgraphs of $G$ that are contained inside and outside of 4-cycles with respect to an arbitrary planar embedding of $G$. These results lead us to a new characterization of planar median graphs in terms of cubesquare-graphs that is, graphs that can be obtained by starting with cubes and square graphs, and iteratively replacing 4-cycle boundaries (relative to some embedding) by cubes or square-graphs. As a corollary we also show that a graph is planar median if and only if it can be obtained from cubes and square-graphs by a sequence of ``square-boundary'' amalgamations. These considerations also lead to an $\mathcal{O}(n\log n)$-time recognition algorithm to compute a decomposition of a planar median graph with $n$ vertices into cubes and square-graphs.

math.CO↗

arXiv2021

A Simple Linear-Time Algorithm for the Common Refinement of Rooted Phylogenetic Trees on a Common Leaf Set

Background. The supertree problem, i.e., the task of finding a common refinement of a set of rooted trees is an important topic in mathematical phylogenetics. The special case of a common leaf set $L$ is known to be solvable in linear time. Existing approaches refine one input tree using information of the others and then test whether the results are isomorphic. Results. A linear-time algorithm, LinCR, for constructing the common refinement $T$ of $k$ input trees with a common leaf set is proposed that explicitly computes the parent function of $T$ in a bottom-up approach. Conclusion. LinCR is simpler to implement than other asymptotically optimal algorithms for the problem and outperforms the alternatives in empirical comparisons. Availability. An implementation of LinCR in Python is freely available at https://github.com/david-schaller/tralda.

cs.DS↗

arXiv2021

Construction of $k$-matchings and $k$-regular subgraphs in graph products

A $k$-matching $M$ of a graph $G=(V,E)$ is a subset $M\subseteq E$ such that each connected component in the subgraph $F = (V,M)$ of $G$ is either a single-vertex graph or $k$-regular, i.e., each vertex has degree $k$. In this contribution, we are interested in $k$-matchings within the four standard graph products: the Cartesian, strong, direct and lexicographic product. As we shall see, the problem of finding non-empty $k$-matchings ($k\geq 3$) in graph products is NP-complete. Due to the general intractability of this problem, we focus on distinct polynomial-time constructions of $k$-matchings in a graph product $G\star H$ that are based on $k_G$-matchings $M_G$ and $k_H$-matchings $M_H$ of its factors $G$ and $H$, respectively. In particular, we are interested in properties of the factors that have to be satisfied such that these constructions yield a maximum $k$-matching in the respective products. Such constructions are also called "well-behaved" and we provide several characterizations for this type of $k$-matchings. Our specific constructions of $k$-matchings in graph products satisfy the property of being weak-homomorphism preserving, i.e., constructed matched edges in the product are never "projected" to unmatched edges in the factors. This leads to the concept of weak-homomorphism preserving $k$-matchings. Although the specific $k$-matchings constructed here are not always maximum $k$-matchings of the products, they have always maximum size among all weak-homomorphism preserving $k$-matchings. Not all weak-homomorphism preserving $k$-matchings, however, can be constructed in our manner. We will, therefore, determine the size of maximum-sized elements among all weak-homomorphims preserving $k$-matching within the respective graph products, provided that the matchings in the factors satisfy some general assumptions.

math.CO↗

arXiv2021

Cayley Graphs of Semigroups Applied to Atom Tracking in Chemistry

While atom tracking with isotope-labeled compounds is an essential and sophisticated wet-lab tool in order to, e.g., illuminate reaction mechanisms, there exists only a limited amount of formal methods to approach the problem. Specifically when large (bio-)chemical networks are considered where reactions are stereo-specific, rigorous techniques are inevitable. We present an approach using the right Cayley graph of a monoid in order to track atoms concurrently through sequences of reactions and predict their potential location in product molecules. This can not only be used to systematically build hypothesis or reject reaction mechanisms (we will use the ANRORC mechanism "Addition of the Nucleophile, Ring Opening, and Ring Closure" as an example), but also to infer naturally occurring subsystems of (bio-)chemical systems. Our results include the analysis of the carbon traces within the TCA cycle and infer subsystems based on projections of the right Cayley graph onto a set of relevant atoms.

q-bio.MN↗

arXiv2021

Combining Orthology and Xenology Data in a Common Phylogenetic Tree

A rooted tree $T$ with vertex labels $t(v)$ and set-valued edge labels $λ(e)$ defines maps $δ$ and $\varepsilon$ on the pairs of leaves of $T$ by setting $δ(x,y)=q$ if the last common ancestor $\text{lca}(x,y)$ of $x$ and $y$ is labeled $q$, and $m\in \varepsilon(x,y)$ if $m\inλ(e)$ for at least one edge $e$ along the path from $\text{lca}(x,y)$ to $y$. We show that a pair of maps $(δ,\varepsilon)$ derives from a tree $(T,t,λ)$ if and only if there exists a common refinement of the (unique) least-resolved vertex labeled tree $(T_δ,t_δ)$ that explains $δ$ and the (unique) least resolved edge labeled tree $(T_{\varepsilon},λ_{\varepsilon})$ that explains $\varepsilon$ (provided both trees exist). This result remains true if certain combinations of labels at incident vertices and edges are forbidden.

math.CO↗

arXiv2021

Generalized Fitch Graphs III: Symmetrized Fitch maps and Sets of Symmetric Binary Relations that are explained by Unrooted Edge-labeled Trees

Binary relations derived from labeled rooted trees play an import role in mathematical biology as formal models of evolutionary relationships. The (symmetrized) Fitch relation formalizes xenology as the pairs of genes separated by at least one horizontal transfer event. As a natural generalization, we consider symmetrized Fitch maps, that is, symmetric maps $\varepsilon$ that assign a subset of colors to each pair of vertices in $X$ and that can be explained by a tree $T$ with edges that are labeled with subsets of colors in the sense that the color $m$ appears in $\varepsilon(x,y)$ if and only if $m$ appears in a label along the unique path between $x$ and $y$ in $T$. We first give an alternative characterization of the monochromatic case and then give a characterization of symmetrized Fitch maps in terms of compatibility of a certain set of quartets. We show that recognition of symmetrized Fitch maps is NP-complete. In the restricted case where $|\varepsilon(x,y)|\leq 1$ the problem becomes polynomial, since such maps coincide with class of monochromatic Fitch maps whose graph-representations form precisely the class of complete multi-partite graphs.

cs.DM↗

arXiv2021

Indirect Identification of Horizontal Gene Transfer

Several implicit methods to infer Horizontal Gene Transfer (HGT) focus on pairs of genes that have diverged only after the divergence of the two species in which the genes reside. This situation defines the edge set of a graph, the later-divergence-time (LDT) graph, whose vertices correspond to genes colored by their species. We investigate these graphs in the setting of relaxed scenarios, i.e., evolutionary scenarios that encompass all commonly used variants of duplication-transfer-loss scenarios in the literature. We characterize LDT graphs as a subclass of properly vertex-colored cographs, and provide a polynomial-time recognition algorithm as well as an algorithm to construct a relaxed scenario that explains a given LDT. An edge in an LDT graph implies that the two corresponding genes are separated by at least one HGT event. The converse is not true, however. We show that the complete xenology relation is described by an rs-Fitch graph, i.e., a complete multipartite graph satisfying constraints on the vertex coloring. This class of vertex-colored graphs is also recognizable in polynomial time. We finally address the question "how much information about all HGT events is contained in LDT graphs" with the help of simulations of evolutionary scenarios with a wide range of duplication, loss, and HGT events. In particular, we show that a simple greedy graph editing scheme can be used to efficiently detect HGT events that are implicitly contained in LDT graphs.

q-bio.PE↗