A least common ancestor (LCA) of two leaves in a directed acyclic graph (DAG) is a vertex that is an ancestor of both leaves and has no proper descendant that is also their common ancestor. LCAs capture hierarchical relationships in rooted trees and, more generally, in DAGs. In 1981, Aho et al. introduced the problem of determining whether a set of pairwise LCA constraints on a set $X$, of the form $(i,j)<(k,l)$ with $i,j,k,l\in X$, can be realized by a rooted tree whose leaf set is $X$, such that whenever $(i,j)<(k,l)$, the LCA of $i,j$ is a descendant of that of $k,l$. They also presented a polynomial-time algorithm, BUILD, to solve this problem. However, many such constraint systems cannot be realized by any tree, prompting the question of whether they can be realized by a more general DAG. We extend Aho et al.'s framework from trees to DAGs, providing both theoretical and algorithmic foundations for reasoning about LCA constraints in this broader setting. Given a collection $R$ of LCA constraints, we define its $+$-closure $R^+$, capturing additional LCA relations implied by $R$. Using $R^+$, we construct a canonical DAG $G_R$ and prove that $R$ is DAG-realizable if and only if it is realized by $G_R$. We further adapt this construction to phylogenetic networks, defining a canonical network $N_R$ and prove that it is regular, i.e., it coincides with the Hasse diagram of its underlying set system. Finally, we show that for any DAG-realizable $R$, its classical closure - comprising all LCA constraints that hold in every DAG realizing $R$ - coincides with its $+$-closure. All constructions are computable in polynomial time, and we provide explicit algorithms for each. All algorithms developed in this paper are implemented in the freely available Python package RealLCA.

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}∈E(G) if and only if the lowest common ancestor lcaT(x,y) of x and y has label “1”. Pseudo-cographs, or, more generally, 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 FGT 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 P4-sparse and P4-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.

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 O(nlogn)-time recognition algorithm to compute a decomposition of a planar median graph with n vertices into cubes and square-graphs.

Abstract. Horizontal gene transfer (HGT) events partition a gene tree [Formula: see text], and thus its leaf set [Formula: see text], into subsets of genes whose evolutionary history is described by speciation and duplication events alone. Two genes thus are xenologs if and only if they belong to two different sets of this partition [Formula: see text]. Indirect phylogenetic methods can be used to infer the partition [Formula: see text] of [Formula: see text] from sequence similarity or evolutionary distances without any a priori knowledge about the underlying tree [Formula: see text]. In this contribution, we assume that a partition [Formula: see text] of the gene set [Formula: see text] and a usually incompletely resolved estimate [Formula: see text] of the original gene tree on [Formula: see text] are known. We then ask to what extent [Formula: see text] and [Formula: see text] can be combined to determine the horizontal transfer edges in [Formula: see text] and thus the orientation of the HGT events that separate the sets of [Formula: see text]. If [Formula: see text] and [Formula: see text] are compatible, it can be decided for each pair of genes [Formula: see text] and [Formula: see text] whether there always exists or never exists a horizontal gene transfer in [Formula: see text] along the path connecting [Formula: see text] and the most recent common ancestor of [Formula: see text] and [Formula: see text], and thus a directed edge [Formula: see text] in the so-called Fitch graph of the gene family. We generalize this result to insufficiently resolved gene trees. We show that the classification of a gene pair [Formula: see text] 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 [Formula: see text] can be recovered unambiguously from the knowledge of the partition [Formula: see text]. All algorithms developed here are implemented and freely available within the Python package AsymmeTree hosted at https://github.com/david-schaller/AsymmeTree .

The modular decomposition of a symmetric map δ:X×X→Υ (or, equivalently, a set of pairwise-disjoint symmetric binary relations, a 2-structure, or an edge-colored undirected graph) is a natural construction to capture key features of δ in terms of a labeled tree. A map δ is explained by a vertex-labeled rooted tree (T,t) if the label δ(x,y) coincides with the label of the lowest common ancestor of x and y in T, i.e., if δ(x,y)=t(lca(x,y)). Only maps whose modular decomposition does not contain prime nodes, i.e., the symbolic ultrametrics, can be explained in this manner. Here we consider rooted median graphs as a generalization of (modular decomposition) trees to explain symmetric maps. We derive a linear-time algorithm that stepwisely resolves prime vertices in the modular decomposition tree to obtain a rooted and labeled median graph that explains a given symmetric map δ.

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}∈E(G) if and only if the lowest common ancestor lcaT(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: polar-cats are a proper subclass of pseudo-cographs which forms a proper subclass of 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 particular, Polar-cats are precisely those graphs that can be explained by an elementary level-1 network (N,t), i.e., (N,t) contains exactly one cycle C that is rooted at the root ρN of N and where ρN has exactly two children while every vertex distinct from ρN has a unique child that is not located in C. Pseudo-cographs are less restrictive and those graphs that can be explained by particular level-1 networks (N,t) that contain at most one cycle. These findings, eventually, help us to characterize the class of all graphs that can be explained by labeled level-1 networks, namely prime polar-cats. Moreover, we show under which conditions there is a unique least-resolved labeled level-1 network that explains a given graph. In addition, we provide linear-time algorithms to recognize all these types of graphs and to construct level-1 networks to explain them.

The question whether a partition P and a hierarchy H or a tree-like split system S are compatible naturally arises in a wide range of classification problems. In the setting of phylogenetic trees, one asks whether the sets of P coincide with leaf sets of connected components obtained by deleting some edges from the tree T that represents H or S, respectively. More generally, we ask whether a refinement T∗ of T exists such that T∗ and 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 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.

Many properties of molecules vary systematically with changes in the structural formula and can thus be estimated from regression models defined on small structural building blocks, usually functional groups. Typically, such approaches are limited to a particular class of compounds and requires hand-curated lists of chemically plausible groups. This limits their use in particular in the context of generative approaches to explore large chemical spaces. Here we overcome this limitation by proposing a generic group contribution method that iteratively identifies significant regressors of increasing size. To this end, LASSO regression is used and the context-dependent contributions are "anchored" around a reference edge to reduce ambiguities and prevent overcounting due to multiple embeddings. We benchmark our approach, which is available as "Context AwaRe Group cOntribution" ( CARGO), on artificial data, typical applications from chemical thermodynamics. As we shall see, this method yields stable results with accuracies comparable to other regression techniques. As a by-product, we obtain interpretable additive contributions for individual chemical bonds and correction terms depending on local contexts.

Best match graphs (BMGs) are vertex-colored directed graphs that were introduced to model the relationships of genes (vertices) from different species (colors) given an underlying evolutionary tree that is assumed to be unknown. In real-life applications, BMGs are estimated from sequence similarity data. Measurement noise and approximation errors usually result in empirically determined graphs that in general violate characteristic properties of BMGs. The arc modification problems for BMGs aim at correcting such violations and thus provide a means to improve the initial estimates of best match data. We show here that the arc deletion, arc completion and arc editing problems for BMGs are NP-complete and that they can be formulated and solved as integer linear programs. To this end, we provide a novel characterization of BMGs in terms of triples (binary trees on three leaves) and a characterization of BMGs with two colors in terms of forbidden subgraphs.

Antiparallel strong traces (ASTs) are a type of walks in graphs which use every edge exactly twice. They correspond to 1-face embeddings in orientable surfaces and can be used to design self-assembling protein or DNA strands. Based on a novel canonical form invariant for ASTs, gap vector, we provide a linear-time isomorphism test for ASTs and thus, also for orientable 1-face embeddings of graphs. Using the canonical form, we develop an algorithm for enumerating all pairwise non-isomorphic 1-face embeddings of graphs. We compare our algorithm with an independent implementation of a recent algebraic approach (Bašić et al., MATCH Commun. Math. Comput. Chem. 78 (3), 2017) on large data sets. Our results yield the first large-scale enumeration of non-isomorphic embeddings and investigation of their properties.

Motivation: In the absence of horizontal gene transfer it is possible to reconstruct the history of gene families from empirically determined orthology relations, which are equivalent to event-labeled gene trees. Knowledge of the event labels considerably simplifies the problem of reconciling a gene tree T with a species trees S, relative to the reconciliation problem without prior knowledge of the event types. It is well-known that optimal reconciliations in the unlabeled case may violate time-consistency and thus are not biologically feasible. Here we investigate the mathematical structure of the event labeled reconciliation problem with horizontal transfer. Results: We investigate the issue of time-consistency for the event-labeled version of the reconciliation problem, provide a convenient axiomatic framework, and derive a complete characterization of time-consistent reconciliations. This characterization depends on certain weak conditions on the event-labeled gene trees that reflect conditions under which evolutionary events are observable at least in principle. We give an O(|V(T)|log(|V(S)|))-time algorithm to decide whether a time-consistent reconciliation map exists. It does not require the construction of explicit timing maps, but relies entirely on the comparably easy task of checking whether a small auxiliary graph is acyclic. The algorithms are implemented in C++ using the boost graph library and are freely available at https://github.com/Nojgaard/tc-recon. Significance: The combinatorial characterization of time consistency and thus biologically feasible reconciliation is an important step towards the inference of gene family histories with hor- izontal transfer from orthology data, i.e., without presupposed gene and species trees. The fast algorithm to decide time consistency is useful in a broader context because it constitutes an attractive component for all tools that address tree reconciliation problems.

The way gene families and genomes evolve can be understood in detail only when the location of gene duplication episodes in the tree of life can be deciphered. Since most genes belong to larger gene families, the analysis of the gene family histories thus plays an important role in the study of genome evolution. Empirically, one frequently observes that the tree that describes the evolution of species, the species tree, is inconsistent with the tree that is obtained from a group of genes of a gene family (the gene tree). Goodman et al. deduced that this inconsistency might be the result of mistaking paralogs for orthologs. Orthologous genes refer to copies of genes that reveal the phylogeny of species, while paralogous genes have been created by duplication events. Phylogeny reconstruction can help to understand how gene families evolved and to identify the chronology of duplications within a gene family of a single species. Several software tools, including GeneTree, DupTree, NOTUNG, and AUGIST have been developed for this task. There is, however, lack of both test data and evaluation procedures to test, compare, and benchmark their performance and results. We present here a simulation environment designed to generate large gene families with complex duplication histories on which reconstruction algorithms can be tested and software tools can be benchmarked. The simulation of gene family histories starts with the generation of species trees. Within these rooted bifurcating trees the nodes represent species and edges their relation. Specifically, internal nodes represent ancient species whereas leaf nodes represent extant species. Given a number of species N, we generate a random tree T under the Age Model described in Keller-Schmidt et al. This model starts with a rooted tree with two leaves. In an iterative process one of the leaves is selected and two new leaves are attached to it until the tree has N leaves. This model makes use of the idea that the longer a leaf has not been involved in a speciation, the less likely it will be in the future. The user will introduce n number of genes (gene families), which will be placed at the root of the generated species tree T. T will then be traversed in a depth first order. For each visited edge a number of events is sampled from a stochastic Poisson Process P λ , l where λ is the probability of the event to happen and l the branch length. The process may generate none, one or a series of these events: one gene gets duplicated (gene duplication), a group of genes gets duplicated (cluster duplication), the whole group of genes gets duplicated (genome duplication) and one gene of the species gets lost (gene loss). After each gene duplication, one of the copies will be lost with a user defined probability θ, based on the fact that when there is a gene duplication, one of the copies might be lost or become nonfunctional. In the case of a cluster or genome duplication, we apply this probability to every gene in the group, since it is known that in the wake of multiple gene duplications and in particular for genome duplications we have to expect that many duplicated genes are rapidly lost again through the formation of pseudogenes. A small example of a gene family history generated by our simulation is shown in Fig. 1 . We also show the gene tree generated from the gene family history embedded in the species tree. Each leaf node represents a gene and each internal node represents an event (speciation or duplication). This tree is typically depicted as the reconciled tree as in Fig. 2 . A one-gene family history: from a node parent to a node child, there could be duplications and losses of genes. The reconciled tree: the gene tree embedded in the species tree. Each internal node represents an event, either a speciation or a gene duplication. Finally, the algorithm will generate one gene tree for each species, i.e. the pruned reconciled tree containing only genes of a certain species. Furthermore, for each gene family the orthology and homology matrices are computed. To generate the orthology matrix, we say that two genes are orthologous if their lowest common ancestor (LCA) in the reconciled tree represents a speciation event. To generate the homology matrix, a gene a from species i is homologous to gene b from species j if for every gene c from species i and every gene d from species j the LCA(a, b) ≤ LCA(c, b) and LCA(a, b) ≤ LCA(a, d). We propose an algorithm that simulates gene family histories akin to real data. This will allow reconstruction algorithms to measure their accuracy and performance. Given a certain reconstruction method one might ask if the orthology matrix could be deduced from the inferred reconciled tree or if the homology relation between the genes was predicted correctly. Furthermore it could be analysed if the method was able to infer the gene duplications and losses. A method that is able to detect large scale duplications will then identify the cluster and genome duplications generated by our algorithm.

Tree reconciliation problems have long been studied in phylogenetics. A particular variant of the reconciliation problem for a gene tree T and a species tree S assumes that for each interior vertex x of T it is known whether x represents a speciation or a duplication. This problem appears in the context of analyzing orthology data. We show that S is a species tree for T if and only if S displays all rooted triples of T that have three distinct species as their leaves and are rooted in a speciation vertex. A valid reconciliation map can then be found in polynomial time. Simulated data shows that the event-labeled gene trees convey a large amount of information on underlying species trees, even for a large percentage of losses. The knowledge of event labels in a gene tree strongly constrains the possible species tree and, for a given species tree, also the possible reconciliation maps. Nevertheless, many degrees of freedom remain in the space of feasible solutions. In order to disambiguate the alternative solutions additional external constraints as well as optimization criteria could be employed.

Graphs are a versatile structure and abstraction for binary relationships between objects. To gain insight into such relationships, their corresponding graph can be visualized. In the past, many classes of graphs have been defined, e.g. trees, planar graphs, directed acyclic graphs, and visualization algorithms were proposed for these classes. Although many graphs may only be classified as "general" graphs, they can contain substructures that belong to a certain class. Archambault proposed the TopoLayout framework: rather than draw any arbitrary graph using one method, split the graph into components that are homogeneous with respect to one graph class and then draw each component with an algorithm best suited for this class. Graph products constitute a class that arises frequently in graph theory, but for which no visualization algorithm has been proposed until now. In this paper, we present an algorithm for drawing graph products and the aesthetic criterion graph product's drawings are subject to. We show that the popular High-Dimensional Embedder approach applied to cartesian products already respects this aestetic criterion, but has disadvantages. We also present how our method is integrated as a new component into the TopoLayout framework. Our implementation is used for further research of graph products in a biological context.