49 research outputs found

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

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    The modular decomposition of a graph GG is a natural construction to capture key features of GG in terms of a labeled tree (T,t)(T,t) whose vertices are labeled as "series" (11), "parallel" (00) or "prime". However, full information of GG is provided by its modular decomposition tree (T,t)(T,t) only, if GG is a cograph, i.e., GG does not contain prime modules. In this case, (T,t)(T,t) explains GG, i.e., {x,y}∈E(G)\{x,y\}\in E(G) if and only if the lowest common ancestor lcaT(x,y)\mathrm{lca}_T(x,y) of xx and yy has label "11". Pseudo-cographs, or, more general, GaTEx graphs GG are graphs that can be explained by labeled galled-trees, i.e., labeled networks (N,t)(N,t) that are obtained from the modular decomposition tree (T,t)(T,t) of GG by replacing the prime vertices in TT 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\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 P4P_4-sparse and P4P_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.Comment: 18 pages, 3 figure

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    LIPIcs, Volume 248, ISAAC 2022, Complete Volume

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volum

    Gene Family Histories: Theory and Algorithms

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    Detailed gene family histories and reconciliations with species trees are a prerequisite for studying associations between genetic and phenotypic innovations. Even though the true evolutionary scenarios are usually unknown, they impose certain constraints on the mathematical structure of data obtained from simple yes/no questions in pairwise comparisons of gene sequences. Recent advances in this field have led to the development of methods for reconstructing (aspects of) the scenarios on the basis of such relation data, which can most naturally be represented by graphs on the set of considered genes. We provide here novel characterizations of best match graphs (BMGs) which capture the notion of (reciprocal) best hits based on sequence similarities. BMGs provide the basis for the detection of orthologous genes (genes that diverged after a speciation event). There are two main sources of error in pipelines for orthology inference based on BMGs. Firstly, measurement errors in the estimation of best matches from sequence similarity in general lead to violations of the characteristic properties of BMGs. The second issue concerns the reconstruction of the orthology relation from a BMG. We show how to correct estimated BMG to mathematically valid ones and how much information about orthologs is contained in BMGs. We then discuss implicit methods for horizontal gene transfer (HGT) inference that 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 an undirected graph, the later-divergence-time (LDT) graph. We explore the mathematical structure of LDT graphs and show how much information about all HGT events is contained in such LDT graphs

    Graph Transversals for Hereditary Graph Classes: a Complexity Perspective

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    Within the broad field of Discrete Mathematics and Theoretical Computer Science, the theory of graphs has been of fundamental importance in solving a large number of optimization problems and in modelling real-world situations. In this thesis, we study a topic that covers many aspects of Graph Theory: transversal sets. A transversal set in a graph G is a vertex set that intersects every subgraph of G that belongs to a certain class of graphs. The focus is on vertex cover, feedback vertex set and odd cycle transversal. The decision problems Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal ask, for a given graph G and an integer k, whether there is a corresponding transversal of G of size at most k. These problems are NP-complete in general and our focus is to determine the complexity of the problems when various restrictions are placed on the input, both for the purpose of finding tractable cases and to increase our understanding of the point at which a problem becomes NP-complete. We consider graph classes that are closed under vertex deletion and in particular H-free graphs, i.e. graphs that do not contain a graph H as an induced subgraph. The first chapter is an introduction to the thesis. There we illustrate the motivation of our work and introduce most of the terminology we have used for our research. In the second chapter, we develop a number of structural results for some classes of H-free graphs. The third chapter looks at the Subset Transversal problems: there we prove that Feedback Vertex Set and Odd Cycle Transversal and their subset variants can be solved in polynomial time for both P_4-free and (sP_1+P_3)-free graphs, while for Subset Vertex Cover we show that it can be solved in polynomial time for (sP_1+P_4)-free graphs. The fourth chapter is entirely dedicated to the Connected Vertex Cover problem. The connectivity constraint requires additional proof techniques. We prove this problem can be solved in polynomial time for (sP_1+P_5)-free graphs, even when weights are given to the vertices of the graph. We continue the research on connected transversals in the fifth chapter: we show that Connected Feedback Vertex Set, Connected Odd Cycle Transversal and their extension variants can be solved in polynomial time for both P_4-free and (sP_1+P_3)-free graphs. In the sixth chapter we study the price of independence: can the size of a smallest independent transversal be bounded in terms of the minimum size of a transversal? We establish complete and almost-complete dichotomies which determine for which graph classes such a bound exists and for which cases such a bound is the identity
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