An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees

The relationship between two important problems in tree pattern matching, the largest common subtree and the smallest common supertree problems, is established by means of simple constructions, which allow one to obtain a largest common subtree of two trees from a smallest common supertree of them, and vice versa. These constructions are the same for isomorphic, homeomorphic, topological, and minor embeddings, they take only time linear in the size of the trees, and they turn out to have a clear algebraic meaning.Comment: 32 page

Finding k-secluded trees faster

We revisit the k-SECLUDED TREE problem. Given a vertex-weighted undirected graph G, its objective is to find a maximum-weight induced subtree T whose open neighborhood has size at most k. We present a fixed-parameter tractable algorithm that solves the problem in time 2 O(klog⁡k)⋅n O(1), improving on a double-exponential running time from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from a single vertex, our algorithm grows a k-secluded tree by branching on vertices in the open neighborhood of the current tree T. To bound the branching depth, we prove a structural result that can be used to identify a vertex that belongs to the neighborhood of any k-secluded supertree T ′⊇T once the open neighborhood of T becomes sufficiently large. We extend the algorithm to enumerate compact descriptions of all maximum-weight k-secluded trees, which allows us to count them as well.</p

Finding k-secluded trees faster

We revisit the k-SECLUDED TREE problem. Given a vertex-weighted undirected graph G, its objective is to find a maximum-weight induced subtree T whose open neighborhood has size at most k. We present a fixed-parameter tractable algorithm that solves the problem in time 2 O(klog⁡k)⋅n O(1), improving on a double-exponential running time from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from a single vertex, our algorithm grows a k-secluded tree by branching on vertices in the open neighborhood of the current tree T. To bound the branching depth, we prove a structural result that can be used to identify a vertex that belongs to the neighborhood of any k-secluded supertree T ′⊇T once the open neighborhood of T becomes sufficiently large. We extend the algorithm to enumerate compact descriptions of all maximum-weight k-secluded trees, which allows us to count them as well.</p

Breath-giving cooperation: critical review of origin of mitochondria hypotheses Major unanswered questions point to the importance of early ecology

The origin of mitochondria is a unique and hard evolutionary problem, embedded within the origin of eukaryotes. The puzzle is challenging due to the egalitarian nature of the transition where lower-level units took over energy metabolism. Contending theories widely disagree on ancestral partners, initial conditions and unfolding of events. There are many open questions but there is no comparative examination of hypotheses. We have specified twelve questions about the observable facts and hidden processes leading to the establishment of the endosymbiont that a valid hypothesis must address. We have objectively compared contending hypotheses under these questions to find the most plausible course of events and to draw insight on missing pieces of the puzzle. Since endosymbiosis borders evolution and ecology, and since a realistic theory has to comply with both domains' constraints, the conclusion is that the most important aspect to clarify is the initial ecological relationship of partners. Metabolic benefits are largely irrelevant at this initial phase, where ecological costs could be more disruptive. There is no single theory capable of answering all questions indicating a severe lack of ecological considerations. A new theory, compliant with recent phylogenomic results, should adhere to these criteria

A survey of parameterized algorithms and the complexity of edge modification

The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

Finding Smallest Supertrees under Minor Containment

The diversity of application areas relying on tree-structured data results in a wide interest in algorithms which determine differences or similarities among trees. One way of measuring the similarity between trees is to find the smallest common superstructure or supertree, where common elements are typically defined in terms of a mapping or embedding. In the simplest case, a supertree will contain exact copies of each input tree, so that for each input tree, each vertex of a tree can be mapped to a vertex in the supertree such that each edge maps to the corresponding edge. More general mappings allow for the extraction of more subtle common elements captured by looser definitions of similarity. We consider supertrees under the general mapping of minor containment, where a graph G is a minor of a graph H if it is possible to map each vertex in G to a connected subgraph of H such that each edge in G maps to a path in H. Minor containment generalizes both subgraph isomorphism and topol..

Finding smallest supertrees under minor containment

The diversity of application areas relying on tree-structured data results in a wide interest in algorithms which determine differences or similarities among trees. One way of measuring the similarity between trees is to find the smallest common superstructure or supertree, where common elements are typically defined in terms of a mapping or embedding. In the simplest case, a supertree will contain exact copies of each input tree, so that for each input tree, each vertex of a tree can be mapped to a vertex in the supertree such that each edge maps to the corresponding edge. More general mappings allow for the extraction of more subtle common elements captured by looser definitions of similarity. We consider supertrees under the general mapping of minor containment. Minor containment generalizes both subgraph isomorphism and topological embedding; as a consequence of this generality, however, it is NP-complete to determine whether or not G is a minor of H, even for general trees. By focusing on trees of bounded degree, we obtain an O(n^3) algorithm which determines the smallest tree T such that both of the input trees are minors of T, even when the trees are assumed to be unrooted and unordered