Related papers: A matroid associated with a phylogenetic tree
Optimal transport provides a metric which quantifies the dissimilarity between probability measures. For measures supported in discrete metric spaces, finding the optimal transport distance has cubic time complexity in the size of the…
Let ${\cal G}=(G,w)$ be a weighted simple finite connected graph, that is, let $G$ be a simple finite connected graph endowed with a function $w$ from the set of the edges of $G$ to the set of real numbers. For any subgraph $G'$ of $G$, we…
The vertex (resp. edge) metric dimension of a connected graph G; denoted by dim(G) (resp. edim(G)), is defined as the size of a smallest set S in V (G) which distinguishes all pairs of vertices (resp. edges) in G: Bounds dim(G) <=…
Finite metric trees are known to have strict 1-negative type. In this paper we introduce a new family of inequalities that quantify the extent of the "strictness" of the 1-negative type inequalities for finite metric trees. These…
Let $T$ be a tree rooted at $r$. Two vertices of $T$ are related if one is a descendant of the other; otherwise, they are unrelated. Two subsets $A$ and $B$ of $V(T)$ are unrelated if, for any $a\in A$ and $b\in B$, $a$ and $b$ are…
Ultrametric matrices have a rich structure that is not apparent from their definition. Notably, the subclass of strictly ultrametric matrices are covariance matrices of certain weighted rooted binary trees. In applications, these matrices…
Based on solid theoretical foundations, we present strong evidences that a number of real-life networks, taken from different domains like Internet measurements, biological data, web graphs, social and collaboration networks, exhibit…
We study two graph parameters defined via tree decompositions: tree-independence number and induced matching treewidth. Both parameters are defined similarly as treewidth, but with respect to different measures of a tree decomposition…
Let M to be a matroid defined on a finite set E. A subset L of E is locked in M if L is 2-connected in M, the complement of L is 2-connected in the dual M*, and min{r(L), r*(complement of L)} is greater than 1. In this paper, we prove that…
A biased graph is a graph with a class of selected circles ("cycles", "circuits"), called "balanced", such that no theta subgraph contains exactly two balanced circles. A biased graph has two natural matroids, the frame matroid and the lift…
Contraction$^*$-depth is a matroid depth parameter analogous to tree-depth of graphs. We establish the matroid analogue of the classical graph theory result asserting that the tree-depth of a graph $G$ is the minimum height of a rooted…
This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a…
Identifiability is a crucial property for a statistical model since distributions in the model uniquely determine the parameters that produce them. In phylogenetics, the identifiability of the tree parameter is of particular interest since…
Matroids and semigraphoids are discrete structures abstracting and generalizing linear independence among vectors and conditional independence among random variables, respectively. Despite the different nature of conditional independence…
Since Darwin, species trees have been used as a simplified description of the relationships which summarize the complicated network $N$ of reality. Recent evidence of hybridization and lateral gene transfer, however, suggest that there are…
Structural parameters of graphs, such as treewidth, play a central role in the study of the parameterized complexity of graph problems. Motivated by the study of parametrized algorithms on phylogenetic networks, scanwidth was introduced…
In this paper we investigate the geometry of a discrete Bayesian network whose graph is a tree all of whose variables are binary and the only observed variables are those labeling its leaves. We provide the full geometric description of…
Minimal spanning trees on infinite vertex sets are investigated. A criterion for minimality of a spanning tree having a finite length is obtained, which generalizes the corresponding classical result for finite sets. It is given an analytic…
In classification and forecasting with tabular data, one often utilizes tree-based models. Those can be competitive with deep neural networks on tabular data and, under some conditions, explainable. The explainability depends on the depth…
We prove that every infinite sequence of skew-symmetric or symmetric matrices M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such that M_i is isomorphic to a principal submatrix of the Schur complement of a…