Divisorial gonality of graphs, the slice rank polynomial method, and tensor products of convex cones

On the Complexity of Finding a Sparse Connected Spanning Subgraph in a Non-Uniform Failure Model

We study a generalization of the classic Spanning Tree problem that allows for a non-uniform failure model. More precisely, edges are either safe or unsafe and we assume that failures only affect unsafe edges. In Unweighted Flexible Graph Connectivity we are given an undirected graph G = (V,E) in which the edge set E is partitioned into a set...

Finding Degree-Constrained Acyclic Orientations

We consider the problem of orienting a given, undirected graph into a (directed) acyclic graph such that the in-degree of each vertex ν is in a prescribed list λ(ν). Variants of this problem have been studied for a long time and with various applications, but mostly without the requirement for acyclicity. Without this requirement, the problem...

Embedding Phylogenetic Trees in Networks of Low Treewidth

Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called Tree Containment, arises when validating networks constructed by phylogenetic inference methods. We present the first algorithm for (rooted) Tree Containment using the treewidth t of the input...

Treewidth is a lower bound on graph gonality

We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. Graphs for which equality holds include the grid graphs and the complete multipartite graphs. We prove that the treewidth lower bound also holds for metric graphs (tropical curves) by constructing for any positive rank divisor on a metric...