Related papers: Packing mixed hyperarborescences
A mixed graph $G$ is a graph that consists of both undirected and directed edges. An orientation of $G$ is formed by orienting all the undirected edges of $G$, i.e., converting each undirected edge $\{u,v\}$ into a directed edge that is…
We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that…
As an extension of a classical tree-partition problem, we consider decompositions of graphs into edge-disjoint (rooted-)trees with an additional matroid constraint. Specifically, suppose we are given a graph $G=(V,E)$, a multiset…
Wu, Zhang and Li [4] conjectured that the set of vertices of any simple graph $G$ can be equitably partitioned into $\lceil(\Delta(G)+1)/2\rceil$ subsets so that each of them induces a forest of $G$. In this note, we prove this conjecture…
We give an algorithm for finding the arboricity of a weighted, undirected graph, defined as the minimum number of spanning forests that cover all edges of the graph, in $\sqrt{n} m^{1+o(1)}$ time. This improves on the previous best bound of…
We determine to within a constant factor the threshold for the property that two random k-uniform hypergraphs with edge probability p have an edge-disjoint packing into the same vertex set. More generally, we allow the hypergraphs to have…
Given a 3-connected biased graph $\Omega$ with a balancing vertex, and with frame matroid $F(\Omega)$ nongraphic and 3-connected, we determine all biased graphs $\Omega'$ with $F(\Omega') = F(\Omega)$. As a consequence, we show that if $M$…
We consider the problem of inferring a matching hidden in a weighted random $k$-hypergraph. We assume that the hyperedges' weights are random and distributed according to two different densities conditioning on the fact that they belong to…
We prove a conjecture of Fox, Huang, and Lee that characterizes directed graphs that have constant density in all tournaments: they are disjoint unions of trees that are each constructed in a certain recursive way.
Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid. As our main tool, we prove for any infinite graph $G$ with vertex sets $A$…
The aim of this paper is to reveal the discrete convexity of the minimum-cost packings of arborescences and branchings. We first prove that the minimum-cost packings of disjoint $k$ branchings (minimum-cost $k$-branchings) induce an…
The {\sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we improve known parameterized algorithms and…
Among all trees on $n$ vertices with a given degree sequence, how do we maximise or minimise the sum over all adjacent pairs of vertices $x$ and $y$ of $f(\mathrm{deg} x, \mathrm{deg} y)$? Here $f$ is a fixed symmetric function satisfying a…
One perspective on tree decompositions is that they display (low-order) separations of the underlying graph or matroid. The separations displayed by a tree decomposition are necessarily nested. In 2013, Clark and Whittle proved the…
An arborescence in a digraph is an acyclic arc subset in which every vertex execpt a root has exactly one incoming arc. In this paper, we reveal the reconfigurability of the union of $k$ arborescences for fixed $k$ in the following sense:…
Equitable list arboricity, introduced by Zhang in 2016, generalizes the notion of equitable list coloring by requiring the subgraph induced by each color class to be acyclic (instead of edgeless) in addition to the usual upper bound on the…
For an $n$-vertex graph $G$, let $z(G;k)$ denote the number of zero forcing sets of size $k$. A conjecture of Boyer et al. asserts that the path $P_n$ maximizes these numbers coefficientwise among all $n$-vertex graphs; equivalently, the…
The Maximum Leaf Spanning Arborescence problem (MLSA) is defined as follows: Given a directed graph $G$ and a vertex $r\in V(G)$ from which every other vertex is reachable, find a spanning arborescence rooted at $r$ maximizing the number of…
We demonstrate that graph-based models are fully capable of representing higher-order interactions, and have a long history of being used for precisely this purpose. This stands in contrast to a common claim in the recent literature on…
We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight…