Related papers: Spanning directed trees with many leaves

A "tree-partition" of a graph $G$ is a partition of $V(G)$ such that identifying the vertices in each part gives a tree. It is known that every graph with treewidth $k$ and maximum degree $\Delta$ has a tree-partition with parts of size…

For any integer $k\geq 2$, a spanning $k$-ended tree is a spanning tree with at most $k$ leaves. In this paper, we provide a tight spectral radius condition for the existence of a spanning $k$-ended tree in $t$-connected graphs, which…

Let $k\geq2$ be an integer. A tree $T$ is called a $k$-tree if $d_T(v)\leq k$ for each $v\in V(T)$, that is, the maximum degree of a $k$-tree is at most $k$. Let $\lambda_1(D(G))$ denote the distance spectral radius in $G$, where $D(G)$…

The dynamical phenomena of complex networks are very difficult to predict from local information due to the rich microstructures and corresponding complex dynamics. On the other hands, it is a horrible job to compute some stochastic…

The decycling number of a graph $G$ is the minimum number of vertices whose removal from $G$ results in an acyclic subgraph. It is known that determining the decycling number of a graph $G$ is equivalent to finding the maximum induced…

We consider the ``minimum degree spanning tree'' problem. As input, we receive an undirected, connected graph $G=(V, E)$ with $n$ nodes and $m$ edges, and our task is to find a spanning tree $T$ of $G$ that minimizes $\max_{u \in V}…

A chief problem in phylogenetics and database theory is the computation of a maximum consistent tree from a set of rooted or unrooted trees. A standard input are triplets, rooted binary trees on three leaves, or quartets, unrooted binary…

The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…

Let $G$ be a simple graph on $n$ vertices. We consider the problem LIS of deciding whether there exists an induced subtree with exactly $i \leq n$ vertices and $\ell$ leaves in $G$. We study the associated optimization problem, that…

The strong thin tree conjecture states that every $k$-edge-connected graph $G$ contains an $O(1/k)$-thin spanning tree, meaning a spanning tree which contains at most an $O(1/k)$ fraction of the edges across each cut in $G$. This conjecture…

We introduce a new Steiner-type problem for directed graphs named \textsc{$q$-Root Steiner Tree}. Here one is given a directed graph $G=(V,A)$ and two subsets of its vertices, $R$ of size $q$ and $T$, and the task is to find a minimum size…

We show two results related to the Hamiltonicity and $k$-Path algorithms in undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10]. First, we demonstrate that the technique used can be generalized to finding some…

For a graph $G$, let $c_k(G)$ be the number of spanning trees of $G$ with maximum degree at most $k$. For $k \ge 3$, it is proved that every connected $n$-vertex $r$-regular graph $G$ with $r \ge \frac{n}{k+1}$ satisfies $$ c_k(G)^{1/n} \ge…

In the laminar-constrained spanning tree problem, the goal is to find a minimum-cost spanning tree which respects upper bounds on the number of times each cut in a given laminar family is crossed. This generalizes the well-studied…

Graphs with bounded treewidth and bounded maximum degree are known to have tree-partitions of bounded width. What can be said if the bounded treewidth assumption is strengthened to bounded pathwidth? We prove that every graph with bounded…

We introduce a graph partitioning problem motivated by computational topology and propose two algorithms that produce approximate solutions. Specifically, given a weighted, undirected graph $G$ and a positive integer $k$, we desire to find…

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected…

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is…

In this paper, we investigate the computational complexity of subgraph reconfiguration problems in directed graphs. More specifically, we focus on the problem of reconfiguring arborescences in a digraph, where an arborescence is a directed…

We give almost-linear-time algorithms for approximating rooted minimum cut and maximum arborescence packing in directed graphs, two problems that are dual to each other [Edm73]. More specifically, for an $n$-vertex, $m$-edge directed graph…