Related papers: Additive Tree $O(\rho\log n)$-Spanners from Tree B…

Computing a directed minimum spanning tree, called arborescence, is a fundamental algorithmic problem, although not as common as its undirected counterpart. In 1967, Edmonds discussed an elegant solution. It was refined to run in…

Let $G$ be a connected graph with vertex set $V(G)$, and denote by $d_G(u,v)$ the distance from $u$ to $v$ in $G$, for any $u,v \in V(G)$. The average distance of an $n$-vertex connected graph $G$, denoted by $\mu(G)$, is defined to be the…

In the length-constrained minimum spanning tree (MST) problem, we are given an $n$-node edge-weighted graph $G$ and a length constraint $h \geq 1$. Our goal is to find a spanning tree of $G$ whose diameter is at most $h$ with minimum…

Let $\tau(G)$ and $\kappa'(G)$ denote the edge-connectivity and the spanning tree packing number of a graph $G$, respectively. Proving a conjecture initiated by Cioaba and Wong, Liu et al. in 2014 showed that for any simple graph $G$ with…

A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. We prove a conjecture of McDiarmid, Steger, and Welsh, that says that…

We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more…

Given an edge-weighted graph $G$ and $\epsilon>0$, a $(1+\epsilon)$-spanner is a spanning subgraph $G'$ whose shortest path distances approximate those of $G$ within a $(1+\epsilon)$ factor. If $G$ is from certain minor-closed graph…

We establish relations between the bandwidth and the treewidth of bounded degree graphs G, and relate these parameters to the size of a separator of G as well as the size of an expanding subgraph of G. Our results imply that if one of these…

We present new approaches to constructing graph sparsifiers --- weighted subgraphs for which every cut has the same value as the original graph, up to a factor of $(1 \pm \epsilon)$. Our first approach independently samples each edge $uv$…

The semi-streaming model is a variant of the streaming model frequently used for the computation of graph problems. It allows the edges of an $n$-node input graph to be read sequentially in $p$ passes using $\tilde{O}(n)$ space. In this…

We construct $n$-node graphs on which any $O(n)$-size spanner has additive error at least $+\Omega(n^{3/17})$, improving on the previous best lower bound of $\Omega(n^{1/7})$ [Bodwin-Hoppenworth FOCS '22]. Our construction completes the…

The main paradigm of smoothed analysis on graphs suggests that for any large graph $G$ in a certain class of graphs, perturbing slightly the edges of $G$ at random (usually adding few random edges to $G$) typically results in a graph having…

Thin spanning trees lie at the intersection of graph theory, approximation algorithms, and combinatorial optimization. They are central to the long-standing \emph{thin tree conjecture}, which asks whether every $k$-edge-connected graph…

For a connected graph, a {\em minimum vertex separator} is a minimum set of vertices whose removal creates at least two connected components. The vertex connectivity of the graph refers to the size of the minimum vertex separator and a…

In a graph, a spanning tree is said to be a tree t-spanner of the graph if the distance between any two vertices in is at most times their distance in . The tree t-spanner has many applications in networks and distributed environments. In…

We give an $O(g^{1/2} n^{3/2} + g^{3/2} n^{1/2})$-size extended formulation for the spanning tree polytope of an $n$-vertex graph embedded on a surface of genus $g$, improving on the known $O(n^2 + g n)$-size extended formulations following…

The resiliency of a network is its ability to remain \emph{effectively} functioning also when any of its nodes or links fails. However, to reduce operational and set-up costs, a network should be small in size, and this conflicts with the…

In length-constrained minimum spanning tree (MST) we are given an $n$-node graph $G = (V,E)$ with edge weights $w : E \to \mathbb{Z}_{\geq 0}$ and edge lengths $l: E \to \mathbb{Z}_{\geq 0}$ along with a root node $r \in V$ and a…

We study the problem of maximizing the number of spanning trees in a connected graph by adding at most $k$ edges from a given candidate edge set. We give both algorithmic and hardness results for this problem: - We give a greedy algorithm…

We present a new algorithm for generating a uniformly random spanning tree in an undirected graph. Our algorithm samples such a tree in expected $\tilde{O}(m^{4/3})$ time. This improves over the best previously known bound of…