Related papers: Lower Bounds on Tree Covers

We define and study analogs of probabilistic tree embedding and tree cover for directed graphs. We define the notion of a DAG cover of a general directed graph $G$: a small collection $D_1,\dots D_g$ of DAGs so that for all pairs of…

Codes over trees were introduced recently to bridge graph theory and coding theory with diverse applications in computer science and beyond. A central challenge lies in determining the maximum number of labelled trees over $n$ nodes with…

The threshold-$k$ metric dimension ($\mathrm{Tmd}_k$) of a graph is the minimum number of sensors -- a subset of the vertex set -- needed to uniquely identify any vertex in the graph, solely based on its distances from the sensors, when the…

Let $s(n)$ be the minimum number of edges in a graph that contains every $n$-vertex tree as a subgraph. Chung and Graham [J. London Math. Soc. 1983] claim to prove that $s(n)\leqslant O(n\log n)$. We point out a mistake in their proof. The…

A {\em tree cover} of a metric space $(X,d)$ is a collection of trees, so that every pair $x,y\in X$ has a low distortion path in one of the trees. If it has the stronger property that every point $x\in X$ has a single tree with low…

Tree-decompositions and treewidth are of fundamental importance in structural and algorithmic graph theory. The "spread" of a tree-decomposition is the minimum integer $s$ such that every vertex lies in at most $s$ bags. A…

Given a graph $G=(V,E)$, a tree cover is a collection of trees $\mathcal{T}=\{T_1,T_2,...,T_q\}$, such that for every pair of vertices $u,v\in V$ there is a tree $T\in\mathcal{T}$ that contains a $u-v$ path with a small stretch. If the…

Given a connected graph $G$ and its vertex $x$, let $U_x(G)$ denote the universal cover of $G$ obtained by unfolding $G$ into a tree starting from $x$. Let $T=T(n)$ be the minimum number such that, for graphs $G$ and $H$ with at most $n$…

The metric dimension of a graph is the minimum size of a set of vertices such that each vertex is uniquely determined by the distances to the vertices of that set. Our aim is to upper-bound the order $n$ of a graph in terms of its diameter…

We show that there exists a graph $G$ with $O(n)$ nodes, where any forest of $n$ nodes is a node-induced subgraph of $G$. Furthermore, for constant arboricity $k$, the result implies the existence of a graph with $O(n^k)$ nodes that…

An out-tree $T$ of a directed graph $D$ is a rooted tree subgraph with all arcs directed outwards from the root. An out-branching is a spanning out-tree. By $l(D)$ and $l_s(D)$ we denote the maximum number of leaves over all out-trees and…

Given a graph $G=(V,E)$ with non-negative real edge lengths and an integer parameter $k$, the Min-Max k-Tree Cover problem seeks to find a set of at most $k$ subtrees of $G$, such that the union of the trees is the vertex set $V$. The…

We prove the first nontrivial worst-case lower bounds for two closely related problems. First, $\Omega(n^{3/2})$ degree-1 reductions, series-parallel reductions, and $\Delta$Y transformations are required in the worst case to reduce an…

We show that for every $n$-vertex graph with at least one edge, its treewidth is greater than or equal to $n \lambda_{2} / (\Delta + \lambda_{2}) - 1$, where $\Delta$ and $\lambda_{2}$ are the maximum degree and the second smallest…

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…

We consider the message complexity of verifying whether a given subgraph of the communication network forms a tree with specific properties both in the KT-$\rho$ (nodes know their $\rho$-hop neighborhood, including node IDs) and the KT-$0$…

A vertex set $S$ is a generalized $k$-independent set if the induced subgraph $G[S]$ contains no tree on $k$ vertices. The generalized $k$-independence number $\alpha_k(G)$ is the maximum size of such a set. For a tree $T$ with $n$…

Let $T$ be a tree on $t$ vertices. We prove that for every positive integer $k$ and every graph $G$, either $G$ contains $k$ pairwise vertex-disjoint subgraphs each having a $T$ minor, or there exists a set $X$ of at most $t(k-1)$ vertices…

A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi states that every $n$-vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$-vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on…

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement…