Related papers: Outer linear measure of connected sets via Steiner…
Various questions related to distances between vertices of simple, finite graphs are of interest to extremal graph theorists. The Steiner distance of a set of $k$ vertices is a natural generalization of the regular distance. We extend…
Measuring the importance of nodes in a network with a centrality measure is a core task in any network application. There are many measures available and it is speculated that many encode similar information. We give an explicit non-linear…
Minimal spanning trees on infinite vertex sets are investigated. A criterion for minimality of a spanning tree having a finite length is obtained, which generalizes the corresponding classical result for finite sets. It is given an analytic…
We introduce the notion of metric semilattice on the metric space and prove the criterion of $\R$-tree as connected geodesic metric space $X$ admitting the partial order, such that $X$ is semilinear metric semilattice. Also we state the…
The tree spanner problem for a graph $G$ is as follows: For a given integer $k$, is there a spanning tree $T$ of $G$ (called a tree $k$-spanner) such that the distance in $T$ between every pair of vertices is at most $k$ times their…
It is a classic result in spectral theory that the limit distribution of the spectral measure of random graphs G(n, p) converges to the semicircle law in case np tends to infinity with n. The spectral measure for random graphs G(n, c/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…
A self-contained account of the theory of structure trees for edge cuts in networks is given. Applications include a generalisation of the Max-Flow Min-Cut Theorem to infinite networks and a short proof of a conjecture of Kropholler. This…
An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error e from 120 degrees.This notion arises in numerical approximations of…
In this paper we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of…
We extend Michel's theorem on the geometry of symmetry breaking [L. Michel, {\it Comptes Rendus Acad. Sci. Paris} {\bf 272-A} (1971), 433-436] to the case of pure gauge theories, i.e. of gauge-invariant functionals defined on the space…
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…
It has been known for 30 years that posets with bounded height and with cover graphs of bounded maximum degree have bounded dimension. Recently, Streib and Trotter proved that dimension is bounded for posets with bounded height and planar…
We expand upon the notion of bottlenecking introduced in our earlier work, characterizing a spectrum of graphs and showing that this naturally extends to a concept of coarse bottlenecking. We show how the notion of bottlenecking provides a…
This is a companion note to our paper 'A relative Szemer\'edi theorem', elaborating on a concluding remark. In that paper, we showed how to prove a relative Szemer\'edi theorem for $(r+1)$-term arithmetic progressions assuming a linear…
In this paper, we present a complete characterization of mutual-visibility sets in trees. It is shown that a subset $S$ is a mutual-visibility set of a tree $T$ if and only if it coincides with the set of leaves of the Steiner subtree…
We study the problem of how well a tree metric is able to preserve the sum of pairwise distances of an arbitrary metric. This problem is closely related to low-stretch metric embeddings and is interesting by its own flavor from the line of…
We prove that if $T_1,\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\dots, T_n$. This shows that the tree packing conjecture of Gy\'arf\'as and Lehel from 1976…
We study the metric Steiner tree problem in the sublinear query model. In this problem, for a set of $n$ points $V$ in a metric space given to us by means of query access to an $n\times n$ matrix $w$, and a set of terminals $T\subseteq V$,…
We give a proof for sharp estimate for the number of spanning trees using linear algebra and generalize this bound to multigraphs. In addition, we show that this bound is tight for complete graphs. In addition, we give estimates for number…