Related papers: Outer linear measure of connected sets via Steiner…
Recent work of the author established dual representation theorems for certain vector spaces that arise in an important article of Allcock and Vaaler. These results constructed an object called a consistent map which acts like a measure on…
Consider the class of k-independent bond, respectively site, percolations with parameter p on an infinite tree T. We derive tight bounds on p for both a.s. percolation and a.s. nonpercolation. The bounds are continuous functions of k and…
For gauge theory, the matrix element for any physical process is independent of the gauge used. Since this is a formal statement and examples are known where gauge invariance is violated, for any specific process this gauge invariance needs…
Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic second-order logic (MSO) can be decided in linear time on any class of graphs of bounded treewidth, or in other words, MSO is fixed-parameter…
We establish a dilation-theoretic characterization of the Choquet order on the space of measures on a compact convex set using ideas from the theory of operator algebras. This yields an extension of Cartier's dilation theorem to the…
In 1990, Thomas proved that every graph admits a tree decomposition of minimum width that additionally satisfies a certain vertex-connectivity condition called leanness [A Menger-like property of tree-width: The finite case. Journal of…
We are interested in measures of central tendency for a population on a network, which is modeled by a metric tree. The location parameters that we study are generalized Fr\'echet means obtained by minimizing the objective function $\alpha…
In 1984 Boshernitzan proved an upper bound on the number of ergodic measures for a minimal subshift of linear block growth and asked if it could be lowered without further assumptions on the shift. We answer this question, showing that…
The complexity of a finite connected graph is its number of spanning trees; for a non-connected graph it is the product of complexities of its connected components. If $G$ is an infinite graph with cofinite free ${\mathbb Z}^d$-symmetry,…
Sparse shortcuttings of trees -- equivalently, sparse 1-spanners for tree metrics with bounded hop-diameter -- have been studied extensively (under different names and settings), since the pioneering works of [Yao82, Cha87, AS87, BTS94],…
The well known bunkbed conjecture about percolation on finite graphs is now resolved; Gladkov, Pak and Zimin, building upon work of Hollom, have constructed a counterexample. We revisit this conjecture and study it in the broader context of…
The concept of $n$-distance was recently introduced to generalize the classical definition of distance to functions of $n$ arguments. In this paper we investigate this concept through a number of examples based on certain geometrical…
In this paper we investigate the use of the concept of tree dimension in Horn clause analysis and verification. The dimension of a tree is a measure of its non-linearity - for example a list of any length has dimension zero while a complete…
Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number of additional points is minimized. We…
In this work we consider the Metric Steiner Forest problem in the sublinear time model. Given a set $V$ of $n$ points in a metric space where distances are provided by means of query access to an $n\times n$ distance matrix, along with a…
We consider a general metric Steiner problem which is of finding a set $\mathcal{S}$ with minimal length such that $\mathcal{S} \cup A$ is connected, where $A$ is a given compact subset of a given complete metric space $X$; a solution is…
A linear forest is a collection of vertex-disjoint paths. The Linear Arboricity Conjecture states that every graph of maximum degree $\Delta$ can be decomposed into at most $\lceil(\Delta+1)/2\rceil$ linear forests. We prove that $\Delta/2…
Given a connected graph $G=(V,E)$ and a $k$-set $S\subseteq V(G)$, the $Steiner$ $distance$ $d_{G}(S)$ of $S$ is defined as the size of a minimum tree including $S$ in $G$. The $Steiner$ $k$-$eccentricity$ of a vertex $v$ in $G$ is the…
Menger's theorem tells us that if $S,T$ are sets of vertices in a graph $G$, then (for $k\ge0$) either there are $k+1$ vertex-disjoint paths between $S$ and $T$, or there is a set of $k$ vertices separating $S$ and $T$. But what if we want…
In the Euclidean Steiner Tree problem, we are given as input a set of points (called terminals) in the $\ell_2$-metric space and the goal is to find the minimum-cost tree connecting them. Additional points (called Steiner points) from the…