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Random planar graphs have been the subject of much recent work. Many basic properties of the standard uniform random planar graph P_{n}, by which we mean a graph chosen uniformly at random from the set of all planar graphs with vertex set…
We study the NP-hard problem of approximating a Minimum Routing Cost Spanning Tree in the message passing model with limited bandwidth (CONGEST model). In this problem one tries to find a spanning tree of a graph $G$ over $n$ nodes that…
We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding…
For a random graph on n vertices where the edges appear with individual rates, we give exact formulas for the expected time at which the number of components has gone down to k and the expected length of the corresponding minimal spanning…
The performance of distributed averaging depends heavily on the underlying topology. In various fields, including compressed sensing, multi-party computation, and abstract graph theory, graphs may be expected to be free of short cycles,…
Random graphs are more and more used for modeling real world networks such as evolutionary networks of proteins. For this purpose we look at two different models and analyze how properties like connectedness and degree distributions are…
We prove a lower bound on the number of spanning two-forests in a graph, in terms of the number of vertices, edges, and spanning trees. This implies an upper bound on the average cut size of a random two-forest. The main tool is an identity…
Let $d \geq 3$ be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is…
An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the…
Consider the setting of \emph{randomly weighted graphs}, namely, graphs whose edge weights are chosen independently according to probability distributions with finite support over the non-negative reals. Under this setting, properties of…
Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider a relaxed version of this problem in the setting of local algorithms. The relaxation is that the constructed subgraph is a sparse spanning…
We consider a variant of so called power-law random graph. A sequence of expected degrees corresponds to a power-law degree distribution with finite mean and infinite variance. In previous works the asymptotic picture with number of nodes…
In general the problem of finding a miminum spanning tree for a weighted directed graph is difficult but solvable. There are a lot of differences between problems for directed and undirected graphs, therefore the algorithms for undirected…
Color-constrained subgraph problems are those where we are given an edge-colored (directed or undirected) graph and the task is to find a specific type of subgraph, like a spanning tree, an arborescence, a single-source shortest path tree,…
We consider a Gibbs distribution over all spanning trees of an undirected, edge weighted finite graph, where, up to normalization, the probability of each tree is given by the product of its edge weights. Defining the weighted degree of a…
In this article, we explicitly derive the limiting degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the asymptotics of the degree distribution for large…
The Spanning Tree Congestion (STC) problem is the following NP-hard problem: given a graph $G$, construct a spanning tree $T$ of $G$ minimizing its maximum edge congestion where the congestion of an edge $e\in T$ is the number of edges $uv$…
We introduce a decomposition method for the distributed calculation of exact Euclidean Minimum Spanning Trees in high dimensions (where sub-quadratic algorithms are not effective), or more generalized geometric-minimum spanning trees of…
This paper studies constructive heuristics for the minimum labelling spanning tree (MLST) problem. The purpose is to find a spanning tree that uses edges that are as similar as possible. Given an undirected labeled connected graph (i.e.,…
By means of analytic techniques we show that the expected number of spanning trees in a connected labelled series-parallel graph on $n$ vertices chosen uniformly at random satisfies an estimate of the form $s \varrho^{-n} (1+o(1))$, where…