Related papers: Property-Testing in Sparse Directed Graphs: 3-Star…
We consider the fine-grained complexity of sparse graph problems that currently have $\tilde{O}(mn)$ time algorithms, where m is the number of edges and n is the number of vertices in the input graph. This class includes several important…
A natural requirement of many distributed structures is fault-tolerance: after some failures, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general…
Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of "low-degree dependencies'' such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We…
In this paper, we study sparse signal detection problems in Degree Corrected Exponential Random Graph Models (ERGMs). We study the performance of two tests based on the conditionally centered sum of degrees and conditionally centered…
Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree…
We determine an asymptotic formula for the number of labelled 2-connected (simple) graphs on $n$ vertices and $m$ edges, provided that $m-n\to\infty$ and $m=O(n\log n)$ as $n\to\infty$. This is the entire range of $m$ not covered by…
The girth of a graph is the length of its shortest cycle. Due to its relevance in graph theory, network analysis and practical fields such as distributed computing, girth-related problems have been object of attention in both past and…
Compressed sensing is a technique for finding sparse solutions to underdetermined linear systems. This technique relies on properties of the sensing matrix such as the restricted isometry property. Sensing matrices that satisfy the…
Erd\H{o}s conjectured that every triangle-free graph $G$ on $n$ vertices contains a set of $\lfloor n/2 \rfloor$ vertices that spans at most $n^2 /50$ edges. Krivelevich proved the conjecture for graphs with minimum degree at least…
A directed graph $D$ is singly connected if for every ordered pair of vertices $(s,t)$, there is at most one path from $s$ to $t$ in $D$. Graph orientation problems ask, given an undirected graph $G$, to find an orientation of the edges…
We study connected graphs with a fixed degree sequence, in the sparse setting where the number of edges grows linearly in the number of vertices. Using the relation to the configuration model, we identify the number of such connected graphs…
We study the query complexity of determining if a graph is connected with global queries. The first model we look at is matrix-vector multiplication queries to the adjacency matrix. Here, for an $n$-vertex graph with adjacency matrix $A$,…
There are many methods to estimate the quasi-stationary infected fraction of the SIS process on (random) graphs. A challenge is to adequately incorporate correlations, which is especially important in sparse graphs. Methods typically are…
We present an algorithm to test whether a given graphical degree sequence is forcibly connected or not and prove its correctness. We also outline the extensions of the algorithm to test whether a given graphical degree sequence is forcibly…
We investigate algorithms for testing whether an image is connected. Given a proximity parameter $\epsilon\in(0,1)$ and query access to a black-and-white image represented by an $n\times n$ matrix of Boolean pixel values, a (1-sided error)…
Graph connectivity is a fundamental combinatorial optimization problem that arises in many practical applications, where usually a spanning subgraph of a network is used for its operation. However, in the real world, links may fail…
Let $H$ be a fixed $k$-vertex graph with $m$ edges and minimum degree $d >0$. We use the learning graph framework of Belovs to show that the bounded-error quantum query complexity of determining if an $n$-vertex graph contains $H$ as a…
Structured sparse optimization is an important and challenging problem for analyzing high-dimensional data in a variety of applications such as bioinformatics, medical imaging, social networks, and astronomy. Although a number of structured…
We study the sample complexity of nondeterministically testable graph parameters and improve existing bounds on it by several orders of magnitude. The technique used would be also of independent interest. We also discuss the special case of…
Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph…