Related papers: Sharp threshold for reconstructing points on the l…

We consider a random graph G(n,p) whose vertex set V has been randomly embedded in the unit square and whose edges are given weight equal to the geometric distance between their end vertices. Then each pair {u,v} of vertices have a distance…

We investigate rigidity-type problems on the real line and the circle in the non-generic setting. Specifically, we consider the problem of uniquely determining the positions of $n$ distinct points $V = {v_1, \ldots, v_n}$ given a set of…

Given a set of $n$ sites from $\mathbb{R}^d$, each having some positive weight factor, the Multiplicatively Weighted Voronoi Diagram is a subdivision of space that associates each cell to the site whose weighted Euclidean distance is…

In this paper, we develop efficient exact and approximate algorithms for computing a maximum independent set in random graphs. In a random graph $G$, each pair of vertices are joined by an edge with a probability $p$, where $p$ is a…

We study the fundamental limits for reconstruction in weighted graph (or matrix) database alignment. We consider a model of two graphs where $\pi^*$ is a planted uniform permutation and all pairs of edge weights $(A_{i,j},…

In the binomial random graph $\mathcal{G}(n,p)$, when $p$ changes from $(1-\varepsilon)/n$ (subcritical case) to $1/n$ and then to $(1+\varepsilon)/n$ (supercritical case) for $\varepsilon>0$, with high probability the order of the largest…

We prove that the family of largest cuts in the binomial random graph exhibits the following stability property: If $1/n \ll p = 1-\Omega(1)$, then, with high probability, there is a set of $n - o(n)$ vertices that is partitioned in the…

Let $P$ be a set of $n$ points in $\Re^2$. For a parameter $\varepsilon\in (0,1)$, a subset $C\subseteq P$ is an \emph{$\varepsilon$-kernel} of $P$ if the projection of the convex hull of $C$ approximates that of $P$ within…

This paper studies the problem of recovering the hidden vertex correspondence between two edge-correlated random graphs. We focus on the Gaussian model where the two graphs are complete graphs with correlated Gaussian weights and the…

The percolated random geometric graph $G_n(\lambda, p)$ has vertex set given by a Poisson Point Process in the square $[0,\sqrt{n}]^2$, and every pair of vertices at distance at most 1 independently forms an edge with probability $p$. For a…

We investigate the maximal size of an increasing subset among points randomly sampled from certain probability densities. Kerov and Vershik's celebrated result states that the largest increasing subset among $N$ uniformly random points on…

Embedding graphs in a geographical or latent space, i.e.\ inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We…

Consider a random graph process with $n$ vertices corresponding to points $v_{i} \sim {Unif}[0,1]$ embedded randomly in the interval, and where edges are inserted between $v_{i}, v_{j}$ independently with probability given by the graphon…

A key issue in network reliability analysis. A graph with $n$ nodes and whose $e$ edges fail independently with probability $p$ is an \emph{Uniformly Most Reliable Graph} (UMRG) if it has the highest reliability among all graphs with the…

Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these…

A graph $H$ is {\em $p$-edge colorable} if there is a coloring $\psi: E(H) \rightarrow \{1,2,\dots,p\}$, such that for distinct $uv, vw \in E(H)$, we have $\psi(uv) \neq \psi(vw)$. The {\sc Maximum Edge-Colorable Subgraph} problem takes as…

Reed conjectured that the chromatic number of any graph is closer to its clique number than to its maximum degree plus one. We consider a recolouring version of this conjecture, with respect to Kempe changes. Namely, we investigate the…

Two graphs $G$ and $H$ are \emph{hypomorphic} if there exists a bijection $\varphi \colon V(G) \rightarrow V(H)$ such that $G - v \cong H - \varphi(v)$ for each $v \in V(G)$. A graph $G$ is \emph{reconstructible} if $H \cong G$ for all $H$…

For a graph $G$ and partition $\mathcal{U}$ of its vertex set, an independent transversal of $(G, \mathcal{U})$ is an independent set of $G$ that contains one vertex from each block of $\mathcal{U}$. Buys, Kang, and Ozeki studied when a…

Let $u_k(G,p)$ be the maximum over all $k$-vertex graphs $F$ of by how much the number of induced copies of $F$ in $G$ differs from its expectation in the binomial random graph with the same number of vertices as $G$ and with edge…