Related papers: The Peculiar Phase Structure of Random Graph Bisec…
We deal with a random graph model where at each step, a vertex is chosen uniformly at random, and it is either duplicated or its edges are deleted. Duplication has a given probability. We analyse the limit distribution of the degree of a…
Subgraphs reveal information about the geometry and functionalities of complex networks. For scale-free networks with unbounded degree fluctuations, we obtain the asymptotics of the number of times a small connected graph occurs as a…
We study both numerically and analytically what happens to a random graph of average connectivity "alpha" when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated…
We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main…
We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs. We show that any data structure which approximately stores the sizes of all cuts in an undirected graph on $n$ vertices up to a $1+\epsilon$ error must…
The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least 0.189n, or at least 0.110n when restricted…
We provide theoretical insights around the cutwidth of a graph and the One-Sided Crossing Minimization (OSCM) problem. OSCM was posed in the Parameterized Algorithms and Computational Experiments Challenge 2024, where the cutwidth of the…
A stable cutset is a set of vertices $S$ of a connected graph, that is pairwise non-adjacent and when deleting $S$, the graph becomes disconnected. Determining the existence of a stable cutset in a graph is known to be NP-complete. In this…
We study random subgraphs of an arbitrary finite connected transitive graph $\mathbb G$ obtained by independently deleting edges with probability $1-p$. Let $V$ be the number of vertices in $\mathbb G$, and let $\Omega$ be their degree. We…
In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is to approximate cuts in balanced directed…
We derive a new lower bound for the bandwidth of a graph that is based on a new lower bound for the minimum cut problem. Our new semidefinite programming relaxation of the minimum cut problem is obtained by strengthening the known…
We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is the Cartesian product of two complete graphs on $n$ vertices. Let $p$ be the edge probability, and write $p=\frac{1+\vep}{2(n-1)}$ for some $\vep\in \R$. In Borgs…
We study graph ordering problems with a min-max objective. A classical problem of this type is cutwidth, where given a graph we want to order its vertices such that the number of edges crossing any point is minimized. We give a $…
The sparsest cut problem consists of identifying a small set of edges that breaks the graph into balanced sets of vertices. The normalized cut problem balances the total degree, instead of the size, of the resulting sets. Applications of…
In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the…
We study the percolation properties of graph partitioning on random regular graphs with N vertices of degree $k$. Optimal graph partitioning is directly related to optimal attack and immunization of complex networks. We find that for any…
We consider a model for random hypergraphs with identifiability, an analogue of connectedness. This model has a phase transition in the proportion of identifiable vertices when the underlying random graph becomes critical. The phase…
We consider a random sparse graph with bounded average degree, in which a subset of vertices has higher connectivity than the background. In particular, the average degree inside this subset of vertices is larger than outside (but still…
We show that an intricate relation of cluster properties and optimal bipartitions, which takes place in undirected random graphs, extends to directed and mixed random graphs. In particular, the satisfability threshold coincides with the…
We study a random graph model in continuous time. Each vertex is partially copied with the same rate, i.e.\ an existing vertex is copied and every edge leading to the copied vertex is copied with independent probability $p$. In addition,…