Related papers: Finding and using expanders in locally sparse grap…

We propose a theoretical framework for training Graph Neural Networks (GNNs) on large input graphs via training on small, fixed-size sampled subgraphs. This framework is applicable to a wide range of models, including popular sampling-based…

A k-clique covering of a simple graph G, is an edge covering of G by its cliques such that each vertex is contained in at most k cliques. The smallest k for which G admits a k-clique covering is called local clique cover number of G and is…

For any $S\subset [n]$, we compute the probability that the subgraph of $\mathcal{G}_{n,d}$ induced by $S$ is a given graph $H$ on the vertex set $S$. The result holds for any $d=o(n^{1/3})$ and is further extended to $\mathcal{G}_{{\bf…

A conjecture of Chung and Graham states that every $K_4$-free graph on $n$ vertices contains a vertex set of size $\lfloor n/2 \rfloor$ that spans at most $n^2/18$ edges. We make the first step toward this conjecture by showing that it…

We present sweeping line graphs, a generalization of $\Theta$-graphs. We show that these graphs are spanners of the complete graph, as well as of the visibility graph when line segment constraints or polygonal obstacles are considered. Our…

In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number $s>2$, we prove that every graph on $n$ vertices with average degree at least $d$ contains a subgraph of average degree at…

Local algorithms on graphs are algorithms that run in parallel on the nodes of a graph to compute some global structural feature of the graph. Such algorithms use only local information available at nodes to determine local aspects of the…

We consider the following two algorithmic problems: given a graph $G$ and a subgraph $H\subseteq G$, decide whether $H$ is an isometric or a geodesically convex subgraph of $G$. It is relatively easy to see that the problems can be solved…

We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $r$ active neighbors, where $r>1$ is the activation…

It has long been known that random regular graphs are with high probability good expanders. This was first established in the 1980s by Bollob\'as by directly calculating the probability that a set of vertices has small expansion and then…

The smallest number of cliques, covering all edges of a graph $ G $, is called the (edge) clique cover number of $ G $ and is denoted by $ cc(G) $. It is an easy observation that for every line graph $ G $ with $ n $ vertices, $cc(G)\leq n…

Let G be a graph with maximum degree \Delta whose vertex set is partitioned into parts V(G) = V_1 \cup ... \cup V_r. A transversal is a subset of V(G) containing exactly one vertex from each part V_i. If it is also an independent set, then…

A (1,{\lambda})-embedded graph is a graph that can be embedded on a surface with Euler characteristic {\lambda} so that each edge is crossed by at most one other edge. A graph G is called {\alpha}-linear if there exists an integral constant…

Graph spanners are sparse subgraphs which approximately preserve all pairwise shortest-path distances in an input graph. The notion of approximation can be additive, multiplicative, or both, and many variants of this problem have been…

Graphical data arises naturally in several modern applications, including but not limited to internet graphs, social networks, genomics and proteomics. The typically large size of graphical data argues for the importance of designing…

We say that a graph $G=(V,E)$ on $n$ vertices is a $\beta$-expander for some constant $\beta>0$ if every $U\subseteq V$ of cardinality $|U|\leq \frac{n}{2}$ satisfies $|N_G(U)|\geq \beta|U|$ where $N_G(U)$ denotes the neighborhood of $U$.…

Many natural computational problems, including e.g. Max Weight Independent Set, Feedback Vertex Set, or Vertex Planarization, can be unified under an umbrella of finding the largest sparse induced subgraph, that satisfies some property…

The Local Structure Theorem (LST) for Graph Minors roughly states that for every $H$-minor-free graph $G$ that contains a sufficiently large wall $W$, there is a small vertex subset $A,$ whose removal yields a graph that admits an "almost…

Consider the setting of sparse graphs on N vertices, where the vertices have distinct "names", which are strings of length O(log N) from a fixed finite alphabet. For many natural probability models, the entropy grows as cN log N for some…

What can be said about the structure of graphs that do not contain an induced copy of some graph H? Rodl showed in the 1980s that every H-free graph has large parts that are very dense or very sparse. More precisely, let us say that a graph…