For a graph G, let f(G) denote the maximum number of edges in a cut of G. For an integer m and for a fixed graph H, let f(m,H) denote the minimum possible cardinality of f(G), as G ranges over all graphs on m edges that contain no copy of H. In this paper we study this function for various graphs H. In particular we show that for any graph H obtained by connecting a single vertex to all vertice...

We previously looked at communities as simply dense subgraphs graphs, or as subgraphs such that each node has a large fraction of its edges inside. Another view, taken in [3], starts from the hubs and authorities model. It argues that a structure of densely linked hubs and authorities is a common feature of communities. Such a core, a dense bipartite graph, can be considered the “signature” of ...

It is well-known that the integrality gap of the usual linear programming relaxation for Maxcut is 2 − ǫ. For general graphs, we prove that for any ǫ and any fixed boundk, adding linear constraints of support bounded by k does not reduce the gap below 2−ǫ. We generalize this to prove that for any ǫ and any fixed bound k, strengthening the usual linear programming relaxation by doing k rounds of...

The Discrete Nodal Domain Theorem states that an eigenfunction of the k-th largest eigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. We show that the number of strong nodal domains cannot exceed the size of a maximal induced bipartite subgraph and that this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodal domains is bounded by th...

We prove that, for every integer k ≥ 1, every shortest-path metric on a graph of pathwidth k embeds into a distribution over random trees with distortion at most c = c(k), independent of the graph size. A well-known conjecture of Gupta, Newman, Rabinovich, and Sinclair [GNRS04] states that for every minor-closed family of graphs F , there is a constant c(F) such that the multi-commodity max-flo...

Laurent & Poljak introduced a class of valid inequalities for the max-cut problem, called gap inequalities, which include many other known inequalities as special cases. The gap inequalities have received little attention and are poorly understood. This paper presents the first ever computational results. In particular, we start presenting a cuttingplane scheme based on an effective heuristic s...

This paper deals with two games defined upon well known generalizations of max cut. We study the existence of a strong equilibrium which is a refinement of the Nash equilibrium. Bounds on the price of anarchy for Nash equilibria and strong equilibria are also given. In particular, we show that the max cut game always admits a strong equilibrium and the strong price of anarchy is 2/3.

We show that the exact computation of a minimum or a maximum cut of a given graph G is out of reach for any one-pass streaming algorithm, that is, for any algorithm that runs over the input stream of G’s edges only once and has a working memory of o(n) bits. This holds even if randomization is allowed.

We consider a generalization of the classical max-cut problem where two objective functions are simultaneously considered. We derive some theorems on the existence and the non-existence of feasible cuts that are at the same time near optimal for both criteria. Furthermore, two approximation algorithms with performance guarantee are presented. The first one is deterministic while the second one ...

The maximum independent sets in the Doob graphs D(m,n) are analogs of the distance-2 MDS codes in Hamming graphs and of the latin hypercubes. We prove the characterization of these sets stating that every such set is semilinear or reducible. As related objects, we study vertex sets with maximum cut (edge boundary) in D(m,n) and prove some facts on their structure. We show that the considered tw...