Related papers: Stability for maximal independent sets

Stability is a fundamental notion in dynamical systems and control theory that, traditionally understood, describes asymptotic behavior of solutions around an equilibrium point. This notion may be characterized abstractly as continuity of a…

We present quantum complexity lower and upper bounds for independent set problems in graphs. In particular, we give quantum algorithms for computing a maximal and a maximum independent set in a graph. We present applications of these…

This paper studies the graph-theoretic conditions for stability of positive monotone systems. Using concepts from input-to-state stability and network small-gain theory, we first establish necessary and sufficient conditions for the…

For $r \geq 2$, we show that every maximal $K_{r+1}$-free graph $G$ on $n$ vertices with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contains a complete $r$-partite subgraph on $(1 - o(1))n$ vertices. We also show that this is…

A {\em maximal inequality} seeks to estimate $\mathbb{E}\max_i X_i$ in terms of properties of the $X_i$. When the latter are independent, the union bound (in its various guises) can yield tight upper bounds. If, however, the $X_i$ are…

Rigid graphs have only finitely many realizations. In the recent years significant progress was made in computing the number of such realizations. With this progress it was also possible for the first time to do computations on large sets…

We use probabilistic methods to find lower bounds on the maximum number, in a graph with domination number \gamma, of dominating sets of size \gamma. We find that we can randomly generate a graph that, w.h.p., is dominated by almost all…

We present a new sufficient condition on stability number and toughness of the graph to have an f-factor.

Computing a maximum independent set (MaxIS) is a fundamental NP-hard problem in graph theory, which has important applications in a wide spectrum of fields. Since graphs in many applications are changing frequently over time, the problem of…

Given a locally finite simple graph so that its degree is not bounded, every self-adjoint realization of the adjacency matrix is unbounded from above. In this note we give an optimal condition to ensure it is also unbounded from below. We…

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…

The larger the distance to instability from a matrix is, the more robustly stable the associated autonomous dynamical system is in the presence of uncertainties and typically the less severe transient behavior its solution exhibits.…

An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number…

Let $Q_n$ be the $n$-dimensional Hamming cube and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically \[2n2^{N/4},\] as was conjectured by Ilinca and the first author in connection with a question of…

In recent years there has been increased interest in extremal problems for "counting" parameters of graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a $d$-regular graph. In the same…

This paper considers pairs of optimization problems that are defined from a single input and for which it is desired to find a good approximation to either one of the problems. In many instances, it is possible to efficiently find an…

We introduce a novel evolutionary formulation of the problem of finding a maximum independent set of a graph. The new formulation is based on the relationship that exists between a graph's independence number and its acyclic orientations.…

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in $O(\Delta + \log^* n)$ communication rounds; here $n$ is the number of nodes and $\Delta$ is the maximum degree. The lower bound by Linial…

We propose two solution concepts for matchings under preferences: robustness and near stability. The former strengthens while the latter relaxes the classic definition of stability by Gale and Shapley (1962). Informally speaking, robustness…

We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane.…