The minimum number of maximal independent sets in graphs with given order and independence number

In this paper, we consider the question of the existence of disjoint maximal independent sets (mis) in graphs and hypergraphs. The question was raised in the 1970's independently by Berge and Payan. They considered the question of characterizing the graphs that admit disjoint mis, and in particular whether regular graphs do. In this paper, we are interested in the existence of disjoint mis in a graph or in its complement, motivated by the fact that most constructions of graphs that do not admit disjoint mis are such that their complement does. We prove that there are disjoint mis in a graph or its complement whenever the graph has diameter at least three or has chromatic number at most four. We also define a graph of chromatic number 5 and diameter 2 which does not admit disjoint mis nor its complement. As our work was first motivated by a more recent work on disjoint mis in hypergraphs by Acharya (2010), we also consider the question of the existence of disjoint mis in hypergraphs. We answer a question by Jose and Tuza (2009) , proving that there exists balanced k-connected hypergraphs admitting no disjoint mis.

In this paper, we give a class of graphs that do not admit disjoint maximum and maximal independent (MMI) sets. The concept of inverse independence was introduced by Bhat and Bhat in [Inverse independence number of a graph, Int. J. Comput. Appl. 42(5) (2012) 9–13]. Let [Formula: see text] be a [Formula: see text]-set in [Formula: see text]. An independent set [Formula: see text] is called an inverse independent set with respect to [Formula: see text]. The inverse independence number [Formula: see text] is the size of the largest inverse independent set in [Formula: see text]. Bhat and Bhat gave few bounds on the independence number of a graph, we continue the study by giving some new bounds and exact value for particular classes of graphs: spider tree, the rooted product and Cartesian product of two particular graphs.

Let F:0,1n⟶0,1n be a parallel dynamical system over an undirected graph with a Boolean maxterm or minterm function as a global evolution operator. It is well known that every periodic point has at most two periods. Actually, periodic points of different periods cannot coexist, and a fixed point theorem is also known. In addition, an upper bound for the number of periodic points of F has been given. In this paper, we complete the study, solving the minimum number of periodic points’ problem for this kind of dynamical systems which has been usually considered from the point of view of complexity. In order to do this, we use methods based on the notions of minimal dominating sets and maximal independent sets in graphs, respectively. More specifically, we find a lower bound for the number of fixed points and a lower bound for the number of 2-periodic points of F. In addition, we provide a formula that allows us to calculate the exact number of fixed points. Furthermore, we provide some conditions under which these lower bounds are attained, thus generalizing the fixed-point theorem and the 2-period theorem for these systems.

Whether or not the problem of finding maximal independent sets (MIS) in hypergraphs is in (R)NC is one of the fundamental problems in the theory of parallel computing. Essentially, the challenge is to design (randomized) algorithms in which the number of processors used is polynomial and the (expected) runtime is polylogarithmic in the size of the input. Unlike the well-understood case of MIS in graphs, for the hypergraph problem, our knowledge is quite limited despite considerable work. It is known that the problem is in RNC when the edges of the hypergraph have constant size. For general hypergraphs with n vertices and m edges, the fastest previously known algorithm works in time O (√ n ) with poly( m,n ) processors. In this article, we give an EREW PRAM randomized algorithm that works in time n o (1) with O ( n + mlog n ) processors on general hypergraphs satisfying m ≤ n o (1) log log n /log log log n . We also give an EREW PRAM deterministic algorithm that runs in time n ϵ on a graph with m ≤ n 1/δ edges, for any constants δ, ϵ; the number of processors is polynomial in m, n for a fixed choice of δ, ϵ. Our algorithms are based on a sampling idea that reduces the dimension of the hypergraph and employs the algorithm for constant dimension hypergraphs as a subroutine.

Abstract A grid graph is the Cartesian product of two path graphs. Enumerating all maximal independent sets in a graph is a well‐known combinatorial problem. For a general graph, it is . In this work, we provide a polynomial‐time algorithm to generate the whole family of maximal independent sets (mis) of complete grid graphs with two rows. The same algorithm is used in two particular cases: chordless paths and cycles. We apply this result to characterize the independent graph (intersection graph of maximal independent sets) of these three classes of graphs. We also present an alternative proof of Euler's result for grid graphs with three rows that can be used for enumerating the family of mis.

The second largest number of maximal independent sets in graphs with at most two cycles

On disjoint maximal independent sets in graphs

Maximal independent sets in bipartite graphs obtained from Boolean lattices

Abstract We find the maximum number of maximal independent sets in two families of graphs. The first family consists of all graphs with n vertices and at most r cycles. The second family is all graphs of the first family which are connected and satisfy n ≥ 3r. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 270–282, 2006

Maximal independent sets in graphs with at most one cycle

Abstract A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. Let i(G) denote the number of maximal independent sets of G. Here, we prove two conjectures, suggested by P. Erdös, that the maximum number of maximal independent sets among all graphs of order n in a family Φ is o(3n/3) if Φ is either a family of connected graphs such that the largest value of maximum degrees among all graphs of order n in Φ is o(n) or a family of graphs such that the approaches infinity as n → ∞.

Parallel algorithms for fractional and maximal independent sets in planar graphs