Constraints on the number of maximal independent sets in graphs

Maximal independent sets in bipartite graphs obtained from Boolean lattices

Parallel algorithms for fractional and maximal independent sets in planar graphs

A locally-optimal structure is a combinatorial structure that cannot be improved by certain (greedy) local moves, even though it may not be globally optimal. An example is a maximal independent set in a graph. It is trivial to construct an independent set in a graph. It is easy to (greedily) construct a maximal independent set. However, it is NP-hard to construct a globally-optimal (maximum) independent set.This situation is typical. Constructing a locally-optimal structure is somewhat more difficult than constructing an arbitrary structure, and constructing a globally-optimal structure is more difficult than constructing a locally-optimal structure. The same situation arises with listing. The differences between the problems become obscured when we move from listing to counting because nearly everything is \(\#\text {P} \)-complete. However, we highlight an interesting phenomenon that arises in approximate counting, where approximately counting locally-optimal structures is apparently more difficult than approximately counting globally-optimal structures. Specifically, we show that counting maximal independent sets is complete for \(\#\text {P} \) with respect to approximation-preserving reductions, whereas counting all independent sets, or counting maximum independent sets is complete for an apparently smaller class, #RH\(\varPi _1\) which has a prominent role in the complexity of approximate counting. Motivated by the difficulty of approximately counting maximal independent sets in bipartite graphs, we also study counting problems involving minimal separators and minimal edge separators (which are also locally-optimal structures). Minimal separators have applications via fixed-parameter-tractable algorithms for constructing triangulations and phylogenetic trees. Although exact (exponential-time) algorithms exist for listing these structures, we show that the counting problems are as hard as they could possibly be. All of the exact counting problems are \(\#\text {P} \)-complete, and all of the approximation problems are complete for \(\#\text {P} \) with respect to approximation-preserving reductions. A full version [14] containing detailed proofs is available at http://arxiv.org/abs/1411.6829. Theorem-numbering here matches the full version.

Let mi(G) be the number of maximal independent sets in a graph G. A graph G is mi-minimal if mi(H) 2. Hence the extremal problem of calculating m(k) = max{IV(G)1: G is a mi-minimal graph with mi(G) = k} has a solution for any k ~ 1 We show that 2(k -1) ~ m(k) ~ k(k -1) for any k ~ andconjecture that m(k) = 2(k - 1). We also prove NP-completeness of some related problems.

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.

Maximal independent sets in graphs with at most one cycle

We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of maximum independent sets; (3) a polynomial-time algorithm for counting the number of independent sets of a fixed size. With similar ideas, we show that enumeration (namely, listing) of the independent sets, the maximum independent sets, and the independent sets of a fixed size in a chordal graph can be done in constant amortized time per output. On the other hand, we prove that the following problems for a chordal graph are # P-complete: (1) counting the number of maximal independent sets; (2) counting the number of minimum maximal independent sets. With similar ideas, we also show that finding a minimum weighted maximal independent set in a chordal graph is nP-hard, and even hard to approximate.

A result of Balas and Yu (1989) states that the number of maximal independent sets of a graph G is at most δp+1, where δ is the number of pairs of vertices in G at distance 2, and p is the cardinality of a maximum induced matching in G. In this paper, we give an analogue of this result for hypergraphs and, more generally, for subsets of vectors ℬ in the product of n lattices ℒ=ℒ1×⋯×ℒn, where the notion of an induced matching in G is replaced by a certain binary tree each internal node of which is mapped into ℬ. We show that our bounds may be nearly sharp for arbitrarily large hypergraphs and lattices. As an application, we prove that the number of maximal infeasible vectors xℒ=ℒ1×⋯×ℒn for a system of polymatroid inequalities \({{f_1(x) \ge t_1,\ldots,f_r(x) \ge t_r}}\) does not exceed max{Q,βlogt/c(2Q,β)}, where β is the number of minimal feasible vectors for the system, \({{Q=|{{\mathcal L}}_1|+\ldots+|{{\mathcal L}}_n|}}\), \({{t=\hbox{max}\{t_1,\ldots,t_r\}}}\), and c(ρ,β) is the unique positive root of the equation 2c(ρc/logβ−1)=1. This bound is nearly sharp for the Boolean case ℒ={0,1}n, and it allows for the efficient generation of all minimal feasible sets to a given system of polymatroid inequalities with quasi-polynomially bounded right-hand sides \({{t_1, \ldots, t_r}}\).

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

Counting the number of independent sets in chordal graphs

Graph Theory A maximal independent set is an independent set that is not a proper subset of any other independent set. Liu [J.Q. Liu, Maximal independent sets of bipartite graphs, J. Graph Theory, 17 (4) (1993) 495-507] determined the largest number of maximal independent sets among all n-vertex bipartite graphs. The corresponding extremal graphs are forests. It is natural and interesting for us to consider this problem on bipartite graphs with cycles. Let \mathscrBₙ (resp. \mathscrBₙ') be the set of all n-vertex bipartite graphs with at least one cycle for even (resp. odd) n. In this paper, the largest number of maximal independent sets of graphs in \mathscrBₙ (resp. \mathscrBₙ') is considered. Among \mathscrBₙ the disconnected graphs with the first-, second-, \ldots, \fracn-22-th largest number of maximal independent sets are characterized, while the connected graphs in \mathscrBₙ having the largest, the second largest number of maximal independent sets are determined. Among \mathscrBₙ' graphs have the largest number of maximal independent sets are identified.

Graphs with the second largest number of maximal independent sets

The main result of this paper is an algorithm counting maximal independent sets in graphs with maximum degree at most 3 in time O *(1.2570n) and polynomial space.KeywordsMaximum DegreeRecursive CallInternal VertexPrimal GraphSparse GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

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