Maximal independent sets in bipartite graphs obtained from Boolean lattices

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.

Parallel algorithms for fractional and maximal independent sets in planar graphs

Maximal independent sets in grid graphs

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.

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 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 → ∞.

Linear-time algorithms for counting independent sets in bipartite permutation 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.

J.Y.-T. Leung (J. Algorithms, no.5, (1984)) presented algorithms for generating all the maximal independent sets in interval graphs and circular-arc graphs. The algorithms take O(n/sup 2/+ beta ) steps, where beta is the sum of the number of nodes in all maximal independent sets. The authors use a new technique to give fast and efficient algorithms for finding all the maximum weight independent sets in interval graphs and circular-arc graphs. The algorithms take O(max(n/sup 2/, beta )) steps in O(n/sup 2/) space, where beta is the sum of the number of nodes in all maximum weight independent sets. The algorithms can be directly applied for finding a maximum weight independent set in these graphs in O(n/sup 2/) steps. Thus, the result is an improvement over the best known result of O(n/sup 2/ log n) for finding the maximum weight independent set in circular-arc graphs. >

Maximal independent sets in graphs with at most one cycle

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

Boolean-width is similar to clique-width, rank-width and NLC-width in that all these graph parameters are constantly bounded on the same classes of graphs. In many classes where these parameters are not constantly bounded, boolean-width is distinguished by its much lower value, such as in permutation graphs and interval graphs where boolean-width was shown to be O(logn) [1]. Together with FPT algorithms having runtime O *(c boolw ) for a constant c this helped explain why a variety of problems could be solved in polynomial-time on these graph classes. In this paper we continue this line of research and establish non-trivial upper-bounds on the boolean-width and linear boolean-width of any graph. Again we combine these bounds with FPT algorithms having runtime O *(c boolw ), now to give a common framework of moderately-exponential exact algorithms that beat brute-force search for several independence and domination-type problems, on general graphs. Boolean-width is closely related to the number of maximal independent sets in bipartite graphs. Our main result breaking the triviality bound of n/3 for boolean-width and n/2 for linear boolean-width is proved by new techniques for bounding the number of maximal independent sets in bipartite graphs.

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 \({{t_1, \ldots, t_r}}\).

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