Maximal independent sets in grid 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.

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.

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. >

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.

Counting independent sets in 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.

Maximal independent sets in graphs with a given matching number

Maximal independent sets in graphs with at most one cycle

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 → ∞.

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

In this lecture and the next we develop a probabilistic NC algorithm of Luby for finding a maximal independent set in an undirected graph. Recall that a set of vertices of a graph is independent if the induced subgraph on those vertices has no edges. A maximal independent set is one contained in no larger independent set. A maximal independent set need not be of maximum cardinality among all independent sets in the graph.

Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well studied for various classes of graphs. When it comes to random graphs, the classic Erdős–Rényi–Gilbert random graph [Formula: see text] has been analyzed and shown to have the largest independent sets of size [Formula: see text] with high probability (w.h.p.) This classic model does not capture any dependency structure between edges that can appear in real-world networks. We define random graphs [Formula: see text] whose existence of edges is determined by a Markov process that is also governed by a decay parameter [Formula: see text]. We prove that w.h.p. [Formula: see text] has independent sets of size [Formula: see text] for arbitrary [Formula: see text]. This is derived using bounds on the terms of a harmonic series, a Turán bound on a stability number, and a concentration analysis for a certain sequence of dependent Bernoulli variables that may also be of independent interest. Because [Formula: see text] collapses to [Formula: see text] when there is no decay, it follows that having even the slightest bit of dependency (any [Formula: see text]) in the random graph construction leads to the presence of large independent sets, and thus, our random model has a phase transition at its boundary value of r = 1. This implies that there are large matchings in the line graph of [Formula: see text], which is a Markov random field. For the maximal independent set output by a greedy algorithm, we deduce that it has a performance ratio of at most [Formula: see text] w.h.p. when the lowest degree vertex is picked at each iteration and also show that, under any other permutation of vertices, the algorithm outputs a set of size [Formula: see text], where [Formula: see text] and, hence, has a performance ratio of [Formula: see text]. Funding: The initial phase of this research was supported by the National Science Foundation [Grant DMS-1913294].