Maximal Independent Sets In Graphs Research Articles

Disjoint maximal independent sets in graphs and hypergraphs

Disjoint maximal independent sets in graphs and hypergraphs

On disjoint maximum and maximal independent sets in graphs and inverse independence number

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 Periodic Points in Parallel Graph Dynamical Systems

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.

On Computing Maximal Independent Sets of Hypergraphs in Parallel

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.

Maximal independent sets in grid graphs

AbstractA 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

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

On disjoint maximal independent sets in graphs

An old problem in graph theory is to characterize the graphs that admit two disjoint maximal independent sets.In this paper, we characterize the class of graphs for which every induced subgraph admits two disjoint maximal independent sets. Moreover, we state a polynomial-time algorithm which, for an arbitrary input graph G, either constructs two disjoint maximal independent sets or puts out an induced subgraph of G from the list of forbidden induced subgraphs.As a second result, we show the following: the computation of two disjoint maximal independent sets of minimum total size, NP-hard for general graphs, can be done in linear time on trees.

Maximal independent sets in bipartite graphs obtained from Boolean lattices

Attempts to enumerate maximal antichains in Boolean lattices give rise to problems involving maximal independent sets in bipartite graphs whose vertex sets are comprised of adjacent levels of the lattice and whose edges correspond to proper containment. In this paper, we find bounds on the numbers of maximal independent sets in these graphs.

Maximal independent sets in graphs with at most r cycles

AbstractWe 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

In this paper, we determine the largest number of maximal independent sets among all connected graphs of order n, which contain at most one cycle. We also characterize those extremal graphs achieving this maximum value. As a consequence, the corresponding results for graphs with at most one cycle but not necessarily connected are also given.

Constraints on the number of maximal independent sets in graphs

AbstractA 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

The problem of computing maximal independent sets in graphs on parallel models of computation has received considerable attention. We present simple efficient parallel algorithms for the maximal independent set problem—and a relaxation that we call the fractional independent set problem—restricted to planar graphs. Our algorithms rely on an efficient parallel algorithm for constructing large independent sets in graphs of bounded degree. The latter is accomplished by a simple reduction to the same problem for lists. Using a linear number of EREW processors, the algorithm identifies a maximal independent set in an arbitrary planar graph in O( logn log∗n) parallel time. A randomized version of the algorithm runs in O(log n) expected parallel time.