scholarly journals Maximal independent sets in bipartite graphs with at least one cycle

2013 ◽  
Vol Vol. 15 no. 2 (Graph Theory) ◽  
Author(s):  
Shuchao Li ◽  
Huihui Zhang ◽  
Xiaoyan Zhang
Keyword(s):  
Graph Theory ◽  
Independent Set ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Proper Subset ◽  
Maximal Independent Set ◽  
Extremal Graphs ◽  
International Audience ◽  
Maximal Independent Sets ◽  
Disconnected Graphs

Graph Theory International audience 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.

scholarly journals Lower Bounds for Maximal Matchings and Maximal Independent Sets

2021 ◽  
Vol 68 (5) ◽  
pp. 1-30
Author(s):  
Alkida Balliu ◽  
Sebastian Brandt ◽  
Juho Hirvonen ◽  
Dennis Olivetti ◽  
Mikaël Rabie ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Independent Set ◽  
Independent Sets ◽  
Deterministic Algorithm ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Maximal Independent Set ◽  
Maximal Matching ◽  
Maximal Independent Sets ◽  
Open Question

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O ( Δ + log * n ) communication rounds; here, n is the number of nodes and Δ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on n is optimal: These problems cannot be solved in o (log * n ) rounds even if Δ = 2. However, the dependency on Δ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that any algorithm that finds a maximal matching or maximal independent set with probability at least 1-1/ n requires Ω (min { Δ , log log n / log log log n }) rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires Ω (min { Δ , log n / log log n }) rounds; this is an improvement over prior lower bounds also as a function of  n .

AN EFFICIENT INCREMENTAL ALGORITHM FOR GENERATING ALL MAXIMAL INDEPENDENT SETS IN HYPERGRAPHS OF BOUNDED DIMENSION

2000 ◽  
Vol 10 (04) ◽  
pp. 253-266 ◽  
Author(s):  
E. BOROS ◽  
V. GURVICH ◽  
K. ELBASSIONI ◽  
L. KHACHIYAN
Keyword(s):  
Parallel Algorithm ◽  
Independent Set ◽  
Independent Sets ◽  
Incremental Algorithm ◽  
Maximal Independent Set ◽  
Bounded Number ◽  
Maximal Independent Sets ◽  
Edge Size

We show that for hypergraphs of bounded edge size, the problem of extending a given list of maximal independent sets is NC-reducible to the computation of an arbitrary maximal independent set for an induced sub-hypergraph. The latter problem is known to be in RNC. In particular, our reduction yields an incremental RNC dualization algorithm for hypergraphs of bounded edge size, a problem previously known to be solvable in polynomial incremental time. We also give a similar parallel algorithm for the dualization problem on the product of arbitrary lattices which have a bounded number of immediate predecessors for each element.

scholarly journals Recognizing Generating Subgraphs Revisited

2021 ◽  
Vol 32 (01) ◽  
pp. 93-114
Author(s):  
Vadim E. Levit ◽  
David Tankus
Keyword(s):  
Weight Function ◽  
Independent Set ◽  
Independent Sets ◽  
Weight Functions ◽  
Polynomial Algorithms ◽  
Maximal Independent Set ◽  
Input Graph ◽  
Bipartite Subgraph ◽  
Np Complete ◽  
Maximal Independent Sets

A graph [Formula: see text] is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function [Formula: see text] is defined on its vertices. Then [Formula: see text] is [Formula: see text]well-covered if all maximal independent sets are of the same weight. For every graph [Formula: see text], the set of weight functions [Formula: see text] such that [Formula: see text] is [Formula: see text]-well-covered is a vector space, denoted as WCW(G). Deciding whether an input graph [Formula: see text] is well-covered is co-NP-complete. Therefore, finding WCW(G) is co-NP-hard. A generating subgraph of a graph [Formula: see text] is an induced complete bipartite subgraph [Formula: see text] of [Formula: see text] on vertex sets of bipartition [Formula: see text] and [Formula: see text], such that each of [Formula: see text] and [Formula: see text] is a maximal independent set of [Formula: see text], for some independent set [Formula: see text]. If [Formula: see text] is generating, then [Formula: see text] for every weight function [Formula: see text]. Therefore, generating subgraphs play an important role in finding WCW(G). The decision problem whether a subgraph of an input graph is generating is known to be NP-complete. In this article we prove NP- completeness of the problem for graphs without cycles of length 3 and 5, and for bipartite graphs with girth at least 6. On the other hand, we supply polynomial algorithms for recognizing generating subgraphs and finding WCW(G), when the input graph is bipartite without cycles of length 6. We also present a polynomial algorithm which finds WCW(G) when [Formula: see text] does not contain cycles of lengths 3, 4, 5, and 7.

scholarly journals Stability for Maximal Independent Sets

10.37236/8530 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Jeff Kahn ◽  
Jinyoung Park
Keyword(s):  
Independent Set ◽  
Independent Sets ◽  
Upper Bounds ◽  
Maximal Independent Set ◽  
Induced Matching ◽  
Maximal Independent Sets ◽  
Answering Questions

Answering questions of Y. Rabinovich, we prove "stability" versions of upper bounds on maximal independent set counts in graphs under various restrictions. Roughly these say that being close to the maximum implies existence of a large induced matching or triangle matching (depending on assumptions). A mild strengthening of one of these results is a key ingredient in a proof (to appear elsewhere) of a conjecture of L. Ilinca and the first author giving asymptotics for the number of maximal independent sets in the graph of the Hamming cube.

Greedy Algorithms

1995 ◽  
Author(s):  
Raymond Greenlaw ◽  
H. James Hoover ◽  
Walter L. Ruzzo
Keyword(s):  
Greedy Algorithms ◽  
Independent Set ◽  
Independent Sets ◽  
Case Finding ◽  
Sequential Algorithm ◽  
Maximal Independent Set ◽  
Maximum Independent Set ◽  
Greedy Method ◽  
Numerical Order ◽  
Selection Of

We consider the selection of two basketball teams at a neighborhood playground to illustrate the greedy method. Usually the top two players are designated captains. All other players line up while the captains alternate choosing one player at a time. Usually, the players are picked using a greedy strategy. That is, the captains choose the best unclaimed player. The system of selection of choosing the best, most obvious, or most convenient remaining candidate is called the greedy method. Greedy algorithms often lead to easily implemented efficient sequential solutions to problems. Unfortunately, it also seems to be that sequential greedy algorithms frequently lead to solutions that are inherently sequential — the solutions produced by these algorithms cannot be duplicated rapidly in parallel, unless NC equals P. In the following subsections we will examine this phenomenon. We illustrate some of the important aspects of greedy algorithms using one that constructs a maximal independent set in a graph. An independent set is a set of vertices of a graph that are pairwise nonadjacent. A maximum independent set is such a set of largest cardinality. It is well known that finding maximum independent sets is NP-hard. An independent set is maximal if no other vertex can be added while maintaining the independent set property. In contrast to the maximum case, finding maxima? independent sets is very easy. Figure 7.1.1 depicts a simple polynomial time sequential algorithm computing a maximal independent set. The algorithm is a greedy algorithm: it processes the vertices in numerical order, always attempting to add the lowest numbered vertex that has not yet been tried. The sequential algorithm in Figure 7.1.1, having processed vertices 1,... , j -1, can easily decide whether to include vertex j. However, notice that its decision about j potentially depends on its decisions about all earlier vertices — j will be included in the maximal independent set if and only if all j' less than j and adjacent to it were excluded.

scholarly journals Recognizing Maximal Unfrozen Graphs with respect to Independent Sets is CO-NP-complete

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Nesrine Abbas ◽  
Joseph Culberson ◽  
Lorna Stewart
Keyword(s):  
Independent Set ◽  
Independent Sets ◽  
Complete Problems ◽  
International Audience ◽  
Np Complete

International audience A graph is unfrozen with respect to k independent set if it has an independent set of size k after the addition of any edge. The problem of recognizing such graphs is known to be NP-complete. A graph is maximal if the addition of one edge means it is no longer unfrozen. We designate the problem of recognizing maximal unfrozen graphs as MAX(U(k-SET)) and show that this problem is CO-NP-complete. This partially fills a gap in known complexity cases of maximal NP-complete problems, and raises some interesting open conjectures discussed in the conclusion.

scholarly journals Max-independent set and the quantum alternating operator ansatz

2020 ◽  
Vol 18 (04) ◽  
pp. 2050011 ◽  
Author(s):  
Zain Hamid Saleem
Keyword(s):  
Graph Theory ◽  
Probability Distribution ◽  
Independent Set ◽  
Independent Sets ◽  
Maximum Independent Set ◽  
Initial State ◽  
Max Cut Problem ◽  
Initial States ◽  
Circuit Depth ◽  
Ring Graph

The maximum-independent set (MIS) problem of graph theory using the quantum alternating operator ansatz is studied. We perform simulations on the Rigetti Forest simulator for the square ring, [Formula: see text], and [Formula: see text] graphs and analyze the dependence of the algorithm on the depth of the circuit and initial states. The probability distribution of observation of the feasible states representing maximum-independent sets is observed to be asymmetric for the MIS problem, which is unlike the Max-Cut problem where the probability distribution of feasible states is symmetric. For asymmetric graphs, it is shown that the algorithm clearly favors the independent set with the larger number of elements even for finite circuit depth. We also compare the approximation ratios for the algorithm when we choose different initial states for the square ring graph and show that it is dependent on the choice of the initial state.

scholarly journals THE TYPICAL STRUCTURE OF MAXIMAL TRIANGLE-FREE GRAPHS

2015 ◽  
Vol 3 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
HONG LIU ◽  
ŠÁRKA PETŘÍČKOVÁ ◽  
MARYAM SHARIFZADEH
Keyword(s):  
Perfect Matching ◽  
Independent Set ◽  
Independent Sets ◽  
Free Graph ◽  
Typical Structure ◽  
Vertex Partition ◽  
Maximal Independent Sets ◽  
Vertex Disjoint ◽  
Removal Lemma

Recently, settling a question of Erdős, Balogh, and Petříčková showed that there are at most $2^{n^{2}/8+o(n^{2})}$$n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here, we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph $G$ admits a vertex partition $X\cup Y$ such that $G[X]$ is a perfect matching and $Y$ is an independent set.Our proof uses the Ruzsa–Szemerédi removal lemma, the Erdős–Simonovits stability theorem, and recent results of Balogh, Morris, and Samotij and Saxton and Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_{3}$s, which is of independent interest.

scholarly journals Independent sets in (P₆, diamond)-free graphs

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Raffaele Mosca
Keyword(s):  
Polynomial Time ◽  
Dominating Set ◽  
Minimum Weight ◽  
Independent Set ◽  
Independent Sets ◽  
Maximum Weight ◽  
Independent Set Problem ◽  
International Audience ◽  
Dominating Set Problem ◽  
Independent Dominating Set

Graphs and Algorithms International audience We prove that on the class of (P6,diamond)-free graphs the Maximum-Weight Independent Set problem and the Minimum-Weight Independent Dominating Set problem can be solved in polynomial time.

scholarly journals A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set

2012 ◽  
Vol Vol. 14 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Serge Gaspers ◽  
Mathieu Liedloff
Keyword(s):  
Dominating Set ◽  
Independent Set ◽  
Time Algorithm ◽  
Worst Case ◽  
Running Time ◽  
Running Time Analysis ◽  
Np Hard Problem ◽  
International Audience ◽  
Maximal Independent Sets ◽  
Independent Dominating Set

Graphs and Algorithms International audience An independent dominating set D of a graph G = (V,E) is a subset of vertices such that every vertex in V \ D has at least one neighbor in D and D is an independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum independent dominating set in a graph is an NP-hard problem. Whereas it is hard to cope with this problem using parameterized and approximation algorithms, there is a simple exact O(1.4423^n)-time algorithm solving the problem by enumerating all maximal independent sets. In this paper we improve the latter result, providing the first non trivial algorithm computing a minimum independent dominating set of a graph in time O(1.3569^n). Furthermore, we give a lower bound of \Omega(1.3247^n) on the worst-case running time of this algorithm, showing that the running time analysis is almost tight.

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