Recognizing Generating Subgraphs Revisited

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.

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

Parallel Processing Letters ◽

10.1142/s0129626400000251 ◽

2000 ◽

Vol 10 (04) ◽

pp. 253-266 ◽

Cited By ~ 35

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.

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Maximal independent sets in bipartite graphs with at least one cycle

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.607 ◽

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.

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Lower Bounds for Maximal Matchings and Maximal Independent Sets

Journal of the ACM ◽

10.1145/3461458 ◽

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 .

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Stability for Maximal Independent Sets

The Electronic Journal of Combinatorics ◽

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.

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Greedy Algorithms

Limits to Parallel Computation ◽

10.1093/oso/9780195085914.003.0011 ◽

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.

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

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.345 ◽

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.

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THE TYPICAL STRUCTURE OF MAXIMAL TRIANGLE-FREE GRAPHS

Forum of Mathematics Sigma ◽

10.1017/fms.2015.22 ◽

2015 ◽

Vol 3 ◽

Cited By ~ 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.

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Infinite Gammoids

The Electronic Journal of Combinatorics ◽

10.37236/4181 ◽

2015 ◽

Vol 22 (1) ◽

Cited By ~ 1

Author(s):

Seyed Hadi Afzali Borujeni ◽

Hiu-Fai Law ◽

Malte Müller

Keyword(s):

Independent Set ◽

Independent Sets ◽

Maximal Independent Sets

Finite strict gammoids, introduced in the early 1970's, are matroids defined via finite digraphs equipped with some set of sinks: a set of vertices is independent if it admits a linkage to these sinks. In particular, an independent set is maximal (i.e. a base) precisely if it is linkable onto the sinks.In the infinite setting, this characterization of the maximal independent sets need not hold. We identify a type of substructure as the unique obstruction. This allows us to prove that the sets linkable onto the sinks form the bases of a (possibly non-finitary) matroid if and only if this substructure does not occur.

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Complexidade Parametrizada de Cliques e Conjuntos Independentes em Grafos Prismas Complementares

10.5753/etc.2018.3166 ◽

2018 ◽

Author(s):

Priscila Camargo ◽

Alan D. A. Carneiro ◽

Uéverton S. Santos

Keyword(s):

Disjoint Union ◽

Perfect Matching ◽

Independent Set ◽

Point Of View ◽

Polynomial Kernel ◽

Input Graph ◽

Fixed Parameter Tractable ◽

Fixed Parameter ◽

Np Complete ◽

Complementary Prism

The complementary prism GG¯ arises from the disjoint union of the graph G and its complement G¯ by adding the edges of a perfect matching joining pairs of corresponding vertices of G and G¯. The classical problems of graph theory, clique and independent set were proved NP-complete when the input graph is a complemantary prism. In this work, we study the complexity of both problems in complementary prisms graphs from the parameterized complexity point of view. First, we prove that these problems have a kernel and therefore are Fixed-Parameter Tractable (FPT). Then, we show that both problems do not admit polynomial kernel.

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Graph Sparsification for Derandomizing Massively Parallel Computation with Low Space

ACM Transactions on Algorithms ◽

10.1145/3451992 ◽

2021 ◽

Vol 17 (2) ◽

pp. 1-27

Author(s):

Artur Czumaj ◽

Peter Davies ◽

Merav Parter

Keyword(s):

Parallel Computation ◽

Independent Set ◽

Massively Parallel ◽

Easy Access ◽

Maximal Independent Set ◽

Input Graph ◽

Single Node ◽

Maximal Matching ◽

Graph Problems ◽

Graph Sparsification

The Massively Parallel Computation (MPC) model is an emerging model that distills core aspects of distributed and parallel computation, developed as a tool to solve combinatorial (typically graph) problems in systems of many machines with limited space. Recent work has focused on the regime in which machines have sublinear (in n , the number of nodes in the input graph) space, with randomized algorithms presented for the fundamental problems of Maximal Matching and Maximal Independent Set. However, there have been no prior corresponding deterministic algorithms. A major challenge underlying the sublinear space setting is that the local space of each machine might be too small to store all edges incident to a single node. This poses a considerable obstacle compared to classical models in which each node is assumed to know and have easy access to its incident edges. To overcome this barrier, we introduce a new graph sparsification technique that deterministically computes a low-degree subgraph, with the additional property that solving the problem on this subgraph provides significant progress towards solving the problem for the original input graph. Using this framework to derandomize the well-known algorithm of Luby [SICOMP’86], we obtain O (log Δ + log log n )-round deterministic MPC algorithms for solving the problems of Maximal Matching and Maximal Independent Set with O ( n ɛ ) space on each machine for any constant ɛ > 0. These algorithms also run in O (log Δ) rounds in the closely related model of CONGESTED CLIQUE, improving upon the state-of-the-art bound of O (log 2 Δ) rounds by Censor-Hillel et al. [DISC’17].

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