scholarly journals Infinite Gammoids

10.37236/4181 ◽  
2015 ◽  
Vol 22 (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.

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.

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 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 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 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 .

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.

On characterization of maximal independent sets via quadratic optimization

2011 ◽  
Vol 19 (4) ◽  
pp. 629-644 ◽  
Author(s):  
Foad Mahdavi Pajouh ◽  
Balabhaskar Balasundaram ◽  
Oleg A. Prokopyev
Keyword(s):  
Independent Sets ◽  
Quadratic Optimization ◽  
Maximal Independent Sets

scholarly journals ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

2021 ◽  
pp. 1-40
Author(s):  
NICK GILL ◽  
BIANCA LODÀ ◽  
PABLO SPIGA
Keyword(s):  
Permutation Group ◽  
Model Theory ◽  
Independent Set ◽  
Maximum Size ◽  
Proper Subset ◽  
Permutation Groups ◽  
Finite Permutation Group ◽  
Pointwise Stabilizer ◽  
Relational Complexity

Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

Sign in / Sign up

Export Citation Format

Share Document