scholarly journals Combinatorial properties of the family of maximum stable sets of a graph

2002 ◽  
Vol 117 (1-3) ◽  
pp. 149-161 ◽  
Author(s):  
Vadim E. Levit ◽  
Eugen Mandrescu
Keyword(s):  
Stable Sets ◽  
Combinatorial Properties ◽  
The Family

scholarly journals Further steps on the reconstruction of convex polyominoes from orthogonal projections

2021 ◽  
Author(s):  
Paolo Dulio ◽  
Andrea Frosini ◽  
Simone Rinaldi ◽  
Lama Tarsissi ◽  
Laurent Vuillon
Keyword(s):  
Discrete Geometry ◽  
Convex Subset ◽  
Convex Sets ◽  
A Priori ◽  
Orthogonal Projections ◽  
Reconstruction Process ◽  
Combinatorial Properties ◽  
The Family ◽  
Discrete Counterpart ◽  
Interior Part

AbstractA remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean convex sets, and combine the constraints of convexity and connectedness. In this paper we study the problem of their reconstruction from orthogonal projections, relying on the approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of its contour, without losing its convexity. To reach this goal we consider convexity in terms of certain combinatorial properties of the boundary word encoding the polyomino. So, we first show some conditions that allow us to extend the kernel maintaining the convexity. Then, we provide examples where the addition of one or two points causes a loss of convexity, which can be restored by adding other points, whose number and positions cannot be determined a priori.

scholarly journals VERY WELL-COVERED GRAPHS OF GIRTH AT LEAST FOUR AND LOCAL MAXIMUM STABLE SET GREEDOIDS

2011 ◽  
Vol 03 (02) ◽  
pp. 245-252 ◽  
Author(s):  
VADIM E. LEVIT ◽  
EUGEN MANDRESCU
Keyword(s):  
Perfect Matching ◽  
Local Maximum ◽  
Stable Set ◽  
Stable Sets ◽  
Maximum Cardinality ◽  
Math Program ◽  
The Family ◽  
Maximum Stable Set ◽  
Vertex Set ◽  
Appl Math

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter Jr. [Vertex packings: structural properties and algorithms, Math. Program.8 (1975) 232–248], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [Levit and Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discrete Appl. Math.124 (2002) 91–101] we have shown that the family Ψ(T) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while Ψ(G) is a greedoid, were analyzed in [Levit and Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math.132 (2004) 163–174], [Levit and Mandrescu, Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids, Discrete Appl. Math.155 (2007) 2414–2425], [Levit and Mandrescu, Well-covered graphs and greedoids, Proc. 14th Computing: The Australasian Theory Symp. (CATS2008), Wollongong, NSW, Conferences in Research and Practice in Information Technology, Vol. 77 (2008) 89–94], respectively. In this paper we demonstrate that if G is a very well-covered graph of girth ≥4, then the family Ψ(G) is a greedoid if and only if G has a unique perfect matching.

scholarly journals Greedoids on vertex sets of B-joins of graphs

2014 ◽  
Vol 30 (3) ◽  
pp. 335-344
Author(s):  
VADIM E. LEVIT ◽  
◽  
EUGEN MANDRESCU ◽  
Keyword(s):  
Bipartite Graph ◽  
Sufficient Conditions ◽  
Local Maximum ◽  
Stable Set ◽  
Stable Sets ◽  
The Family ◽  
Maximum Stable Set ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
Vertex Sets

Let Ψ(G) be the family of all local maximum stable sets of graph G, i.e., S ∈ Ψ(G) if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. It was shown that Ψ(G) is a greedoid for every forest G [15]. The cases of bipartite graphs, triangle-free graphs, and well-covered graphs, were analyzed in [16, 17, 18, 19, 20, 24]. If G1, G2 are two disjoint graphs, and B is a bipartite graph having E(B) as an edge set and bipartition {V (G1), V (G2)}, then by B-join of G1, G2 we mean the graph B (G1, G2) whose vertex set is V (G1) ∪ V (G2) and edge set is E(G1) ∪ E(G2) ∪ E (B). In this paper we present several necessary and sufficient conditions for Ψ(B (G1, G2)) to form a greedoid, an antimatroid, and a matroid, in terms of Ψ(G1), Ψ(G2) and E (B).

scholarly journals Combinatorics of one-dimensional simple Toeplitz subshifts

2018 ◽  
Vol 40 (6) ◽  
pp. 1673-1714
Author(s):  
DANIEL SELL
Keyword(s):  
Systematic Study ◽  
Henri Poincaré ◽  
Explicit Formulas ◽  
De Bruijn Graphs ◽  
One Dimensional ◽  
Combinatorial Properties ◽  
Complexity Function ◽  
Schreier Graphs ◽  
The Family ◽  
De Bruijn

This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the repetitivity function are proved. Moreover, a complete description of the de Bruijn graphs of the subshifts is given. Finally, the Boshernitzan condition is characterized in terms of combinatorial quantities, based on a recent result of Liu and Qu [Uniform convergence of Schrödinger cocycles over simple Toeplitz subshift. Ann. Henri Poincaré12(1) (2011), 153–172]. Particular simple characterizations are provided for simple Toeplitz subshifts that correspond to the orbital Schreier graphs of the family of Grigorchuk’s groups, a class of subshifts that serves as the main example throughout the paper.

CROSS RATIO GRAPHS

2001 ◽  
Vol 64 (2) ◽  
pp. 257-272 ◽  
Author(s):  
A. GARDINER ◽  
CHERYL E. PRAEGER ◽  
SANMING ZHOU
Keyword(s):  
Automorphism Groups ◽  
Projective Line ◽  
Cross Ratio ◽  
Transitive Graph ◽  
Combinatorial Properties ◽  
The Family ◽  
The Cross ◽  
Transitive Graphs ◽  
Ordered Pairs

A family of arc-transitive graphs is studied. The vertices of these graphs are ordered pairs of distinct points from a finite projective line, and adjacency is defined in terms of the cross ratio. A uniform description of the graphs is given, their automorphism groups are determined, the problem of isomorphism between graphs in the family is solved, some combinatorial properties are explored, and the graphs are characterised as a certain class of arc-transitive graphs. Some of these graphs have arisen as examples in studies of arc-transitive graphs with complete quotients and arc-transitive graphs which ‘almost cover’ a 2-arc transitive graph.

scholarly journals A new Greedoid: the family of local maximum stable sets of a forest

2002 ◽  
Vol 124 (1-3) ◽  
pp. 91-101 ◽  
Author(s):  
Vadim E. Levit ◽  
Eugen Mandrescu
Keyword(s):  
Local Maximum ◽  
Stable Sets ◽  
The Family

scholarly journals The effect of edge and vertex deletion on omega invariant

2020 ◽  
pp. 46-46 ◽  
Author(s):  
Sadik Delen ◽  
Muge Togan ◽  
Aysun Yurttas ◽  
Ugur Ana ◽  
Ismail Cangu
Keyword(s):  
Euler Characteristic ◽  
Degree Sequence ◽  
Graph Invariant ◽  
Combinatorial Properties ◽  
Number Of Components ◽  
Vertex Deletion ◽  
The Family

Recently the first and last authors defined a new graph characteristic called omega related to Euler characteristic to determine several topological and combinatorial properties of a given graph. This new characteristic is defined in terms of a given degree sequence as a graph invariant and gives a lot of information on the realizability, number of realizations, connectedness, cyclicness, number of components, chords, loops, pendant edges, faces, bridges etc. of the family of realizations. In this paper, the effect of the deletion of vertices and edges from a graph on omega invariant is studied.

scholarly journals An order property for families of sets

1988 ◽  
Vol 44 (3) ◽  
pp. 294-310
Author(s):  
N. H. Williams
Keyword(s):  
Pairwise Disjoint ◽  
Chain Condition ◽  
Infinite Cardinal ◽  
Order Property ◽  
Disjoint Family ◽  
Combinatorial Properties ◽  
The Family ◽  
Almost Disjoint ◽  
A Chain ◽  
Almost Disjoint Family

AbstractWe develop the idea of a θ-ordering (where θ is an infinite cardinal) for a family of infinite sets. A θ-ordering of the family A is a well ordering of A which decomposes A into a union of pairwise disjoint intervals in a special way, which facilitates certain transfinite constructions. We show that several standard combinatorial properties, for instance that of the family A having a θ-transversal, are simple consequences of A possessing a θ-ordering. Most of the paper is devoted to showing that under suitable restrictions, an almost disjoint family will have a θ-ordering. The restrictions involve either intersection conditions on A (the intersection of every λ-size subfamily of A has size at most κ) or a chain condition on A.

scholarly journals Macaulay-like marked bases

2017 ◽  
Vol 16 (05) ◽  
pp. 1750100 ◽  
Author(s):  
Cristina Bertone ◽  
Francesca Cioffi ◽  
Margherita Roggero
Keyword(s):  
Polynomial Ring ◽  
Open Subset ◽  
Hilbert Scheme ◽  
Combinatorial Properties ◽  
Ideal Formula ◽  
Complexity Results ◽  
The Family ◽  
Stable Ideal ◽  
Affine Scheme

We define marked sets and bases over a quasi-stable ideal [Formula: see text] in a polynomial ring on a Noetherian [Formula: see text]-algebra, with [Formula: see text] a field of any characteristic. The involved polynomials may be non-homogeneous, but their degree is bounded from above by the maximum among the degrees of the terms in the Pommaret basis of [Formula: see text] and a given integer [Formula: see text]. Due to the combinatorial properties of quasi-stable ideals, these bases behave well with respect to homogenization, similarly to Macaulay bases. We prove that the family of marked bases over a given quasi-stable ideal has an affine scheme structure, is flat and, for large enough [Formula: see text], is an open subset of a Hilbert scheme. Our main results lead to algorithms that explicitly construct such a family. We compare our method with similar ones and give some complexity results.

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