scholarly journals NBC Complexes of Convex Geometries

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Kenji Kashiwabara ◽  
Masataka Nakamura
Keyword(s):  
Convex Geometry ◽  
Decomposition Theorem ◽  
International Audience ◽  
Generating Theorem ◽  
Convex Geometries ◽  
Broken Circuit

International audience We introduce a notion of a $\textit{broken circuit}$ and an $\textit{NBC complex}$ for an (abstract) convex geometry. Based on these definitions, we shall show the analogues of the Whitney-Rota's formula and Brylawski's decomposition theorem for broken circuit complexes on matroids for convex geometries. We also present an Orlik-Solomon type algebra on a convex geometry, and show the NBC generating theorem.

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

scholarly journals The Prime Stems of Rooted Circuits of Closure Spaces and Minimum Implicational Bases

10.37236/3068 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Masataka Nakamura ◽  
Kenji Kashiwabara
Keyword(s):  
Convex Geometry ◽  
Closure Operators ◽  
Optimal Basis ◽  
Closed Set ◽  
Closed Sets ◽  
Closure Systems ◽  
Closure Spaces ◽  
Convex Geometries ◽  
Necessary And Sufficient ◽  
Special Case

A rooted circuit is firstly introduced for convex geometries (antimatroids). We generalize it for closure systems or equivalently for closure operators. A rooted circuit is a specific type of a pair $(X,e)$ of a subset $X$, called a stem, and an element $e\not\in X$, called a root. We introduce a notion called a 'prime stem', which plays the key role in this article. Every prime stem is shown to be a pseudo-closed set of an implicational system. If the sizes of stems are all the same, the stems are all pseudo-closed sets, and they give rise to a canonical minimum implicational basis. For an affine convex geometry, the prime stems determine a canonical minimum basis, and furthermore  gives rise to an optimal basis. A 'critical rooted circuit' is a special case of a rooted circuit defined for an antimatroid. As a precedence structure, 'critical rooted circuits' are necessary and sufficient to fix an antimatroid whereas critical rooted circuits are not necessarily sufficient to restore the original antimatroid as an implicational system. It is shown through an example.

scholarly journals A new two-variable generalization of the chromatic polynomial

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Klaus Dohmen ◽  
André Poenitz ◽  
Peter Tittmann
Keyword(s):  
Chromatic Polynomial ◽  
Complete Graphs ◽  
Matching Polynomial ◽  
Independence Polynomial ◽  
Decomposition Formula ◽  
International Audience ◽  
Complete Bipartite Graphs ◽  
Chromatic Symmetric Function ◽  
Broken Circuit ◽  
Edge Decomposition

International audience We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph. This new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings. We show that the new polynomial may be considered as a specialization of Stanley's chromatic symmetric function. We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth.

scholarly journals Convexity in Ordered Matroids and the Generalized External Order

10.37236/7582 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Bryan R. Gillespie
Keyword(s):  
Distributive Lattice ◽  
Generating Function ◽  
Convex Sets ◽  
Convex Geometry ◽  
Tutte Polynomial ◽  
Oriented Matroids ◽  
Geometric Lattice ◽  
Discrete Convexity ◽  
New Type ◽  
Convex Geometries

In 1980, Las Vergnas defined a notion of discrete convexity for oriented matroids, which Edelman subsequently related to the theory of anti-exchange closure functions and convex geometries. In this paper, we use generalized matroid activity to construct a convex geometry associated with an ordered, unoriented matroid. The construction in particular yields a new type of representability for an ordered matroid defined by the affine representability of its corresponding convex geometry. The lattice of convex sets of this convex geometry induces an ordering on the matroid independent sets which extends the external active order on matroid bases. We show that this generalized external order forms a supersolvable meet-distributive lattice refining the geometric lattice of flats, and we uniquely characterize the lattices isomorphic to the external order of a matroid. Finally, we introduce a new trivariate generating function generalizing the matroid Tutte polynomial.

scholarly journals Lattices of regular closed subsets of closure spaces

2014 ◽  
Vol 24 (07) ◽  
pp. 969-1030 ◽  
Author(s):  
Luigi Santocanale ◽  
Friedrich Wehrung
Keyword(s):  
Hyperplane Arrangement ◽  
Convex Geometry ◽  
Macneille Completion ◽  
Closed Set ◽  
Closed Sets ◽  
Closure Spaces ◽  
Infinite Collection ◽  
Convex Geometries ◽  
Poset Of Regions ◽  
Regular Closed

For a closure space (P, φ) with φ(ø) = ø, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg (P, φ), extending the poset Clop (P, φ) of all clopen subsets. If (P, φ) is a finite convex geometry, then Reg (P, φ) is pseudocomplemented. The Dedekind–MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, hence it is pseudocomplemented. The lattice Reg (P, φ) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For finite such closure spaces,• Reg (P, φ) satisfies an infinite collection of stronger and stronger quasi-identities, weaker than both meet- and join-semidistributivity. Nevertheless it may fail semidistributivity.• If Reg (P, φ) is semidistributive, then it is a bounded homomorphic image of a free lattice.• Clop (P, φ) is a lattice if and only if every regular closed set is clopen.The extended permutohedron R (G) on a graph G and the extended permutohedron Reg S on a join-semilattice S, are both defined as lattices of regular closed sets of suitable closure spaces. While the lattice of all regular closed sets is, in the semilattice context, always the Dedekind–MacNeille completion of the poset of clopen sets, this does not always hold in the graph context, although it always does so for finite block graphs and for cycles. Furthermore, both R (G) and Reg S are bounded homomorphic images of free lattices.

scholarly journals An Abstraction of Whitney's Broken Circuit Theorem

10.37236/4356 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Klaus Dohmen ◽  
Martin Trinks
Keyword(s):  
Finite Lattice ◽  
Chromatic Polynomial ◽  
Möbius Function ◽  
Least Common Multiple ◽  
Mobius Function ◽  
Finite Set ◽  
Domination Polynomial ◽  
Riemann Zeta ◽  
Convex Geometries ◽  
Broken Circuit

We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of type $\sum_{A\subseteq S} f(A)$ where $S$ is a finite set and $f$ is a mapping from the power set of $S$ into an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polynomial and Crapo's beta invariant of a matroid, and the principle of inclusion-exclusion. Thus, we discover several known and new results in a concise and unified way. As further applications of our main result, we derive a new generalization of the maximums-minimums identity and of a theorem due to Blass and Sagan on the Möbius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical Möbius function, both Euler's totient function and its Dirichlet inverse, and the reciprocal of the Riemann zeta function we obtain new expansions involving the greatest common divisor resp. least common multiple. We finally establish an even broader generalization of Whitney's broken circuit theorem in the context of convex geometries (antimatroids).

scholarly journals Cycle transversals in bounded degree graphs

2011 ◽  
Vol Vol. 13 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Marina Groshaus ◽  
Pavol Hell ◽  
Sulamita Klein ◽  
Loana Tito Nogueira ◽  
Fábio Protti
Keyword(s):  
Approximation Algorithms ◽  
Polynomial Time ◽  
Maximum Degree ◽  
Decomposition Theorem ◽  
Np Hard ◽  
Input Graph ◽  
Bounded Degree ◽  
Time Approximation ◽  
International Audience ◽  
Bounded Degree Graphs

Graphs and Algorithms International audience In this work we investigate the algorithmic complexity of computing a minimum C(k)-transversal, i.e., a subset of vertices that intersects all the chordless cycles with k vertices of the input graph, for a fixed k \textgreater= 3. For graphs of maximum degree at most three, we prove that this problem is polynomial-time solvable when k \textless= 4, and NP-hard otherwise. For graphs of maximum degree at most four, we prove that this problem is NP-hard for any fixed k \textgreater= 3. We also discuss polynomial-time approximation algorithms for computing C(3)-transversals in graphs of maximum degree at most four, based on a new decomposition theorem for such graphs that leads to useful reduction rules.

scholarly journals Resolutions of Convex Geometries

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Domenico Cantone ◽  
Jean-Paul Doignon ◽  
Alfio Giarlotta ◽  
Stephen Watson
Keyword(s):  
Convex Geometry ◽  
Learning Spaces ◽  
Combinatorial Structures ◽  
Open Problems ◽  
Set Systems ◽  
Convex Geometries

Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces.  We define an operation of resolution for convex geometries, which replaces each element of a base convex geometry by a fiber convex geometry.  Contrary to what happens for similar constructions–compounds of hypergraphs, as in Chein, Habib and Maurer (1981), and compositions of set systems, as in Möhring and Radermacher)–, resolutions of convex geometries always yield a convex geometry.   We investigate resolutions of special convex geometries: ordinal and affine.  A resolution of ordinal convex geometries is again ordinal, but a resolution of affine convex geometries may fail to be affine.  A notion of primitivity, which generalize the corresponding notion for posets, arises from resolutions: a convex geometry is primitive if it is not a resolution of smaller ones.  We obtain a characterization of affine convex geometries that are primitive, and compute the number of primitive convex geometries on at most four elements.  Several open problems are listed. 

scholarly journals Improved inclusion-exclusion identities via closure operators

2000 ◽  
Vol Vol. 4 no. 1 ◽  
Author(s):  
Klaus Dohmen
Keyword(s):  
Closure Operator ◽  
Chromatic Polynomial ◽  
Finite Family ◽  
Exclusion Principle ◽  
Closure Operators ◽  
Base Property ◽  
Power Set ◽  
International Audience ◽  
Broken Circuit

International audience Let (A_v)_v ∈ V be a finite family of sets. We establish an improved inclusion-exclusion identity for each closure operator on the power set of V having the unique base property. The result generalizes three improvements of the inclusion-exclusion principle as well as Whitney's broken circuit theorem on the chromatic polynomial of a graph.

scholarly journals Weak Positional Games on Hypergraphs of Rank Three

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Martin Kutz
Keyword(s):  
Polynomial Time ◽  
Optimal Strategies ◽  
Decomposition Theorem ◽  
Positional Games ◽  
Almost Disjoint ◽  
International Audience ◽  
New Type

International audience In a weak positional game, two players, Maker and Breaker, alternately claim vertices of a hypergraph until either Maker wins by getting a complete edge or all vertices are taken without this happening, a Breaker win. For the class of almost-disjoint hypergraphs of rank three (edges with up to three vertices only and edge-intersections on at most one vertex) we show how to find optimal strategies in polynomial time. Our result is based on a new type of decomposition theorem which might lead to a better understanding of weak positional games in general.

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