## On Robust Cycle Bases (0)

Venue: | Proc. 9th Quadrennial International Conference on Graph Theory and Computer Science, Y. Alavi et al., Eds., SIAM |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Kainen_onrobust,

author = {Paul C. Kainen},

title = {On Robust Cycle Bases},

booktitle = {Proc. 9th Quadrennial International Conference on Graph Theory and Computer Science, Y. Alavi et al., Eds., SIAM},

year = {}

}

### OpenURL

### Abstract

Two types of robust cycle bases are defined via recursively nice arrangements; complete and bipartite complete graphs are shown to have such bases. It is shown that a diagram in a groupoid is commutative up to natural equivalence (cutne) if for each cycle in a robust basis of the graph underlying the diagram, the composition of the morphisms is naturally equivalent to the identity. For a hypercube Qn , it is shown that the commutativity (or cutne) of a particular subset of asymptotically 4=n of the square faces forces commutativity (or cutne) of the entire diagram. Keywords: cycle bases, coherence, diagrams, hypercubes, groupoids, commutativity up to natural equivalence, approximate commutativity, cognitive science 1 Introduction We study the elementary properties of a new type of cycle basis for graphs based on the possibility of ordering those basis cycles which sum to a given cycle in such a way that some recursive condition holds. A cycle in some graph is a subset of the...

### Citations

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(Show Context)
Citation Context ... meeting the previous partial sum in a nice way. A diagram is a directed graph that lives in a category; i.e., the vertices are objects of the category and the arcs are morphisms. See, e.g., Mac Lane =-=[7]-=-; a brief review is in section 3 below. We apply robustness to questions of diagram coherence related to Mac Lane [6] and Stasheff [11]. They asked when the commutativity of all diagrams of a certain ... |

870 |
Algebraic Topology
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(Show Context)
Citation Context ...of. Let B be a basis of commutative cycles for the diagram D; that is, for the underlying graph G. The fundamental group of the topological realization of G is the free group on B. See, e.g., Spanier =-=[10]-=-. Hence, any cycle Z in G can be expressed as a word in the generators and their inverses. Since the letters all evaluate to the identity because of our assumption about B, the words must also evaluat... |

333 |
Homotopy associativity of H-spaces
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(Show Context)
Citation Context ...s of the category and the arcs are morphisms. See, e.g., Mac Lane [7]; a brief review is in section 3 below. We apply robustness to questions of diagram coherence related to Mac Lane [6] and Stasheff =-=[11]-=-. They asked when the commutativity of all diagrams of a certain type is implied by that of a few particular cases just as associativity for n-term expressions follows from that of the 3-term expressi... |

267 |
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(Show Context)
Citation Context ...aph can be constructed by beginning with a cycle and adding on new cycles in such a way that each new cycle intersects the union of those that have preceded it in a nontrivial path. See, e.g., Lovasz =-=[4]-=-. The condition seems natural and the following definition uses an analogous nontrivial-path-intersection property to constrain the way a cycle basis represents the cycles. A cycle basis B is robust i... |

61 |
Natural associativity and commutativity
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(Show Context)
Citation Context ...rtices are objects of the category and the arcs are morphisms. See, e.g., Mac Lane [7]; a brief review is in section 3 below. We apply robustness to questions of diagram coherence related to Mac Lane =-=[6]-=- and Stasheff [11]. They asked when the commutativity of all diagrams of a certain type is implied by that of a few particular cases just as associativity for n-term expressions follows from that of t... |

22 |
Picture theory
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- 1994
(Show Context)
Citation Context ...s a robust basis but as yet have no proof. For n equal to 3, this basis consists of 5 of the 6 faces. The following is a special case of the category theory result known as the cube lemma; see, e.g., =-=[8]-=-. Lemma 2 Let D by any diagram in a groupoid subcategory of CAT with underlying graph Q 3 . If five of the six faces commute up to a natural equivalence, then so does the sixth. This lemma is a conseq... |

9 |
Coherence for the tensor product of 2-categories, and braid groups, Algebra, Topology, and Category Theory (a collection of papers in honour of Samue l Eilenberg
- Gray
- 1976
(Show Context)
Citation Context ...appings, it is easily checked that the resulting inverse transformation must be natural, so they are actually natural equivalences. A still more general notion than category is 2-category (see, e.g., =-=[1]-=-), where multiple laws of composition also obey rules of mutual consistency. However, we shall now return to the connection of these ideas with cycles. Let D be any diagram in a groupoid subcategory o... |

9 |
Approximate isometries of the space of continuous functions
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- 1947
(Show Context)
Citation Context ...ations are reversible and hence invertible. Also, neural networks have been used to model invertible maps [9]. Robustness is a general theme which appears in many areas of mathematics. Hyers and Ulam =-=[2]-=- discovered that if two spaces are within a sufficiently small tolerance of being isometric, then they are in fact isometric (and the isometry is globally close to the nearisometry) . A robustness phe... |

5 | System modeling using composition networks
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(Show Context)
Citation Context ...cludes many interesting special cases. For example, in the case of quantum computing, all operations are reversible and hence invertible. Also, neural networks have been used to model invertible maps =-=[9]-=-. Robustness is a general theme which appears in many areas of mathematics. Hyers and Ulam [2] discovered that if two spaces are within a sufficiently small tolerance of being isometric, then they are... |

4 |
Quasiorthogonal dimension of Euclidean spaces
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Citation Context ...f being isometric, then they are in fact isometric (and the isometry is globally close to the nearisometry) . A robustness phenomenon also occurs with respect to orthogonality. We showed with Kurkova =-=[3]-=- that in n-dimensional Euclidean space, if any two of a set of unit vectors have dot product not exceeding 1=n in absolute value, then the set cannot have more than n members. We wonder if these resul... |

1 |
A structural characterization of planar graphs
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- 1937
(Show Context)
Citation Context .... In 1937, Mac Lane showed that for a 2-connected graph, planarity is equivalent to the existence of a 2-basis, which is a cycle basis in which every edge belongs to at most two cycles from the basis =-=[5]-=-. For any 2-connected plane graph, the cycles determined by the boundaries of the finite regions constitute a 2-basis. Every 2-connected graph can be constructed by beginning with a cycle and adding o... |