L’axiologie d’Ingarden

Cet article met en lumière la formation des valeurs esthétiques entendues au sens de la philosophie de Roman Ingarden. L'auteur présente et commente trois notions essentielles de cette pensée : il s'agit de la concrétisation, de la valeur esthétique et du noyau de l'objet esthétique. Bien que la critique s'intéresse depuis longtemps à l'axiologie ingardénienne, la perspective de l'auteur, mettant en valeur la notion de noyau esthétique, indique clairement que la pensée d'Ingarden chevauche deux approches différentes en ce qui concerne la formation des valeurs. Lorsqu'il s'agit de la création de nouvelles valeurs, par exemple, la soi-disant objectivité de la philosophie d'Ingarden semble masquer le côté subjectif de leur formation.The objective of this article isfirst to present the major notions of Ingarden's philosophy with respect to axiology : concretization, aesthetic value and the core of the aesthetic object, and second, to show that Ingarden's so-called objective aesthetics covers up or masks the more subjective elements of value theory. Critics have long been interested in Ingarden's aesthetics, but the author's perspective, insisting on the importance of the core of the aesthetic objet, clearly reveals that Ingarden's thought actually involves two different approaches in aesthetic value theory, an empirical approach, and a more subjective, relative one which seems to remain hidden

Hypomorphisms, orbits, and reconstruction

AbstractGraphs G and H are hypomorphic if there is a bijection φ: V(G) → V(H) such that G − u ≅ H − φ(u), for all u ∈ V(G). The reconstruction conjecture states that hypomorphic graphs are isomorphic, if G has at least three vertices. We investigate properties of the isomorphisms G − u ≅ H − φ(u), and their relation to the reconstructibility of G

An algebraic formulation of the graph reconstruction conjecture

The graph reconstruction conjecture asserts that every finite simple graph on at least three vertices can be reconstructed up to isomorphism from its deck - the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important tool in graph reconstruction. Roughly speaking, given the deck of a graph $G$ and any finite sequence of graphs, it gives a linear constraint that every reconstruction of $G$ must satisfy. Let $\psi(n)$ be the number of distinct (mutually non-isomorphic) graphs on $n$ vertices, and let $d(n)$ be the number of distinct decks that can be constructed from these graphs. Then the difference $\psi(n) - d(n)$ measures how many graphs cannot be reconstructed from their decks. In particular, the graph reconstruction conjecture is true for $n$-vertex graphs if and only if $\psi(n) = d(n)$. We give a framework based on Kocay's lemma to study this discrepancy. We prove that if $M$ is a matrix of covering numbers of graphs by sequences of graphs, then $d(n) \geq \mathsf{rank}_\mathbb{R}(M)$. In particular, all $n$-vertex graphs are reconstructible if one such matrix has rank $\psi(n)$. To complement this result, we prove that it is possible to choose a family of sequences of graphs such that the corresponding matrix $M$ of covering numbers satisfies $d(n) = \mathsf{rank}_\mathbb{R}(M)$.Comment: 12 pages, 2 figure

On null 3-hypergraphs

International audienceGiven a 3-uniform hypergraph H consisting of a set V of vertices, and T ⊆ V 3 triples, a null labelling is an assignment of ±1 to the triples such that each vertex is contained in an equal number of triples labelled +1 and −1. Thus, the signed degree of each vertex is zero. A necessary condition for a null labelling is that the degree of every vertex of H is even. The Null Labelling Problem is to determine whether H has a null labelling. It is proved that this problem is NP-complete. Computer enumerations suggest that most hypergraphs which satisfy the necessary condition do have a null labelling. Some constructions are given which produce hypergraphs satisfying the necessary condition, but which do not have a null labelling. A self complementary 3-hypergraph with this property is also constructed

Characterization and enumeration of toroidal K_{3,3}-subdivision-free graphs

We describe the structure of 2-connected non-planar toroidal graphs with no K_{3,3}-subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on a refinement of the algorithmic results for graphs containing a fixed K_5-subdivision in [A. Gagarin and W. Kocay, "Embedding graphs containing K_5-subdivisions'', Ars Combin. 64 (2002), 33-49]. It allows to recognize these graphs in linear-time and makes possible to enumerate labelled 2-connected toroidal graphs containing no K_{3,3}-subdivisions and having minimum vertex degree two or three by using an approach similar to [A. Gagarin, G. Labelle, and P. Leroux, "Counting labelled projective-planar graphs without a K_{3,3}-subdivision", submitted, arXiv:math.CO/0406140, (2004)].Comment: 18 pages, 7 figures and 4 table

The bondage number of graphs on topological surfaces and Teschner's conjecture

The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of the graph, and show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus. Also, we provide stronger upper bounds for graphs with no triangles and graphs with the number of vertices larger than a certain threshold in terms of the graph genera. This settles Teschner's Conjecture in positive for almost all graphs.Comment: 21 pages; Original version from January 201