Switching Reconstruction of Digraphs

Switching about a vertex in a digraph means to reverse the direction of every edge incident with that vertex. Bondy and Mercier introduced the problem of whether a digraph can be reconstructed up to isomorphism from the multiset of isomorphism types of digraphs obtained by switching about each vertex. Since the largest known non-reconstructible oriented graphs have 8 vertices, it is natural to ask whether there are any larger non-reconstructible graphs. In this paper we continue the investigation of this question. We find that there are exactly 44 non-reconstructible oriented graphs whose underlying undirected graphs have maximum degree at most 2. We also determine the full set of switching-stable oriented graphs, which are those graphs for which all switchings return a digraph isomorphic to the original

Reconstruction of Finite Truncated Semi-Modular Lattices

AbstractIn this paper, we prove reconstruction results for truncated lattices. The main results are that truncated lattices that contain a 4-crown and truncated semi-modular lattices are reconstructible. Reconstruction of the truncated lattices not covered by this work appears challenging. Indeed, the remaining truncated lattices possess very little lattice-typical structure. This seems to indicate that further progress on the reconstruction of truncated lattices is closely correlated with progress on reconstructing ordered sets in general

On a stronger reconstruction notion for monoids and clones

Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further give a characterization of automatic homeomorphicity for transformation monoids on arbitrary carriers with a dense group of invertibles having automatic homeomorphicity. We then show how to lift automatic action compatibility from groups to monoids and from monoids to clones under fairly weak assumptions. We finally employ these theorems to get automatic action compatibility results for monoids and clones over several well-known countable structures, including the strictly ordered rationals, the directed and undirected version of the random graph, the random tournament and bipartite graph, the generic strictly ordered set, and the directed and undirected versions of the universal homogeneous Henson graphs.Comment: 29 pp; Changes v1-->v2::typos corr.|L3.5+pf extended|Rem3.7 added|C. Pech found out that arg of L5.3-v1 solved Probl2-v1|L5.3, C5.4, Probl2 of v1 removed|C5.2, R5.4 new, contain parts of pf of L5.3-v1|L5.2-v1 is now L5.3,merged with concl of C5.4-v1,L5.3-v2 extends C5.4-v1|abstract, intro updated|ref[24] added|part of L5.3-v1 is L2.1(e)-v2, another part merged with pf of L5.2-v1 => L5.3-v

Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..

Classificação dos digrafos semicompletos hamiltonianos

Orientador: Claudina Izepe RodriguesDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientificaResumo: 0 objetivo principal deste trabalho é apresentar uma classificação para os dígrafos semicompletos hamiltonianos, extendendo os resultados obtidos para os torneios. Para isso utilizamos da teoria da homotopia regular de grafos de Davide C. Demaria, apresentando resultados sobre torneios simplemente desconexos, a caracterização de torneios por 3-ciclos e o conceito de ciclo conado e não-conado para dígrafos, introduzido por Kiihl e Tironi. Com a noção de ciclo minimal e característico para dígrafo uma classificação para os dígrafos semicompletos hamiltonianos surge então naturalmente. Esses resultados, quando encontrados para torneios, proporcionaram a obtenção de uma classe de torneios reconstrutíveis (torneios normais) e pesquisa nesse sentido deve ser efetuada para dígrafos. Apresentamos em apêndice a matriz de um dígrafo, os torneios de moon, normais e, brevemente, o problema da reconstrução de grafosAbstract: The main target in this work is to present a classification for the hamiltonian semicomplete digraphs, extending the results previously obtained for the tournaments. In this way we apply the regular homotopy of finite directed graphs theory developed by Davide G. Demaria, presenting results on simply disconnected tournaments, on the caracterization of tournaments by 3-cicles and the concept of coned and non-coned cicle for digraphs, introduced by Kiihl and Tironi. With the notion of minimal and caracteristic cicle we naturally get a classification of the semicomplete hamiltonian digraphs. These results, when used for tournaments led to a new class of reconstructible ones (named normal) and future research on the extension of these results for digraphs in general seems to be interesting. We present in appendixes the array of a digraph, the tournaments of Moon, Normal and, briefly, the reconstruction problem for graphsMestradoMestre em Matemátic

Half-graphs, other non-stable degree sequences, and the switch Markov chain

One of the simplest methods of generating a random graph with a given degree sequence is provided by the Monte Carlo Markov Chain method using switches. The switch Markov chain converges to the uniform distribution, but generally the rate of convergence is not known. After a number of results concerning various degree sequences, rapid mixing was established for so-called $P$-stable degree sequences (including that of directed graphs), which covers every previously known rapidly mixing region of degree sequences. In this paper we give a non-trivial family of degree sequences that are not $P$-stable and the switch Markov chain is still rapidly mixing on them. This family has an intimate connection to Tyshkevich-decompositions and strong stability as well.Comment: Generalized the main theorem, paper extended with a number of corroborating result

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic