Combinatorial reconstruction problems, Hopf algebras and graph posets

• Autor(a) principal: Deisiane Lopes Gonçalves
• Data de Publicação: 2023
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: http://hdl.handle.net/1843/58291
• topic: Graph reconstruction; Graph posets; Graph homomorphisms; Hopf algebras; Kelly's lemma; Matemática; Reconstrução de grafos; Homomorfismos (Matemática); Hopf, lgebra de

The vertex reconstruction conjecture asserts that every finite simple undirected graph on three or more vertices is determined, up to isomorphism, by its collection of unlabelled vertex-deleted subgraphs. The edge reconstruction conjecture asserts that every finite simple undirected graph with four or more edges is determined, up to isomorphism, by its collection of unlabelled edge-deleted subgraphs. We consider analogous problems of reconstructing an arbitrary graph G up to isomorphism from its abstract edge‐subgraph poset, its abstract induced subgraph poset and its abstract bond lattice. We show that if a graph has no isolated vertices, then its abstract bond lattice and the abstract induced subgraph poset can be constructed from the abstract edge-subgraph poset except for the families of graphs that we characterise. We also study other relational structures obtained by considering different types of homomorphisms (e.g., general homomorphisms, monomorphisms, epimorphisms, etc.) and questions about constructions relating these structures, (for example, which structures can be constructed from which other structures). These questions are motivated by the following conjecture of Thatte. Let G be the set of all unlabelled graphs. Let f:G -> G be a bijection such that for all G,H in G, the number of homomorphisms from G to H is equal to the number of homomorphisms from f(G) to f(H). Then, f is the identity map. The conjecture is weaker than the edge reconstruction conjecture. Next we construct a subalgebra of the algebra UGQSym studied by Borie. The elements of this algebra are formal power series which can be evaluated on graphs, and count occurrences of blocks. In this formulation, we obtain an algebraic proof of a result of Whitney. Given the vertex-deck of a graph G, all vertex-proper subgraphs of G can be counted using a basic result on graph reconstruction, known as Kelly's lemma. We consider the problem of refining the lemma to count rooted subgraphs such that the root vertex coincides the deleted vertex. We show that such counting is not possible in general unless the vertex reconstruction conjecture is true, but a multiset of rooted subgraphs of a fixed height k can be constructed from the vertex-deck of G provided G has radius more than k. We prove analogous result for the edge reconstruction problem.