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- W2282804969 abstract "A vertex-deleted subgraph (or simply a card) of graph G is an induced subgraphof G containing all but one of its vertices. The deck of G is the multisetof its cards. One of the best-known unanswered questions of graph theoryasks whether G can be reconstructed in a unique way (up to isomorphism)from its deck. The likely positive answer to this question is known as theReconstruction Conjecture.In the first part of the thesis two basic equivalence relations are consideredon the set of vertices of the graph G to be reconstructed. The one is cardequivalence, better known as removal equivalence, by which two vertices areequivalent if their removal results in isomorphic cards. The other equivalenceis similarity, also called automorphism equivalence. Two vertices u and vare automorphism-equivalent (similar) if there exists an automorphism of Gtaking u to v. These relations are analyzed on various examples with specialattention to vertices that are card-equivalent but not similar. Such verticesare called pseudo-similar, and they have been studied very extensively in theliterature. The first result of the thesis is a structural characterization ofcard equivalence in terms of automorphism equivalence. A similar result wasobtained by Godsil and Kocay in 1982 on the characterization of pseudosimilarvertices, which result is proved in the thesis as a corollary to thecharacterization theorem on card equivalence.In the second part of the thesis, the concept of relative degree-sequenceis introduced for subgraphs of a graph G. By “relative” it is meant that eachdegree in the degree-sequence of the subgraph is coupled up with the originaldegree of the corresponding vertex in G. A new conjecture is formulated,which says that G is uniquely determined (up to isomorphism) by the multisetof the relative degree-sequences of its induced subgraphs. The new conjectureis then related to the Reconstruction Conjecture in a natural way.The third part of the thesis contains an original new result on graphreconstruction. Card-minimal graphs are investigated, the deck of which isa set. Thus, the deck of such graphs is free from duplicate cards. It is shownthat every card-minimal graph G is reconstructible, provided that G doesnot have pseudo-similar couples of vertices. This condition is recognizable,that is, it can be checked by looking at the deck of G only.The results of this thesis have been partially published in [1]." @default.
- W2282804969 created "2016-06-24" @default.
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- W2282804969 date "2015-02-01" @default.
- W2282804969 modified "2023-09-27" @default.
- W2282804969 title "The graph reconstruction conjecture: some new results and observations" @default.
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