A vertex-deleted subgraph (or simply a card) of graph G is an induced subgraph
of G containing all but one of its vertices. The deck of G is the multiset
of its cards. One of the best-known unanswered questions of graph theory
asks 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 the
Reconstruction Conjecture.
In the first part of the thesis two basic equivalence relations are considered
on the set of vertices of the graph G to be reconstructed. The one is card
equivalence, better known as removal equivalence, by which two vertices are
equivalent if their removal results in isomorphic cards. The other equivalence
is similarity, also called automorphism equivalence. Two vertices u and v
are automorphism-equivalent (similar) if there exists an automorphism of G
taking u to v. These relations are analyzed on various examples with special
attention to vertices that are card-equivalent but not similar. Such vertices
are called pseudo-similar, and they have been studied very extensively in the
literature. The first result of the thesis is a structural characterization of
card equivalence in terms of automorphism equivalence. A similar result was
obtained by Godsil and Kocay in 1982 on the characterization of pseudosimilar
vertices, which result is proved in the thesis as a corollary to the
characterization theorem on card equivalence.
In the second part of the thesis, the concept of relative degree-sequence
is introduced for subgraphs of a graph G. By “relative” it is meant that each
degree in the degree-sequence of the subgraph is coupled up with the original
degree of the corresponding vertex in G. A new conjecture is formulated,
which says that G is uniquely determined (up to isomorphism) by the multiset
of the relative degree-sequences of its induced subgraphs. The new conjecture
is then related to the Reconstruction Conjecture in a natural way.
The third part of the thesis contains an original new result on graph
reconstruction. Card-minimal graphs are investigated, the deck of which is
a set. Thus, the deck of such graphs is free from duplicate cards. It is shown
that every card-minimal graph G is reconstructible, provided that G does
not 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]
