1 research outputs found
On some problems in reconstruction
A graph is {\it reconstructible} if it is determined by its {\it deck} of unlabeled subgraphs obtained by deleting one vertex; a {\it card} is one of these subgraphs. The {\it Reconstruction Conjecture} asserts that all graphs with at least three vertices are reconstructible.
In Chapter we consider -deck reconstruction of graphs. The {\it -deck} of a graph is its multiset of -vertex induced subgraphs. We prove a generalization of a result by Bollob\'as concerning the -deck reconstruction of almost all graphs, showing that when , the probability than an -vertex graph is reconstructible from some of the graphs in the -deck tends to as tends to .
We determine the smallest such that all graphs with maximum degree are -deck reconstructible. We prove for that whether a graph is connected is determined by its -deck. We prove that if is a complete -partite graphs, then is -deck reconstructible (the same holds for ).
In Chapter we consider degree-associated reconstruction. An -vertex induced subgraph accompanied with the degree of the missing vertex is called a {\it dacard}. The {\it degree-associated reconstruction number} of a graph is the fewest number of dacards needed to determine . We provide a tool for reconstructing some graphs from two dacards. We prove that certain families of trees and disconnected graphs can be reconstructed from two dacards. We also determine the degree-associated reconstruction number for complete multipartite graphs and their complements. For such graphs, we also determine the least such that {\it every} set of dacards determine the graph.
In Chapter we consider the reconstruction of matrices from principal submatrices. A -by- principal submatrix is a submatrix formed by deleting rows and columns symmetrically. The {\it matrix reconstruction threshold} is the minimum integer such that for all -by- matrices are reconstructible from their deck of -by- principal submatrices. We prove