30 research outputs found
Hypomorphy of graphs up to complementation
Let be a set of cardinality (possibly infinite). Two graphs and
with vertex set are {\it isomorphic up to complementation} if is
isomorphic to or to the complement of . Let be a
non-negative integer, and are {\it -hypomorphic up to
complementation} if for every -element subset of , the induced
subgraphs and are isomorphic up to
complementation. A graph is {\it -reconstructible up to complementation}
if every graph which is -hypomorphic to up to complementation is in
fact isomorphic to up to complementation. We give a partial
characterisation of the set of pairs such that two graphs
and on the same set of vertices are equal up to complementation
whenever they are -hypomorphic up to complementation. We prove in particular
that contains all pairs such that . We
also prove that 4 is the least integer such that every graph having a
large number of vertices is -reconstructible up to complementation; this
answers a question raised by P. Ill
A note on the edge-reconstruction of K1,m-free graphs
AbstractWe show that there exists an absolute constant c such that any K1,m-free graph with the maximum degree Δ > cm(log m)12 is edge reconstructible
Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs
In this thesis we investigate three different aspects of graph theory.
Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdos, Ko and Rado to graphs.
Our main results are a proof of an Erdos-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber.
Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically perfect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses.
Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a “dual” version of the reconstruction conjecture
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