2 research outputs found
Gaps in full homomorphism order
We characterise gaps in the full homomorphism order of graphs.Comment: 9 pages, extended abstract for Eurocomb 201
Phylogenetic trees and homomorphisms
In Chapter 1 we fully characterise pairs of finite graphs which form a gap in
the full homomorphism order. This leads to a simple proof of the existence of
generalised duality pairs. We also discuss how such results can be carried to
relational structures with unary and binary relations. In Chapter 2 we show a
very simple and versatile argument based on divisibility which immediately
yields the universality of the homomorphism order of directed graphs and
discuss three applications. In chapter 3, we show that every interval in the
homomorphism order of finite undirected graphs is either universal or a gap.
Together with density and universality this "fractal" property contributes to
the spectacular properties of the homomorphism order. In Chapter 4 we analyze
the phylogenetic information content from a combinatorial point of view by
considering the binary relation on the set of taxa defined by the existence of
a single event separating two taxa. We show that the graph-representation of
this relation must be a tree. Moreover, we characterize completely the
relationship between the tree of such relations and the underlying phylogenetic
tree.Comment: dissertation, School of Mathematical Sciences, Shanghai Jiaotong
University, Sep 201