20,285 research outputs found
A cactus theorem for end cuts
Dinits-Karzanov-Lomonosov showed that it is possible to encode all minimal
edge cuts of a graph by a tree-like structure called a cactus. We show here
that minimal edge cuts separating ends of the graph rather than vertices can be
`encoded' also by a cactus. We apply our methods to finite graphs as well and
we show that several types of cuts can be encoded by cacti.Comment: 19 page
Cutting up graphs revisited - a short proof of Stallings' structure theorem
This is a new and short proof of the main theorem of classical structure tree
theory. Namely, we show the existence of certain automorphism-invariant
tree-decompositions of graphs based on the principle of removing finitely many
edges. This was first done in "Cutting up graphs" by M.J. Dunwoody. The main
ideas are based on the paper "Vertex cuts" by M.J. Dunwoody and the author. We
extend the theorem to a detailed combinatorial proof of J.R. Stallings' theorem
on the structure of finitely generated groups with more than one end.Comment: 12 page
Extremal Infinite Graph Theory
We survey various aspects of infinite extremal graph theory and prove several
new results. The lead role play the parameters connectivity and degree. This
includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
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