37 research outputs found
A note on forbidding clique immersions
Robertson and Seymour proved that the relation of graph immersion is
well-quasi-ordered for finite graphs. Their proof uses the results of graph
minors theory. Surprisingly, there is a very short proof of the corresponding
rough structure theorem for graphs without -immersions; it is based on the
Gomory-Hu theorem. The same proof also works to establish a rough structure
theorem for Eulerian digraphs without -immersions, where
denotes the bidirected complete digraph of order
The structure of graphs not admitting a fixed immersion
We present an easy structure theorem for graphs which do not admit an
immersion of the complete graph. The theorem motivates the definition of a
variation of tree decompositions based on edge cuts instead of vertex cuts
which we call tree-cut decompositions. We give a definition for the width of
tree-cut decompositions, and using this definition along with the structure
theorem for excluded clique immersions, we prove that every graph either has
bounded tree-cut width or admits an immersion of a large wall
The structure of graphs not admitting a fixed immersion
We present an easy structure theorem for graphs which do not admit an immersion of the complete graph. The theorem motivates the definition of a variation of tree decompositions based on edge cuts instead of vertex cuts which we call tree-cut decompositions. We give a definition for the width of tree-cut decompositions, and using this definition along with the structure theorem for excluded clique immersions, we prove that every graph either has bounded tree-cut width or admits an immersion of a large wall
Immersion of complete digraphs in Eulerian digraphs
A digraph \emph{immerses} a digraph if there is an injection and a collection of pairwise edge-disjoint directed paths
, for , such that starts at and ends at .
We prove that every Eulerian digraph with minimum out-degree immerses a
complete digraph on vertices, thus answering a question of DeVos,
Mcdonald, Mohar, and Scheide.Comment: 17 pages; fixed typo
Unavoidable Immersions and Intertwines of Graphs
The topological minor and the minor relations are well-studied binary relations on the class of graphs. A natural weakening of the topological minor relation is an immersion. An immersion of a graph H into a graph G is a map that injects the vertex set of H into the vertex set of G such that edges between vertices of H are represented by pairwise-edge-disjoint paths of G. In this dissertation, we present two results: the first giving a set of unavoidable immersions of large 3-edge-connected graphs and the second on immersion intertwines of infinite graphs. These results, along with the methods used to prove them, are analogues of results on the graph minor relation. A conjecture for the unavoidable immersions of large 3-edge-connected graphs is also stated with a partial proof