99 research outputs found
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
Displaying blocking pairs in signed graphs
A signed graph is a pair (G, S) where G is a graph and S is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of S. A blocking pair of (G, S) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts in an auxiliary graph. We discuss the relevance of this result to the problem of recognizing even cycle matroids and to the problem of characterizing signed graphs with no odd-K5 minor
Finding topological subgraphs is fixed-parameter tractable
We show that for every fixed undirected graph , there is a
time algorithm that tests, given a graph , if contains as a
topological subgraph (that is, a subdivision of is subgraph of ). This
shows that topological subgraph testing is fixed-parameter tractable, resolving
a longstanding open question of Downey and Fellows from 1992. As a corollary,
for every we obtain an time algorithm that tests if there is
an immersion of into a given graph . This answers another open question
raised by Downey and Fellows in 1992
K_6 minors in 6-connected graphs of bounded tree-width
We prove that every sufficiently big 6-connected graph of bounded tree-width
either has a K_6 minor, or has a vertex whose deletion makes the graph planar.
This is a step toward proving that the same conclusion holds for all
sufficiently big 6-connected graphs. Jorgensen conjectured that it holds for
all 6-connected graphs.Comment: 33 pages, 8 figure
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