1,568 research outputs found
The random k-matching-free process
Let be a graph property which is preserved by removal of edges,
and consider the random graph process that starts with the empty -vertex
graph and then adds edges one-by-one, each chosen uniformly at random subject
to the constraint that is not violated. These types of random
processes have been the subject of extensive research over the last 20 years,
having striking applications in extremal combinatorics, and leading to the
discovery of important probabilistic tools. In this paper we consider the
-matching-free process, where is the property of not
containing a matching of size . We are able to analyse the behaviour of this
process for a wide range of values of ; in particular we prove that if
or if then this process is likely to
terminate in a -matching-free graph with the maximum possible number of
edges, as characterised by Erd\H{o}s and Gallai. We also show that these bounds
on are essentially best possible, and we make a first step towards
understanding the behaviour of the process in the intermediate regime
Isomorph-free generation of 2-connected graphs with applications
Many interesting graph families contain only 2-connected graphs, which have
ear decompositions. We develop a technique to generate families of unlabeled
2-connected graphs using ear augmentations and apply this technique to two
problems. In the first application, we search for uniquely K_r-saturated graphs
and find the list of uniquely K_4-saturated graphs on at most 12 vertices,
supporting current conjectures for this problem. In the second application, we
verifying the Edge Reconstruction Conjecture for all 2-connected graphs on at
most 12 vertices. This technique can be easily extended to more problems
concerning 2-connected graphs.Comment: 15 pages, 3 figures, 4 table
Reconfiguring Independent Sets in Claw-Free Graphs
We present a polynomial-time algorithm that, given two independent sets in a
claw-free graph , decides whether one can be transformed into the other by a
sequence of elementary steps. Each elementary step is to remove a vertex
from the current independent set and to add a new vertex (not in )
such that the result is again an independent set. We also consider the more
restricted model where and have to be adjacent
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