12,204 research outputs found
Editing to a Graph of Given Degrees
We consider the Editing to a Graph of Given Degrees problem that asks for a
graph G, non-negative integers d,k and a function \delta:V(G)->{1,...,d},
whether it is possible to obtain a graph G' from G such that the degree of v is
\delta(v) for any vertex v by at most k vertex or edge deletions or edge
additions. We construct an FPT-algorithm for Editing to a Graph of Given
Degrees parameterized by d+k. We complement this result by showing that the
problem has no polynomial kernel unless NP\subseteq coNP/poly
Amenability of Groupoids Arising from Partial Semigroup Actions and Topological Higher Rank Graphs
We consider the amenability of groupoids equipped with a group valued
cocycle with amenable kernel . We prove a general result
which implies, in particular, that is amenable whenever is amenable and
if there is countable set such that for all . We show that our result is applicable to groupoids arising from
partial semigroup actions. We explore these actions in detail and show that
these groupoids include those arising from directed graphs, higher rank graphs
and even topological higher rank graphs. We believe our methods yield a nice
alternative groupoid approach to these important constructions.Comment: Revised as suggested by a very helpful referee. In particular, a gap
in the proof of Theorem 5.13 has been repaired resulting in a much improved
version (with fewer hypotheses
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