4,537 research outputs found
Detecting wheels
A \emph{wheel} is a graph made of a cycle of length at least~4 together with
a vertex that has at least three neighbors in the cycle. We prove that the
problem whose instance is a graph and whose question is "does contains
a wheel as an induced subgraph" is NP-complete. We also settle the complexity
of several similar problems
Random walks on dynamic configuration models: a trichotomy
We consider a dynamic random graph on vertices that is obtained by
starting from a random graph generated according to the configuration model
with a prescribed degree sequence and at each unit of time randomly rewiring a
fraction of the edges. We are interested in the mixing time of a
random walk without backtracking on this dynamic random graph in the limit as
, when is chosen such that . In [1] we found that, under mild regularity
conditions on the degree sequence, the mixing time is of order
when . In the present paper we investigate
what happens when . It turns out that the mixing time is
of order , with the scaled mixing time exhibiting a one-sided cutoff
when and a two-sided cutoff when . The
occurrence of a one-sided cutoff is a rare phenomenon. In our setting it comes
from a competition between the time scales of mixing on the static graph, as
identified by Ben-Hamou and Salez [4], and the regeneration time of first
stepping across a rewired edge.Comment: 14 pages, 5 figure
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