21,885 research outputs found
A New Framework for Network Disruption
Traditional network disruption approaches focus on disconnecting or
lengthening paths in the network. We present a new framework for network
disruption that attempts to reroute flow through critical vertices via vertex
deletion, under the assumption that this will render those vertices vulnerable
to future attacks. We define the load on a critical vertex to be the number of
paths in the network that must flow through the vertex. We present
graph-theoretic and computational techniques to maximize this load, firstly by
removing either a single vertex from the network, secondly by removing a subset
of vertices.Comment: Submitted for peer review on September 13, 201
Explicit expanders with cutoff phenomena
The cutoff phenomenon describes a sharp transition in the convergence of an
ergodic finite Markov chain to equilibrium. Of particular interest is
understanding this convergence for the simple random walk on a bounded-degree
expander graph. The first example of a family of bounded-degree graphs where
the random walk exhibits cutoff in total-variation was provided only very
recently, when the authors showed this for a typical random regular graph.
However, no example was known for an explicit (deterministic) family of
expanders with this phenomenon. Here we construct a family of cubic expanders
where the random walk from a worst case initial position exhibits
total-variation cutoff. Variants of this construction give cubic expanders
without cutoff, as well as cubic graphs with cutoff at any prescribed
time-point.Comment: 17 pages, 2 figure
Shortest paths between shortest paths and independent sets
We study problems of reconfiguration of shortest paths in graphs. We prove
that the shortest reconfiguration sequence can be exponential in the size of
the graph and that it is NP-hard to compute the shortest reconfiguration
sequence even when we know that the sequence has polynomial length. Moreover,
we also study reconfiguration of independent sets in three different models and
analyze relationships between these models, observing that shortest path
reconfiguration is a special case of independent set reconfiguration in perfect
graphs, under any of the three models. Finally, we give polynomial results for
restricted classes of graphs (even-hole-free and -free graphs)
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