Galaxy cutsets in graphs
Given a network G =( V , E ), we say that a subset of vertices S ⊆ V has radius r if it is spanned by a tree of depth at most r . We are interested in determining whether G has a cutset that can be written as the union of k sets of radius r . This generalizes the notion of k -vertex connectivity,...
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Published in | Journal of combinatorial optimization Vol. 19; no. 3; pp. 415 - 427 |
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Main Authors | , |
Format | Journal Article Conference Proceeding |
Language | English |
Published |
Boston
Springer US
01.04.2010
Springer |
Subjects | |
Online Access | Get full text |
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Summary: | Given a network
G
=(
V
,
E
), we say that a subset of vertices
S
⊆
V
has radius
r
if it is spanned by a tree of depth at most
r
. We are interested in determining whether
G
has a cutset that can be written as the union of
k
sets of radius
r
. This generalizes the notion of
k
-vertex connectivity, since in the special case
r
=0, a set spanned by a tree of depth at most
r
is a single vertex.
Our motivation for considering this problem is that it constitutes a simple model for virus-like malicious attacks on
G
: An attack occurs at a subset of
k
vertices and begins to spread through the network. Any vertex within distance
r
of one of the initially attacked vertices may become infected. Thus an attack corresponds to a subset of vertices that is spanned by
k
trees of depth at most
r
. The question we focus on is whether a given network has a cutset of this particular form.
The main results of this paper are the following. If
r
=1, an attack corresponds to a subset of vertices which is the union of at most
k
stars. We call such a set a
galaxy
of order
k
. We show that it is NP-hard to determine whether a given network contains a cutset which is a galaxy of order
k
, if
k
is part of the input. This is in stark contrast to the case
r
=0, since testing whether a graph is
k
-vertex connected can be done in polynomial time, using standard maxflow-mincut type results.
On the positive side, testing whether a graph can be disconnected by a single attack (i.e.
k
=1) can be done efficiently for any
r
. Such an attack corresponds to a single set of vertices spanned by a tree of depth at most
r
. We present an
O
(
rnm
) algorithm that determines if a given network contains such a set as a cutset. |
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ISSN: | 1382-6905 1573-2886 |
DOI: | 10.1007/s10878-009-9258-1 |