The Evolution of a Hard Graph Theory Problem – Secure Sets
Ron Dutton
Computer Science
University of Central Florida
A Secure Set of a graph is a set S of vertices with the property that for
every subset X of S, N[X]
∩
S contains as many vertices as there are
neighbors of X not in S. This is a special kind of Defensive Alliance,
which only requires the property to hold for all singleton subsets of S.
The question of whether a graph G has a Defensive Alliance of no more
than k vertices, for arbitrary G and k, is known to be NPComplete. The
similar question for Secure Sets seems to be much harder than any NP
Complete problem. In fact, if P ≠ NP, it is probably not even in the set
NP. On the other hand, if P = NP, it can be solved by polynomial
algorithms.
How this all comes about and at least part of what it means will be
the topic of this presentation.
1. Introduction
Many real world problems involve a collection of entities, for example, individuals,
businesses, or countries, that compete for resources. A group of these entities can reach
mutual agreements, or pacts, to simultaneously attack, or defend against, a common
adversary. Informally, a group is secure when they can successfully defend themselves
against any possible takeover or attack. Under certain conditions, this type of problem
can be modeled as a graph with a group reaching an accord corresponding to a Secure Set
in the graph. To date the complexity of determining when a graph has a secure set of
some given size or, even for a given a set, determining whether it is or is not secure is
unknown.
Most of the literature on secure sets attempts to bound the size of a minimum secure
set either for special classes of graphs or in terms of other graphical invariants. Section 3
shows formally that determining if a give set of vertices is a secure set is a difficult
problem, which indicates the former problem may be even more difficult. Section 4
presents results that may prove helpful in developing algorithms and heuristics for these
and related problems. At the least, these results provide insight into the structure and
properties of secure sets.
1
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2. Background
Basic graph theory terminology can be found in [ ], while that involving complexity
theory can be found in Garey and Johnson [ ]. Throughout, assume G = (V, E) is a
connected graph. For a set S
⊆
V, a vertex in N[S]–S
⊆
V–S can "attack" any one of it's
neighbors in S, while a vertex in S can "defend" itself or any one of its neighbors in S. An
attack
on S is an assignment for each vertex in N[S]–S to attack exactly one of its
neighbors in S. A
defense
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 Fall '10
 Dutton
 Computer Science, Graph Theory, Computational complexity theory, secure sets, secure set, Defensive Alliance

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