Galaxy cutsets in graphs

• Published in: Journal of combinatorial optimization Vol. 19; no. 3; pp. 415 - 427
• Main Authors: Sonnerat, Nicolas, Vetta, Adrian
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Boston Springer US 01.04.2010; Springer
• Subjects: Applied sciences; Combinatorics; Computer science; control theory; systems; Convex and Discrete Geometry; Exact sciences and technology; Information retrieval. Graph; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Memory and file management (including protection and security); Memory organisation. Data processing; Operations Research/Decision Theory; Optimization; Software; Theoretical computing; Theory of Computation; Star-cutsets; Graph connectivity; Complexity; Immunopathology; Tree(graph); Vertex(graph); Graph theory; Modeling; Polynomial time; Computer virus; Motivation; NP hard problem; Safety; Computer attack
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-009-9258-1

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.