Network Decontamination with a Single Agent 1 Yessine Daadaa 1 , Asif Jamshed 2 , and Mudassir Shabbir 2 2 1 School of Electrical Engineering and Computer Science, 3 Faculty of Engineering, 4 University of Ottawa, 5 Ottawa, Canada 6 2 Department of Computer Science, Rutgers University, NJ, USA 7 ydaadaa@site.uottawa.ca, 8 {ajamshed,mudassir}@cs.rutgers.edu 9 Abstract. Faults and viruses often spread in networked environments by propa- 10 gating from site to neighboring site. We model this process of network contamina- 11 tion by graphs. Consider a graph G =(V,E), whose vertex set is contaminated 12 and our goal is to decontaminate the set V (G) using mobile decontamination 13 agents that traverse along the edge set of G. Temporal immunity τ (G) ≥ 0 is 14 defined as the time that a decontaminated vertex of G can remain continuously 15 exposed to some contaminated neighbor without getting infected itself. The im- 16 munity number of G, ι k (G), is the least τ that is required to decontaminate G 17 using k agents. We study immunity number for some classes of graphs corre- 18 sponding to network topologies and present upper bounds on ι1(G), in some 19 cases with matching lower bounds. Variations of this problem have been exten- 20 sively studied in literature, but proposed algorithms have been restricted to mono- 21 tone strategies, where a vertex, once decontaminated, may not be recontaminated. 22 We exploit nonmonotonicity to give bounds which are strictly better than those 23 derived using monotone strategies. 24 1 Introduction 25 Faults and viruses often spread in networked environments by propagating from site 26 to neighboring site. The process is called network contamination. Once contaminated, 27 a network node might behave incorrectly, and it could cause its neighboring node to 28 become contaminated as well, thus propagating faulty computations. The propagation 29 patterns of faults can follow different dynamics, depending on the behavior of the af- 30 fected node, and topology of the network. At one extreme we have a full spread be- 31 havior: when a site is affected by a virus or any other malfunction, such a malfunction 32 can propagate to all its neighbors; other times, faults propagate only to sites that are 33 susceptible to be affected; the definition of susceptibility depends on the application but 34 oftentimes it is based on local conditions, for example, a node could be vulnerable to 35 contamination if a majority of its neighbors are faulty, and immune otherwise (e.g., see 36 [14], [15], [18]); or it could be immune to contamination for a certain amount of time 37 after being repaired (e.g., see [8], [12]). 38 In this paper we consider a propagation of faults based on what we call temporal 39 immunity: a clean node is allowed to be exposed to contaminated nodes for a predefined 40 arXiv:1307.7307v1 [math.CO] 27 Jul 2013