In this paper, we present a distributed algorithm to compute the node search number in trees. This algorithm extends the centralized algorithm proposed by Ellis et al. [9]. It can be executed in an asynchronous environment, requires an overall computation time of O(nlogn), and n messages of log3 n + 1 bits each. The main contribution of this work is in the data structure chosen to design our algorithm, the hierarchical decomposition. It is very simple and its flexibility allows us to: propose distributed algorithms for updating the node search number after addition or deletion of any tree-edges; propose a distributed incremental algorithm to compute this parameter of a tree for which edges are added sequentially and in any order; compute other graph invariants such as the process number and the edge search number, by changing only initialization rules; extend our algorithms for trees and forests of unknown size (using messages of less than 2log3 n + 3 bits); compute the node search number in unicyclic graphs.

Abstract. The graph search problem asks for a strategy that enables a minimum sized team of searchers to capture a “fugitive ” while it evades and potentially multiplies through a network. It is motivated by the need to eliminate fast spreading viruses and other malicious software agents in computer networks. The current work improves on previous results with a self-stabilizing algorithm that clears an n node tree network using only 1+log n searchers and O(n log n) moves after initialization. Since Θ(log n) searchers are required to clear some tree networks even in the sequential case, this is the best that any self-stabilizing algorithm can do. The algorithm is based on a novel multi-layer traversal of the network. 1

This paper tackles the well known graph searching problem, where a team of searchers aims at capturing an intruder in a network, modeled as a graph. All variants of this problem assume that any node can be simultaneously occupied by several searchers. This assumption may be unrealistic, e.g., in the case of searchers modeling physical searchers, or may require each individual node to provide additional resources, e.g., in the case of searchers modeling software agents. We thus investigate exclusive graph searching, in which no two or more searchers can occupy the same node at the same time, and, as for the classical variants of graph searching, we study the minimum number of searchers required to capture the intruder. This number is called the exclusive search number of the considered graph. Exclusive graph searching appears to be considerably more complex than classical graph searching, for at least two reasons: (1) it does not satisfy the monotonicity property, and (2) it is not closed under minor. Nevertheless, we design a polynomial-time algorithm which, given any tree T, computes the exclusive search number of T. Moreover, for any integer k, we provide a characterization of the trees T with exclusive search number at most k. This characterization allows us to describe a special type of exclusive search strategies, that can be executed in a distributed environment, i.e., in a framework in which the searchers are limited to cooperate in a distributed manner.