Consider a team of mobile software agents deployed to capture a (possibly hostile) intruder in a network. All agents, including the intruder move along the network links; the intruder could be arbitrarily fast, and aware of the positions of all the agents. The problem is to design the agents' strategy for capturing the intruder. The main eciency parameter is the size of the team. This is an instance of the well known graph-searching problem whose many variants have been extensively studied in the literature. In all existing solutions, and in all the variants of the problem, it is assumed that agents can be removed from their current location and placed in another network site arbitrarily and at any time. As a consequence, the existing optimal strategies cannot be employed in situations for which agents cannot access the network at any point, or cannot "jump" across the network, or cannot reach an arbitrary point of the network via an internal travel through insecure zones. This motivates the contiguous search problem in which agents cannot be removed from the network, and clear links must form a connected sub-network at any time, providing safety of movements. This new problem is NP-complete in general. We study it for tree networks, and we consider its more general version, the weighted case, which arises naturally when considering networks whose nodes and links are of different nature and thus require a different number of agents to be explored. We give a linear-time algorithm that computes, for any tree T , the minimum number of agents to capture the intruder, and the corresponding search strategy. Beside its optimality in time, our algorithm is naturally distributed: if T is a processor-network...

This paper is concerned with the graph searching game. The search number s(G) of a graph G is the smallest number of searchers required to "clear" G. A search strategy is monotone (m) if no recontamination ever occurs. It is connected (c) if the set of clear edges always forms a connected subgraph. It is internal (i) if the removal of searchers is not allowed. The di#culty of the "connected" version and of the "monotone internal" version of the graph searching problem comes from the fact that, as shown in the paper, none of these problems is minor closed for arbitrary graphs, as opposed to all known variants of the graph searching problem. Motivated by the fact that connected graph searching, and monotone internal graph searching are both minor closed in trees, we provide a complete characterization of the set of trees that can be cleared by a given number of searchers. In fact, we show that, in trees, there is only one obstruction for monotone internal search, as well as for connected search, and this obstruction is the same for the two problems. This allows us to prove that, for any tree T , mis(T ) = cs(T ). For arbitrary

. We present O(n 3 ) embedding algorithms (generalizing subgraph isomorphism) for classes of graphs of bounded pathwidth, where n is the number of vertices in the graph. These include the rst polynomialtime algorithm for minor containment and the rst O(n c ) algorithm (c a constant independent of k) for topological embedding of graphs from subclasses of partial k-trees. Of independent interest are structural properties of k-connected graphs of bounded pathwidth on which our algorithms are based. We also describe special cases which reduce to various generalizations of string matching, permitting more ecient solutions. 1 Introduction Many fundamental problems in a diverse set of research areas can be characterized as graph embedding problems, where data is represented as graphs and patterns can be detected by nding smaller graphs in larger ones. Classic patternmatching problems make use of the subgraph isomorphism problem, namely, the problem of determining whether ther...

Abstract. Consider a tree network that has been contaminated by a persistent and active virus: when infected, a network site will continuously attempt to spread the virus to all its neighbours. The decontamination problem is that of disinfecting the entire network using a team of mobile antiviral system agents, called cleaners, avoiding any recontamination of decontaminated areas. A cleaner is able to decontaminate any infected node it visits; once the cleaner departs, the decontaminated node is immune for t ≥ 0 time units to viral attacks from infected neighbours. After the immunity time t is elapsed, re-contamination can occur. The primary research objective is to determine the minimum team size, as well as the solution strategy, that is the protocol that would enable such a minimal team of cleaners to perform the task. The network decontamination problem has been extensively investigated in the literature, and a very large number of studies exist on the subject. However, all the existing work is limited to the special case t = 0. In this paper we examine the tree decontamination problem for any value t ≥ 0. We determine the minimum team size necessary to disinfect any

We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2−ǫ) n m O(1) time, we show that for any ǫ> 0; • INDEPENDENT SET cannot be solved in (2 − ǫ) tw(G) |V (G) | O(1) time, • DOMINATING SET cannot be solved in (3 − ǫ) tw(G) |V (G) | O(1) time, • MAX CUT cannot be solved in (2 − ǫ) tw(G) |V (G) | O(1) time, • ODD CYCLE TRANSVERSAL cannot be solved in (3 − ǫ) tw(G) |V (G) | O(1) time, • For any q ≥ 3, q-COLORING cannot be solved in (q − ǫ) tw(G) |V (G) | O(1) time, • PARTITION INTO TRIANGLES cannot be solved in (2 − ǫ) tw(G) |V (G) | O(1) time. Our lower bounds match the running times for the best known algorithms for the problems, up to the ǫ in the base.

Abstract. In this paper, we propose a new search model, called strongmixed search, which is a generalization of the mixed search. We show that the strong-mixed search number of a graph equals the pathwidth of the graph. We also describe relationships between the strong-mixed search number and other search numbers.

In this paper we examine the edge searching problem on pseudo 3-sided solid orthoconvex grids. We define a similar problem which we call modified edge searching, and we derive a relation among the original edge searching problem and the modified one. Then, for the modified edge searching problem, we show that there are searching strategies that possess several properties regarding the way the grid is searched. These strategies allow us to obtain a closed formula that expresses the minimum number of searchers required to search a pseudo 3-sided solid orthoconvex grid. From that formula and a rather straight forward algorithm we can show that the problem is in P. We obtain a parallel version of that algorithm that places the problem in NC. For the case of sequential algorithms, we derive an optimal algorithm that solves the problem in O(m) time where m is the number of points necessary to describe the orthoconvex grid. Another important feature of our method is that it also suggests an ...