Searching Graphs with Large Clique Number

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 difficulty 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 graphs, we prove that there is a unique chain of inequalities linking all the search numbers above. More precisely, for any graph G, s(G) = is(G) = ms(G) ≤ mis(G) ≤ cs(G) = ics(G) ≤ mcs(G) = mics(G). The first two inequalities can be strict. In the case of trees, we have mics(G) ≤ 2 s(T) − 2, that is there are exactly 2 different search numbers in trees, and these search numbers differ by a factor of 2 at most.

In graph searching, a team of mobile agents aims at clearing the edges of a contaminated graph. To clear an edge, an agent has to slide along it, however, an edge can be recontaminated if there is a path without agents from a contaminated edge to a clear edge. To goal of graph searching is to clear the graph, i.e., all edges are clear simultaneously, using the fewest number of agents. We study this problem in the minimal CORDA model of distributed computation. This model has very weak hypothesis: network nodes and agents are anonymous, have no memory of the past, and agents have no common sense of orientation. Moreover, all agents execute the same algorithm in the Look-Compute-Move manner and in an asynchronous environment. One interest of this model is that, if the clearing can be done by the agents starting from arbitrary positions (e.g., after faults or recontamination), the lack of memory implies that the clearing is done perpetually and then provides a first approach of fault-tolerant graph searching. Constraints due to the minimal CORDA model lead us to define a new variant of graph searching, called graph searching without collisions, where more than one agent cannot occupy the same node. We show that, in a centralized setting, this variant does not have the same behavior as classical graph searching. For instance, it not monotonous nor close by subgraph. We show that, in a graph with maximum degree ∆, the smallest number of agents required to clear a graph without collisions is at most ∆ times the number of searchers required when collisions are allowed. Moreover, this bound is tight up to a constant ratio. Then, we fully characterize graph searching without collisions in trees. In a distributed setting, i.e., in the minimal CORDA model, the question we ask is the following. Given a graph G, does there exist an algorithm that clears G, whatever be the initial positions of the agents on distinct vertices. In the case of a path network, we show that it is not possible is the number of agents is even in a path of odd order, or if there are at most two agents in a path with at least three vertices. We present an algorithm that clears all paths in all remaining cases. Finally, we propose an algorithm that clears any tree using a sufficient number of agents.

In graph searching game the opponents are a set of searchers and a fugitive in a graph. The searchers try to capture the fugitive by applying some sequence moves that include placement, removal, or sliding of a searcher along an edge. The fugitive tries to avoid capture by moving along unguarded paths. The search number of a graph is the minimum number of searchers required to guarantee the capture of the fugitive. In this paper, we initiate the study of this game under the natural restriction of connectivity where we demand that in each step of the search the locations of the graph that are clean (i.e. non-accessible to the fugitive) remain connected. We give evidence that many of the standard mathematical tools used so far in the classic graph searching fail under the connectivity requirement. We also settle the question on "the price of connectivity" that is how many searchers more are required for searching a graph when the connectivity demand is imposed. We make estimations of the price of connectivity on general graphs and we provide tight bounds for the case of trees. In particular for an n-vertex graph the ratio between the connected searching number and the non-connected one is O(log n) while for trees this ratio is always at most 2. We also conjecture that this constant-ratio upper bound for trees holds also for all graphs. Our combinatorial results imply a complete characterization of connected graph searching on trees. It is based on a forbidden-graph characterization of the connected search number. We prove that the connected search game is monotone for trees, i.e. restricting search strategies to only those where the clean territories increase monotonically does not require more searchers. A consequence of our results is that the connected search number can be computed in polynomial time on trees, moreover, we show how to make this algorithm distributed. Finally, we reveal connections of this parameter to other invariants on trees such as the Horton-Stralher number.

International Journal of Game Theory, 2001

Consider a search game with an immobile hider in a graph. A Chinese postman tour is a closed trajectory which visits all the points of the graph and has minimal length. We show that encircling the Chinese postman tour in a random direction is an optimal search strategy if and only if the graph is weakly Eulerian (i.e it consists of several Eulerian curves connected in a treelike structure).

Discrete Applied Mathematics, 2009

In traditional edge searching one tries to clean all of the edges in a graph employing the least number of searchers. It is assumed that each edge of the graph initially has a weight equal to one. In this paper we modify the problem and introduce the Weighted Edge Searching Problem by considering graphs with arbitrary positive integer weights assigned to its edges. We give bounds on the weighted search number in terms of related graph parameters including pathwidth. We characterize the graphs for which two searchers are sufficient to clear all edges. We show that for every weighted graph the minimum number of searchers needed for a not-necessarily-monotonic weighted edge search strategy is enough for a monotonic weighted edge search strategy, where each edge is cleaned only once. This result proves the NP-completeness of the problem.

Theoretical Computer Science, 2008

We consider time constraints for four models of searching graphs for intruders. One model is the standard cops and robber vertex-searching model with complete visibility. The second model differs from the preceding one only in that none of the searchers can see the intruder. The third model is a vertex-searching model in which searchers and an intruder move simultaneously and none of the searchers can see the intruder. The fourth model is simultaneous edge searching with an arbitrarily fast intruder.

Information Processing Letters, 1991

A variation of the graph search problem is presented. The assumption is that each edge can be searched by any searcher in a single step and the fugitive can move with unbounded speed. The problem is to search a graph in a single step such that no fugitive can hide between the searchers. A sufficient and necessary condition is presented for a graph G = (V,E) to be single step searchable with ⫫;E⫫ searchers. If ⫫E⫫ searchers are not enough, it can be shown that to determine the minimum number of extra searchers needed is intractable. An O(⫫E⫫) algorithm to single step search an interval graph of ⫫E⫫ edges with the minimum number of extra searchers is also presented.

Discrete Applied Mathematics, 2013

In the edge searching problem, searchers move from vertex to vertex in a graph to capture an invisible, fast intruder that may occupy either vertices or edges. Fast searching is a monotonic internal model in which, at every move, a new edge of the graph G must be guaranteed to be free of the intruder. That is, once all searchers are placed the graph G is cleared in exactly |E(G)| moves. Such a restriction obviously necessitates a larger number of searchers. We examine this model, and characterize graphs for which 2 or 3 searchers are sufficient. We prove that the corresponding decision problem is NP-complete.

2000

This papers surveys some of the work done on trying to capture an intruder in a graph. If the intruder may be located only at vertices, the term searching is employed. If the intruder may be located at vertices or along edges, the term sweeping is employed. There are a wide variety of applications for searching and sweeping. Old results,

SODA, 1998

We define a new type of search problem called "mutual search", where k players arbitrarily spread over n nodes are required to locate each other by sending "Anybody at node i?" query messages (for example processes in a computer network). If the messages are not delivered in the order they were sent (for example when the communication delay time is arbitrary) then two players require at least n-1 messages. In an asynchronous network, where the messages are delivered in the order they were sent, 0.88n messages suffice. In a synchronous network 0.586n messages suffice and 0.536n messages are required in the worst case. We exhibit a simple randomized algorithm with expected worst-case cost of 0.5n messages, and a deterministic algorithm for k 2' .: 2 players with a cost well below n for all k = o(vfn). The graph-theoretic framework we formulate for expressing and analyzing algorithms for this problem may be of independent interest.