Searching the Nodes of a Graph: Theory and Algorithms

One or more searchers must capture an invisible evader hiding in the nodes of a graph. We study this version of the graph search problem under additional restrictions, such as monotonicity and connectedness. We emphasize that we study node search, i.e., the capture of a node-located evader; this problem has so far received much less attention than edge search, i.e., the capture of an edge-located evader. We show that in general graphs the problem of node search is easier than that of edge search, Namely, every edge clearing search is also node clearing, but the converse does not hold in general (however node search is NP-complete, just like edge search). Then we concentrate on the internal monotone connected (IMC) node search of trees and show that it is essentially equivalent to IMC edge search; hence Barriere’s tree search algorithm [2], originally designed for edge search, can also be used for node search. We return to IMC node search on general graphs and present (several variants of) a new algorithm: GSST (Guaranteed Search by Spanning Tree). GSST clears a graph G by performing all its clearing moves along a spanning tree T of G. Because spanning trees can be generated and cleared very quickly, GSST can test a large number of spanning trees and find one which clears G with a small (though not necessarily minimal) number of searchers. We prove the existence of probabilistically complete variants of GSST (i.e., these variants are guaranteed to find a minimal IMC node clearing schedule if run for sufficiently long time). Our experiments also indicate that GSST can efficiently node-clear large graphs given only a small running time. An implementation of GSST (running on Windows and Linux computers) is also provided and made publicly available.