We give a new proof of the result, due to A. LaPaugh, that a graph may be optimally "searched" without clearing any edge twice. o 19~~ Academic PWS, IIIC. Let us regard a graph as a system of tunnels containing a (lucky, invisible, fast) fugitive. We desire to capture this fugitive by "searching" all edges of the graph, in a sequence of discrete steps, while using the fewest possible "guards." This problem was introduced by Breisch [2] and Parsons [6] . In the version of graph searching considered in [51 (which we call edge-searching, using terminology from [3]) a search step consists of placing a guard at a vertex, or removing a guard from a vertex, or sliding a guard along an edge. Further, an edge (u, v} is cleared by sliding a guard from u to u, while shielding u from contaminated (that is, uncleared) edges with appropriately placed guards (for example, by keeping another guard at u). If, at any point in time, there is a path from a contaminated edge e to a cleared edge e' that is not blocked by guards, e' becomes instantaneously recontaminated and must be cleared again. Our objective is to reach a state in which all edges are simultaneously cleared, so that the maximum number of guards used at any step is minimized. Any strategy that achieves this result is called optimal, and the optimal number of guards is the edge-search number of the graph. LaPaugh [5] proved that there always exists an optimal strategy that is monotone (without recontamination). One implication of this important result is that there is an optimal strategy that terminates after a linear number of steps.
For any graph G, there is an optimal winning monotone (mixed-search) strategy REF .
Monotonicity in Graph Searching
Mathematics Computer Science