Distributed graph searching with a sense of direction

In this work we consider the edge searching problem for vertex-weighted graphs with arbitrarily fast and invisible fugitive. The weight function ω provides for each vertex v the minimum number of searchers required to guard v , i.e., the fugitive may not pass through v without being detected only if at least ω(v) searchers are present at v . This problem is a generalization of the classical edge searching problem, in which one has ω≡ 1 . We assume that with a graph G to be searched, there is associated a partition (V_1,… ,V_t) of its vertex set such that edges are allowed only within each V_i and between two consecutive V_i ’s. We provide an algorithm for distributed monotone connected edge searching of such graphs, where the searchers are initially placed on an arbitrary vertex of G and have no a priori knowledge on G , but they have a sense of direction that lets them recognize whether an edge incident to already explored vertex in V_i leads to a vertex in one of V_i-1, V_i or V_i+1 . Starting from any vertex the algorithm uses at most 3·max _i=1,… ,tω(V_i)+1 searchers, where ω(V_i) = ∑ _v∈ V_iω(v) . We also prove that this algorithm is best possible up to a small additive constant, that is, each distributed searching algorithm in worst case must use 3·max _i=1,… ,tω(V_i)-1 searchers for some graphs.