The search and the node-search number of dual graphs (Q2782692)

Consider a connected graph \(G\) and a ``running'' vertex in it. Using some \(k\) searchers we want to catch the running vertex. In the first set of rules we are allowed to (1) put a searcher at a vertex of \(G\), (2) remove a searcher from a vertex of \(G\); and (3) move a searcher from a vertex to an adjacent vertex.NEWLINENEWLINENEWLINEThe minimum \(k\) for which we can guarantee to catch the running vertex is called the search number \(\text{s}(G)\) of \(G\). If only the operations (1) and (2) are allowed, we speak of the node search number \(\text{ns}(G)\) of \(G\). The authors conjecture that \(\text{s}(G)= \text{ns}(G^*)\) for a planar 2-connected graph \(G\), where \(G^*\) is the dual to \(G\). Some support of the conjecture is given.