Defending Planar Graphs against Star-Cutsets

We consider the problem of protecting edges in a graph so that the graph remains connected after the removal of any set of vertices that form a star consisting of unprotected edges. We show that the problem of finding a minimal set of edges to protect (the network protection problem) is NP-complete in general graphs, and present an O ( log n ) -approximation algorithm, where the O ( log n ) factor is essentially best possible. Our major focus, though, is on the special case of planar graphs. We analyse in detail the structure of minimal star-cutsets in planar graphs, and exploit this structure to construct a polynomial time approximation scheme for the network protection problem.