The complexity of the locally connected spanning tree problem Original Research Article

A locally connected spanning tree T of a graph G is a spanning tree of G with the following property: for every vertex, its neighbourhood in T induces a connected subgraph in G. The existence of such a spanning tree in a network ensures, in case of site and line failures, effective communication amongst operative sites as long as these failures are isolated. We prove that the problem of determining whether a graph contains a locally connected spanning tree is NP-complete, even when input graphs are restricted to planar graphs or split graphs. On the other hand, we obtain a linear-time algorithm for finding a locally connected spanning tree in a directed path graph, and a linear-time algorithm for adding fewest edges to a graph to make a given spanning tree of the graph a locally connected spanning tree of the augmented graph.