Independent Tree Spanners

Abstract

For any fixed parameter t ≥ 1, a tree t-spanner of a graph G is a spanning tree T of G such that the distance between every pair of vertices in T is at most t times their distance in G. In this paper, we incorporate a concept of fault-tolerance by examining independent tree t-spanners. Given a root vertex r, this is a pair of tree t-spanners, such that the two paths from any vertex to r are edge (resp., internally vertex) disjoint. It is shown that a pair of independent tree 2-spanners can be found in linear time, whereas the problem for arbitrary t ≥ 4 is $\cal NP$-complete.

As a less restrictive concept, we treat tree t-root-spanners, where the distance constraint is relaxed. Here, we show that the problem of finding an independent pair of such subgraphs is $\cal NP$-complete for all t. As a special case, we then consider direct tree t-root-spanners. These are tree t-root-spanners where paths from any vertex to the root have to be detour-free. In the edge independent case, a pair of these can be found in linear time for all t, whereas the vertex independent case remains $\cal NP$-complete.