Degree-Bounded Spanners

Given a graph G, a spanning subgraph H of G is a t-spanner if for every edge xy of G, the distance in H between x and y is at most t. Spanners have applications in communication networks, distributed systems, parallel computation, and other areas. This paper is concerned with the complexity of determining whether a graph contains a t-spanner whose maximum degree is at most Δ, where Δ, t≥2 are fixed integers. The problem is shown to be NP-complete for any fixed Δ≥4 and t≥2, and linear for Δ=2 and any fixed t≥2. The problem for Δ=3 is open. Furthermore, if the t-spanner in the problem is required to be a tree, then the problem becomes linear. The problem also becomes linear when input graphs are partial k-trees for some fixed integer k.