Spanners in Sparse Graphs

Abstract

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t -spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs the tree t -spanner problem (the problem to decide whether G admits a tree t-spanner) is polynomial time solvable for t ≤ 3 and is NP-complete as long as t is part of the input. They also left as an open problem whether the tree t-spanner problem is polynomial time solvable for every fixed t ≥ 4. In this work we resolve this open problem and extend the solution in several directions. We show that for every fixed t, it is possible in polynomial time not only to decide if a planar graph G has a tree t-spanner, but also to decide if G has a t-spanner of bounded treewidth. Moreover, for every fixed values of t and k, the problem, for a given planar graph G to decide if G has a t-spanner of treewidth at most k, is not only polynomial time solvable, but is fixed parameter tractable (with k and t being the parameters). In particular, the running time of our algorithm is linear with respect to the size of G. We extend this result from planar to a much more general class of sparse graphs containing graphs of bounded genus. An apex graph is a graph obtained from a planar graph G by adding a vertex and making it adjacent to some vertices of G. We show that the problem of finding a t-spanner of treewidth k is fixed parameter tractable on graphs that do not contain some fixed apex graph as a minor, i.e. on apex-minor-free graphs. Graphs of bounded treewidth are sparse graphs and our technique can be used to settle the complexity of the parameterized version of the sparse t -spanner problem, where for given t and m one asks if a given n-vertex graph has a t-spanner with at most n − 1 + m edges. Our results imply that the sparse t-spanner problem is fixed parameter tractable on apex-minor-free graphs with t and m being the parameters. Finally we show that the tractability border of the t-spanner problem cannot be extended beyond the class of apex-minor-free graphs. In particular, we prove that for every t ≥ 4, the problem of finding a tree t-spanner is NP-complete on K ₆ -minor-free graphs. Thus our results are tight, in a sense that the restriction of input graph being apex-minor-free cannot be replaced by H-minor-free for some non-apex fixed graph H.