Spanners in Sparse Graphs

• Authors: Feodor F. Dragan;Fedor V. Fomin;Petr A. Golovach
• Venue: ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
• Year: 2008

A t-spanner of a graph G is a spanningsubgraph S in which the distance between every pair ofvertices is at most t times their distance in G.If S is required to be a tree then S is called atree t -spanner of G. In 1998, Feketeand Kremer showed that on unweighted planar graphs the treet -spanner problem (the problem to decide whetherG admits a tree t-spanner) is polynomial timesolvable for t ≤ 3 and is NP-complete as long ast is part of the input. They also left as an open problemwhether the tree t-spanner problem is polynomial timesolvable for every fixed t ≥ 4. In this work we resolvethis open problem and extend the solution in several directions. Weshow that for every fixed t, it is possible in polynomialtime not only to decide if a planar graph G has a treet-spanner, but also to decide if G has at-spanner of bounded treewidth. Moreover, forevery fixed values of t and k, the problem, for agiven planar graph G to decide if G has at-spanner of treewidth at most k, is not onlypolynomial time solvable, but is fixed parameter tractable(with k and t being the parameters). Inparticular, the running time of our algorithm is linear withrespect to the size of G. We extend this result fromplanar to a much more general class of sparse graphs containinggraphs of bounded genus. An apex graph is a graph obtainedfrom a planar graph G by adding a vertex and making itadjacent to some vertices of G. We show that the problemof finding a t-spanner of treewidth k is fixedparameter tractable on graphs that do not contain some fixed apexgraph as a minor, i.e. on apex-minor-free graphs. Graphsof bounded treewidth are sparse graphs and our technique can beused to settle the complexity of the parameterized version of thesparse t -spanner problem, where for givent and m one asks if a given n-vertexgraph has a t-spanner with at most n - 1 +m edges. Our results imply that the sparset-spanner problem is fixed parameter tractable onapex-minor-free graphs with t and m being theparameters. Finally we show that the tractability border of thet-spanner problem cannot be extended beyond the class ofapex-minor-free graphs. In particular, we prove that for everyt ≥ 4, the problem of finding a tree t-spanneris NP-complete on K6 -minor-freegraphs. Thus our results are tight, in a sense that therestriction of input graph being apex-minor-free cannot be replacedby H-minor-free for some non-apex fixed graphH.