Spanners in Sparse Graphs

Abstract

A t-spannerof a graph Gis a spanningsubgraph Sin which the distance between every pair ofvertices is at most ttimes their distance in G.If Sis required to be a tree then Sis called atree t-spannerof G. In 1998, Feketeand Kremer showed that on unweighted planar graphs the treet-spanner problem(the problem to decide whetherGadmits a tree t-spanner) is polynomial timesolvable for t≤ 3 and is NP-complete as long astis 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 Ghas a treet-spanner, but also to decide if Ghas at-spanner of bounded treewidth. Moreover, forevery fixed values of tand k, the problem, for agiven planar graph Gto decide if Ghas at-spanner of treewidth at most k, is not onlypolynomial time solvable, but is fixed parameter tractable(with kand tbeing 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 graphis a graph obtainedfrom a planar graph Gby adding a vertex and making itadjacent to some vertices of G. We show that the problemof finding a t-spanner of treewidth kis 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 giventand mone asks if a given n-vertexgraph has a t-spanner with at most n- 1 +medges. Our results imply that the sparset-spanner problem is fixed parameter tractable onapex-minor-free graphs with tand mbeing 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.