Parameterized Complexity of the Spanning Tree Congestion Problem - Bodlaender, Hans L.; Fomin, Fedor V.; Golovach, Petr A.; Otachi, Yota; van Leeuwen, Erik Jan (Algorithmica, 2012)
We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k . We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d . In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k ≥8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe that for k ≤3 the problem becomes polynomially time solvable.