A lower bound on the tree-width of graphs with irrelevant vertices

For their famous algorithm for the disjoint paths problem, Robertson and Seymour proved that there is a function f such that if the tree-width of a graph G with k pairs of terminals is at least f (k), then G contains a solution-irrelevant vertex (Graph Minors. XXII., JCTB 2012). We give a single-exponential lower bound on f . This bound even holds for planar graphs.