An Efficient Algorithm to Compute the Toughness in Graphs with Bounded Treewidth

Let t be a positive real number. A graph is called t - tough if the removal of any vertex set S that disconnects the graph leaves at most | S |/ t components. The toughness of a graph is the largest t for which the graph is t -tough. We prove that toughness is fixed-parameter tractable parameterized with the treewidth. More precisely, we give an algorithm to compute the toughness of a graph G with running time O(|V(G)|³· tw(G)2tw(G)) O ( | V ( G ) | 3 · tw ( G ) 2 tw ( G ) ) where tw(G) tw ( G ) is the treewidth. If the treewidth is bounded by a constant, then this is a polynomial algorithm.