Removable edges and chords of longest cycles in 3-connected graphs

We verify two special cases of Thomassen’s conjecture of 1976 stating that every longest cycle in a 3-connected graph contains a chord. We prove that Thomassen’s conjecture is true for two classes of 3-connected graphs that have a bounded number of removable edges on or off a longest cycle. Here an edge e of a 3-connected graph G is said to be removable if G-e is still 3-connected or a subdivision of a 3-connected (multi)graph. We give examples to show that these classes are not covered by previous results.