On the number of contractible triples in 3-connected graphs

McCuaig and Ota proved that every 3-connected graph G on at least 9 vertices admits a contractible triple, i.e. a connected subgraph H on three vertices such that G − V ( H ) is 2-connected. Here we show that every 3-connected graph G on at least 9 vertices has more than | V ( G ) | / 10 many contractible triples. If, moreover, G is cubic, then there are at least | V ( G ) | / 3 many contractible triples, which is best possible.