On non-essential edges in 3-connected graphs (Q1586584)

An edge \(e\) in a simple 3-connected graph is deletable (contractible), if the deletion \(G-e\) (contraction \(G/e\)) is still simple and 3-connected. Let \(a,b\), and \(c\) be three non-negative integers. If there exists a simple 3-connected graph with exactly \(a\) edges which are deletable but not contractible, exactly \(b\) edges which are contractible but not deletable, and exactly \(c\) edges which are both deletable and contractible, then the triple \((a,b,c)\) is called realizable, and such a graph is said to be an \((a,b,c)\)-graph. Tutte's classical wheels theorem [Nederl. Akad. Wet., Proc., Ser. A 64, 441-455 (1961; Zbl 0101.40903)] says that the only (0, 0, 0)-graphs are the wheels. Using Tutte's wheels theorem, the authors characterize the \((a,b,c)\) realizable triples for which at least one of \(a+b\leq 2\), \(c=0\), and \(c\geq 16\) holds.