Removable Cycles in Planar Graphs
Herbert Fleischner, Bill Jackson · Journal of the London Mathematical Society · 1985
All graphs considered are finite and loopless, but may contain multiple edges. By a simple graph we shall mean a graph without multiple edges. It follows easily from a result of Mader [4, Theorem 1] that if G is a ^-connected simple graph of minimum degree at least k+2, then G contains a cycle C such that G-E(C) is ^-connected. Stronger results exist for the special case of 2-connected simple graphs. THEOREM 1 [3]. Let G be a 2-connected simple graph of minimum degree 6^4. Then G contains a cycle C, of length at least d—\\, such that G — E(C) is 2-connected. THEOREM 2 [5]. Let G be a 2-connected simple graph of minimum degree at least four. Then G contains a cycle C such that G—V(C) is connected and G — E(C) is 2-connected. Analogous results do not hold for graphs containing multiple edges, as can be seen from the graph in [3, Figure 1]. The purpose of this note is to show that a related result remains valid, however, for planar graphs. THEOREM 3. Let G be a planar, 2-connected graph of minimum degree at least four. Then G contains a cycle C such that G—E(C) is 2-connected.