Abstract :
Let G be a connected graph with minimum degree at least 3. We prove that there exists an even circuit C in G such that G−E(C) is either connected or contains precisely two components one of which is isomorphic to a 1-bond. We further prove sufficient conditions for there to exist an even circuit C in a 2-connected simple graph G such that G−E(C) is 2-connected. As a consequence of this, we obtain sufficient conditions for there to exist an even circuit C in a 2-connected graph G for which G−E(C) is 2-connected.
