On Removable Cycles in Connected Graphs

We call a cycle C of a graph G removable if G - E(C) is connected. In this paper, we obtain sufficient conditions for the existence of a removable cycle in a connected graph G which is edge-disjoint from a connected subgraph of G. Also, a characterization of connected graphs of minimum degree at least 3 having two edge-disjoint removable cycles is obtained in terms of forbidden graphs. We provide sufficient conditions for the existence of removable even cycles, and also for the existence of odd cycles. Further, we handle the problem of determining when a given edge of a connected graph can be guaranteed to lie in some removable cycle.