Before revision: Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or (for nontrivial cactus ) in which every block (maximal subgraph without a cut-vertex) is an edge or a cycle.
After revision: Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or (for nontrivial cacti ) in which every block (maximal subgraph without a cut-vertex) is an edge or a cycle.