Cactus graph
In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. 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.
Cacti are outerplanar graphs. Every pseudotree is a cactus. A nontrivial graph is a cactus if and only if every block is either a simple cycle or a single edge.
The family of graphs in which each component is a cactus is downwardly closed under graph minor operations. This graph family may be characterized by a single forbidden minor, the four-vertex diamond graph formed by removing an edge from the complete graph K4.
Some facility location problems which are NP-hard for general graphs, as well as some other graph problems, may be solved in polynomial time for cacti.
Since cacti are special cases of outerplanar graphs, a number of combinatorial optimization problems on graphs may be solved for them in polynomial time.
Cacti represent electrical circuits that have useful properties. An early application of cacti was associated with the representation of op-amps.
...
Wikipedia