A cactus graph, sometimes also called a cactus tree, a mixed Husimi tree, or a polygonal cactus with bridges, is a connected graph in which
any two graph cycles have no edge in common. Equivalently,
it is a connected graph in which any two (simple)
cycles have at most one vertex in common. Cactus graphs may also be defined as a
connectedplanar graph
in which every block is a cycle or an edge (White 2001,
p. 57).
Every cycle of a cactus graph is a chordless. However, there exist graphs (e.g., the -graph and Pasch graph)
whose cycles are all chordless but which are not cactus graphs.
The inequality
where
is the circuit rank and is the total number of undirected graph
cycles holds for a connected graph iff it is a cactus graph (Volkmann 1996).
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-graph and Pasch graph)
whose cycles are all chordless but which are not cactus graphs.
is the circuit rank and 
is the total number of undirected graph
cycles holds for a connected graph 
iff it is a cactus graph (Volkmann 1996).
-ary Cacti." Adv. Appl. Math. 24,
22-56, 2000.Chao, C. Y. and Whitehead, E. G. Jr. "On
Chromatic Equivalence of Graphs." In