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).
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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