k-cacti exclude large complete minors and thus have edge density O((log k / sqrt(log log k)) n), tight up to a sqrt(log log k) factor.
Mac Lane,A structural characterization of planar combinatorial graphs, Duke Math
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The density of $k$-cacti via excluding minors
k-cacti exclude large complete minors and thus have edge density O((log k / sqrt(log log k)) n), tight up to a sqrt(log log k) factor.