 Lovász and Plummer consider pairs of partitions of the vertices and edges of the given graph into subsets, with the property that every triangle of the graph either has two vertices in a single class of the vertex partition or all three edges in a single class of the edge partition; they call a pair of partitions with this property "valid".Then the number of triangles in the largest triangular cactus equals the maximum, over pairs of valid partitions formula_1 and formula_2, of
where formula_4 is the number of vertices in the given graph and formula_5 is the number of vertex classes met by edge class formula_6.Recently, a tight extremal bound was proven which showed that given any plane graph formula_7, there always exist a cactus subgraph formula_8 containing at least formula_9 fraction of the triangular faces of formula_7.