Fig. 16

Abundance of single-leaf splits for pairs of BMGs \((\vec {G}_{\text {orig}},\sigma )\) and disturbed digraphs \((\vec {G},\sigma )\) (both with vertex set V). The partition \({\mathcal {V}}_{\text {orig}}\) corresponds to the connected components of the Aho graph \(H_\text {orig}{:}{=}[{\mathcal {R}}(\vec {G}_{\text {orig}},\sigma ), V]\) and, hence, to the partition induced by the subtrees of the children of the root of the LRT \((T,\sigma )\) of \((\vec {G}_{\text {orig}},\sigma )\) (cf. Prop. 10). The partition \({\mathcal {V}}_\text {heur}\) corresponds to the partition of V as determined by one of the partitioning methods (based on \(H{:}{=}[{\mathcal {R}}(\vec {G},\sigma ), V]\)). The gray parts of the bars comprise those instances for which H is disconnected. The light and dark red bars indicate the amount of digraphs for which only \({\mathcal {V}}_{\text {orig}}\) or \({\mathcal {V}}_\text {heur}\), resp., is a single-leaf split, while light and dark green bars represent instances for which both and none of the two partitions, resp., are single-leaf splits. Note that the partitions were not compared explicitly, in particular, the identified singletons in \({\mathcal {V}}_\text {heur}\) in the light green instances may deviate from those in \({\mathcal {V}}_{\text {orig}}\) in some cases. Example plot for \(|V|=30\) vertices and \(|\sigma (V)|=10\) colors in each digraph. 200 generated digraph pairs per combination of arc insertion (ins.) and deletion (del.) probabilities