Fig. 4

a Example of a shared-content graph \(\mathcal {C}({\mathbb {A}},{\mathbb {B}})\) with omitted edge scores. The genomes \({\mathbb {A}}\) and \({\mathbb {B}}\) have linear segments \(A_{1{..}{}5}\) and \(B_{1{..}{}5}\), respectively. The capping of \(F\!F\!R({\mathbb {A}},{\mathbb {B}},\mathcal {S})\) induced by \(\mathcal {C}\) is invalid. b Transformation of \(\mathcal {C}\) into a perfect shared-content graph \({\widehat{\mathcal {C}}}({\mathbb {A}},{\mathbb {B}})\): vertex sets \(S_1\) and \(S_2\) represent Hall violators (among other possibilities) that demand the creation of dummy segments \(\varphi _{\mathbb {B}}^1\) and \(\varphi _{\mathbb {A}}^1\), respectively. Dotted edges represent those that are non-matchable and must be removed from \({\widehat{\mathcal {C}}}\) after the completion is finished. Notice that the component with vertices \(A_1, A_2, A_3, B_1, \varphi _{\mathbb {B}}^1 \text { and } B_2\) is not a complete bipartite subgraph. (In both (a, b), we give an abstract illustration of the capped \(F\!F\!R\) where only cap vertices, cap edges and dummy adjacencies are represented explicitly, while vertices of gene extremities between cap vertices are represented by a line with small dots. In addition, colored solid edges represent a maximum cardinality matching between cap vertices, while the cap edges not in the matching are dashed grey)