Table 1 Examples of a singular, a duplicated and two doubled genomes, with their sets of families and their multisets of adjacencies. Note that the doubled genomes \(\mathbb {B}_1\) and \(\mathbb {B}_2\) have exactly the same adjacencies and telomeres

Singular genome

(each family occurs once)

\(\mathbb {S}\!=\!\{({\texttt {1}}\,\,\overline{{\texttt {3}}}\,\,{\texttt {2}})\,\,({\texttt {4}})\,\,[{\texttt {5}}\,\,\overline{{\texttt {6}}}]\}\)

\({\left\{ \begin{array}{ll}\mathcal {F}(\mathbb {S})\!=\!\{{\texttt {1}},{\texttt {2}},{\texttt {3}},{\texttt {4}},{\texttt {5}},{\texttt {6}}\}\\ \mathcal {A}(\mathbb {S})\!=\!\{\texttt{1}^h\texttt{3}^h, \texttt{3}^t\texttt{2}^t, \texttt{2}^h\texttt{1}^t, \texttt{4}^h\texttt{4}^t, \texttt{5}^h\texttt{6}^h\}\\ \mathcal {T}(\mathbb {S})\!=\!\{\texttt{5}^t, \texttt{6}^t\}\end{array}\right. }\)

Duplicated genome

(each family occurs twice)

\(\mathbb {D}\!=\!\{(\mathtt {1\,2}\,\overline{\texttt{3}}\,\texttt{1})\,\,[\overline{\texttt{3}}\,\texttt{2}]\}\)

\({\left\{ \begin{array}{ll}\mathcal {F}(\mathbb {D})\!=\!\{{\texttt {1}},{\texttt {2}},{\texttt {3}}\}\\ \mathcal {A}(\mathbb {D})\!=\!\{\texttt{1}^h\texttt{2}^t, \texttt{2}^h\texttt{3}^h, \texttt{3}^t\texttt{1}^t, \texttt{1}^h\texttt{1}^t, \texttt{3}^t\texttt{2}^t\} \\ \mathcal {T}(\mathbb {D})\!=\!\{\texttt{3}^h, \texttt{2}^h\}\end{array}\right. }\)

Doubled genomes

(each adjacency or telomere occurs twice)

\(\mathbb {B}_1\!=\!\{({\texttt {1}}\,{\texttt {2}})\,\,({\texttt {1}}\,{\texttt {2}})\,\,[{\texttt {3}}\,{\texttt {4}}]\,\,[{\texttt {3}}\,{\texttt {4}}]\}\)

\(\mathbb {B}_2\!=\!\{({\texttt {1}}\,{\texttt {2}}\,{\texttt {1}}\,{\texttt {2}})\,\,[{\texttt {3}}\,{\texttt {4}}]\,\,[{\texttt {3}}\,{\texttt {4}}]\}\)

\({\left\{ \begin{array}{ll}\mathcal {F}(\mathbb {B}_i)\!=\! \{{\texttt {1}},{\texttt {2}},{\texttt {3}},{\texttt {4}}\}\\ \mathcal {A}(\mathbb {B}_i)\!=\!\{\texttt{1}^h\!\texttt{2}^t\!, \texttt{2}^h\!\texttt{1}^t\!, \texttt{1}^h\!\texttt{2}^t\!, \texttt{2}^h\!\texttt{1}^t\!, \texttt{3}^h\!\texttt{4}^t\!, \texttt{3}^h\!\texttt{4}^t\}\\ \mathcal {T}(\mathbb {B}_i)\!=\!\{\texttt{3}^t,\texttt{4}^h,\texttt{3}^t, \texttt{4}^h\}\end{array}\right. }\)