Let a tree be a subgraph of a cubic graph . The graph excision is the graph resulting from removing the tree, then merging the edges. For example, if in the Tutte 8-cage (left figure) the tree formed by the 6 interior points (middle figure) is excised, the McGee graph (right figure) results. Similarly, excising the Heawood graph gives the Petersen graph, and excising the generalized hexagon (i.e., the unique 12-cage graph) gives the Balaban 11-cage (Biggs 1998).

The reverse of excision is insertion. Both operations are used in the analysis of cages.

The following table gives some cubic symmetric graphs with named edge-excised graphs, illustrated above.

graph | edge-excised graph |

utility graph | tetrahedral graph |

cubical graph | 3-prism graph |

Petersen graph | 4-Möbius ladder |

Heawood graph | 12-cubic graph 84 |

Möbius-Kantor graph | 14-cubic graph 503 |

dodecahedral graph | cubic polyhedral graph Cp34 |