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Then the two Tait graphs are the embedded graphs obtained by checkerboard colouring F and forming the whiteface and blackface graphs. We let T (F) denote the set of Tait graphs of F. 13. Historically Tait graphs were considered for 4-regular graphs cellularly embedded in the plane R2 . In this case we can canonically checkerboard colour the graph by insisting that the unbounded region be white. In the literature, the term Tait graph is often used to refer to the (unique) blackface graph of F formed with respect to this canonical checkerboard colouring.

11. 9 is given in Fig. 9. 9b shows the structure obtained by adding discs to G (shown in Fig. 9a) that follow the boundary components of (V (G), A), where A = {2, 3}. 9c shows the ribbon graph obtained by deleting the vertices of G. The ribbon graph Gδ (A) has been redrawn in Fig. 9d. 3 provides a local construction of partial duals: change part of an arrow presentation as indicated in Fig. 4. In contrast, all of the other constructions of partial duals given above are global in that they involve the whole of the embedded graph.

Thus H ∈ C (Gm ). 1 Arrow presentations of twisted duals at an edge e 1(e) d(e) t (e) td(e) d t(e) d t d(e) Conversely, if H ∈ C (Gm ), then it is obtained by replacing each vertex of Gm with one of the arrow marked states shown in Fig. 3. 1, so H ∈ Orb(Gm ), completing the proof. 8 states that we can obtain all of the cycle family graphs of a medial graph as a set of twisted duals. For example, if G is the non-orientable ribbon graph consisting of one vertex and one edge, then its medial graph Gm consists of a single vertex and two alternating non-orientable loops.

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