Figure 1. Fully connected 4-node map, represented by a graph on the left, and overlaid with traditional country-area representation on the right. Figure 2. Illustration of adding induction node “y” — solid lines are an existing map, dashed lines are added at the induction step, and ellipses indicates additional similar features. Figure 3. Examples of identification of possible new edges and nodes (new elements not in existing map are dashed). Presumption is all new structures connecting to existing map (solid lines) are shown. In part (a) several new nodes w, u, v connect to one existing node A, which would create a doubly exposed boundary segment (see Figure 4), however by fully connecting the map (dotted edge with arrows) this is avoided. In part (b) each new node w, u, v connects to only one existing node A, B or C, so that if connected one at a time again a doubly exposed boundary segment would result. Again by fully connecting the map (dotted double arrow edge) this is avoided. Figure parts (a) and (b) correspond to text cases a. and b See text for case c. Step 3 involves two criteria. Recalling that we require maps to be fully connected, circumnavigate an existing boundary to identify possible new edges, and the outside nodes to which they connect. These all qualify as innermost new nodes. Fig- ure 3 shows an example for identifying possible new edges and their associated nodes. The possible new nodes and edges are shown with dashed lines. We must now show there is at least one node which has, or can have, more than one con- nection to the existing map in order to satisfy the remaining condition of step 3. Figure 4. Doubly exposed boundary segment illustration. If node x or y or the edge between were hidden on the top they are still exposed below. Thus when adding interior edges in step 4 they will never connect to an existing boundary node. 3.3. Flexibility Independence and Revised Rules Table 1. Color combinations possible for three nodes. combinations possible for three nodes 1, 2 and 3 are shown in Table 1. We now introduce the notion independent flexibility whereby a node or group of nodes has a flexibility that is independent of other nodes, so that the flexibilities can be multiplied together to get a minimum map flexibility. The actual map flexibility might be larger. We are only concerned with an easily computable value which is not greater than the actual flexibility. We define the following re- vised-flexibility-rules for computation on which arguments about independence will be based:
