# Transfer-Efficient Face Routing Using the Planar Graphs of Neighbors in High Density WSNs
## 4 Transfer-Efficient Face Routing
### Remote Node Selection and Message Forwarding  
The last step in the TEFR method is to discover the node located at the most distant hop on a routing path within radio range using the information on the planar graphs of the neighbors at the message transfer node, and then send the message. The message transfer node generates an edge list  
Figure 5: Topological changes that do not affect the planar graph: (a) changed local network graph, and (b) (a)’s planar graph. Figure 6: Topological changes that affect the planar graph: (a) changed local network graph, and (b) (a)’s planar graph. The second case arises when a face change occurs while the message is forwarded within radio range because the line from the source to the destination and a face boundary edge intersect. In this case, TEFR transmits the message to the node where the intersecting line occurs, not the most distant hop node within radio range. After the selected node receives the message, it changes the face and routing is continued. Figure 8b shows the case where the face is changed because the intersection between line \(\overline{st}\) and \(edge(y,z)\) occurs while traveling to the face within radio range of message transfer node \(x\). If the message is sent to the most distant node \(z\) without considering the face change at message transfer node \(x\), a loop occurs because the face change does not execute. Therefore, message transfer node \(x\) forwards the message to node \(y\) where the face change occurs, after which node \(y\) changes the face and then continues routing. The third case happens when it is necessary to switch to greedy routing while traveling to a face within radio range. TEFR immediately switches from face routing to greedy routing when greedy routing is enabled and it forwards a message according to greedy routing. Figure 8c shows the case where the routing mode needs to be switched from face routing to greedy routing during face traversal within radio range. Message transfer node \(a\) forwards the message to the first node \(b\) closer to the destination (\(t\)) than node \(s\) where face routing started without sending a message to the farthest hop node \(c\) on the face boundary within radio range. After receiving the message, node \(b\) switches to greedy routing and performs message forwarding. The last case occurs when the message transfer node has low residual power. A message is sent sequentially along the face boundary instead of being forwarded to the remote node. Because energy consumption increases as the distance to be transmitted increases in wireless communication [27], the message transfer node sends the message to its nearest neighbor when the energy level is low to minimize power consumption. This extends the lifetime of the node and consequently prolongs the lifetime of the WSN. By selectively transmitting a message to the remote node or nearby nodes according to the remaining power, transfer efficiency can be improved while using the energy of each node in a balanced manner. Figure 8d shows the case where the message transfer node has less energy than the threshold for direct transmission. Message transfer node \(c\) can forward a message directly to remote node \(y\), but it sends the message to nearest node \(x\) to minimize energy consumption. Since TEFR can determine the routing path in advance within radio range using the full GG edge list, it can establish the most remote node where a message can be transferred most efficiently, and a message can be sent directly to the selected node. That is to say, TEFR can directly send a message without journeying via intermediate nodes by simulating routing internally using the edge list as in Table 3 and selecting a remote node in the same way as actually traveling the nodes constituting the face within radio range. Figure 8: Four cases of node selection: (**a**) routing within the same face, (**b**) a face change occurs, (**c**) return to greedy mode, and (**d**) sequential forwarding.