Yes, there is no road split between Cheatham and South Davidson, but one more road split between North Davidson and Cheatham, so why one less road split with your revision of my map?
I am confused again. In all events, as an factual matter, there really is no greater erosity or lack of connectedness, but until I understand your program clearly, there is no point going there.
My general rule is that when a county is chopped, each chop part now acts as a new county for the purpose of connections and thence erosity. Connections between counties depend on a path of numbered federal and state highways between population nodes. The population node is determined by the largest urbanized area (
census maps) or the county seat if there is no urbanized area, or the largest population incorporated or census designated place if none of the other choices are available. Within an urban area the largest population within the county is used for the node. This can be either incorporated or unincorporated, or the downtown within a large city (typically including city hall).
It's easier to start with examples from eastern TN. This is my split of Grainger. There are no urbanized areas in Grainger so county sear Rutledge is the population node. It stays in the northern chop and Blaine is the largest place in the southern chop, so it is the population node for determining connections.
Grainger (Rutledge) is connected by state or federal highways to to Claiborne, Hancock, Hawkins, Hamblen, Jefferson, Knox, and Union. The split of the southern part around Blaine is only connected to Knox, and the remaining northern part is connected to the remaining counties, but not Knox. This is an ideal chop that has no increase in erosity.
This is my split of Blount. Much of the population of the county is within the Knoxville urban area which doesn't show on the DRA map. Maryville is the largest place and within that urban area and is the population node for the county which is connected to Monroe, Loudon, Knox, and Sevier. The northern part including Maryville is only connected to Knox and Sevier.
The southern part has the largest population in the urban area in unincorporated Binfield, and the largest place is the much smaller Friendsville. Binfield has the larger population in the urban area and is used as the node connecting to Monroe and Loudon. Friendsville only links to Loudon, but a connection that exists at the county level cannot vanish by the means of a district line, so the Monroe connection would have to remain there as well. This chop also separates the county connection in a way that does not increase their number.
Davidson county is problematic to define since Nashville includes all of the county not in the few small communities. The northern part has the downtown so that defines its node. The southern part has more population in Nashville than in the smaller separate cities and the largest part is along I-24 so that defines the node for connections.
Davidison connects to Cheatham at the county level, but doesn't need to have connections to both the northern and southern chops of Davidson depending on where the line is drawn. By following the river at the Cheatham line you provide a connection to the south Davidson node along TN-1, TN-100 and TN-254. By moving the line south as I do I remove the link from Cheatham to south Davidson. If I had controls I'd make my cut more cleanly than the course VTDs in DRA.
Yes, it may seem arbitrary that our two plans seem reasonably equal by eye in Nashville, yet differ by this measure. The difference is that without my constraint on erosity, a mapper is free to place the line at Cheatham anywhere they want and that opens the door just a bit to gerrymandering. The idea is that a reasonable set of constraints can limit gerrymandering yet provide flexibility.