# The Hamiltonian Path Graph is Connected for Simple s,t Paths in Rectangular Grid Graphs

A simple s,t path P in a rectangular grid graph 𝔾 is a Hamiltonian path from the top-left corner s to the bottom-right corner t such that each internal subpath of P with both endpoints a and b on the boundary of 𝔾 has the minimum number of bends needed to travel from a to b (i.e., 0, 1, or 2 bends, depending on whether a and b are on opposite, adjacent, or the same side of the bounding rectangle). Here, we show that P can be reconfigured to any other simple s,t path of 𝔾 by switching 2× 2 squares, where at most 5|𝔾|/4 such operations are required. Furthermore, each square-switch is done in O(1) time and keeps the resulting path in the same family of simple s,t paths. Our reconfiguration result proves that the Hamiltonian path graph G for simple s,t paths is connected and has diameter at most 5|𝔾|/4 which is asymptotically tight.

READ FULL TEXT