Spanning 2-Forests and Resistance Distance in 2-Connected Graphs

Wayne Barrett, Emily J. Evans, Amanda E. Francis, Mark Kempton, John Sinkovic

Published 2018-12-31Version 1

A spanning 2-forest separating vertices $u$ and $v$ of an undirected connected graph is a spanning forest with 2 components such that $u$ and $v$ are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance distance between vertices $u$ and $v$ in a graph representing an electrical circuit with unit resistance on each edge is the number of spanning 2-forests separating $u$ and $v$ divided by the number of spanning trees in the graph. It is well-known that the number of these spanning 2-forests separating $u$ and $v$ in a graph is equal to the determinant of the matrix obtained from the combinatorial Laplacian matrix of the graph by deleting the rows and columns corresponding to $u$ and $v$. For most interesting graphs, neither of these quantities can be easily found. For any connected graph $G$ with a 2-separator and its associated decomposition which separates $u$ and $v$, we show that the number of spanning trees and spanning 2-forests separating $u$ and $v$ can be expressed in terms of the number of spanning trees and 2-forests in the smaller graphs in the decomposition, which makes computation significantly more tractable. An important special case is the preservation of the number of spanning 2-forests if $u$ and $v$ are on the "same side" of the decomposition. We apply these results to derive an exact formula for the number of spanning 2-forests separating $u$ and $v$ for any linear 2-tree with a single bend. We focus in particular on the case in which $u$ and $v$ are the end (degree 2) vertices of the linear 2-tree, interpret these as resistance distances, and analyze these values with respect to the location of the bend.