Structural characterization of biconnected planar resistance-nonnegative graphs

Characterize which biconnected planar graphs are resistance nonnegative, extending beyond the known resistance-nonnegative grid graphs and non-resistance-nonnegative complete bipartite graphs K_{2,m} for integers m>3.

Background

The paper studies resistance nonnegative (RN) graphs, which are graphs admitting positive edge conductances for which resistance curvature is nonnegative at every vertex. It establishes that every grid graph P_m square P_n is RN; these graphs are biconnected and planar.

The authors contrast this with the fact that K_{2,m} is biconnected and planar but not RN when m>3. Thus, biconnectivity and planarity alone do not determine resistance nonnegativity, and a broader structural characterization of biconnected planar RN graphs remains unresolved.