Edge Contractions in $3$-Connected Graphs

Abstract

For $v\ge 4$ we determine the largest number $f(v)$, such that every simple $3$-connected graph on $v$ vertices has $f(v)$ edge contractions which result in a smaller $3$-connected graph. We also characterize those simple $3$-connected graphs on $v$ vertices which have exactly $f(v)$ such edge contractions.