A note on the maximum weight planar subgraph problem

The problem of finding the maximum-weight, planar subgraph of a finite, simple graph with nonnegative real edge weights is well known in industrial and electrical engineering, systems biology, sociology and finance. As the problem is known to be NP-hard, much research effort has been devoted over the years to attempt to improve a given approximate solution to the problem by using local moves applied to a planar embedding of the solution. It has long been established that any feasible solution to the problem, a maximal planar graph, can be transformed into any other (having the same vertex set) in a finite sequence of local moves based on: (i) edge substitution and (ii) vertex relocation and it has been conjectured that moves of only type (i) are sufficient. In this note we settle this conjecture in the affirmative. We hope this result will be useful in the design of future approximate methods for the problem, as not having to consider vertex relocation means significantly less work has to be done. 2000 AMS Subject Classification: 05C10, 05C05 and 05C76.