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The problem of finding the maximum-weight, planar subgraph of a finite,simple graph with nonnegative real edge weights is well known in industrial andelectrical engineering, systems biology, sociology and finance. As the problemis known to be NP-hard, much research effort has been devoted over the years toattempt to improve a given approximate solution to the problem by using localmoves applied to a planar embedding of the solution. It has long beenestablished that any feasible solution to the problem, a maximal planar graph,can be transformed into any other (having the same vertex set) in a finitesequence of local moves of based on: (i) edge substitution and (ii) vertexrelocation and it has been conjectured that moves of only type (i) aresufficient. In this note we settle this conjecture in the affirmative.Furthermore, contrary to recent supposition, we demonstrate that any maximalspanning tree of the original graph is not necessarily a part of any optimalsolution to the problem. We hope these results will be useful in the design offuture approximate methods for the problem.

Results for the maximum weight planar subgraph problem