Independent Trees and Branchings in Planar Multigraphs

Abstract.

 The following statement is proved: Let G be a finite directed or undirected planar multigraph and s be a vertex of G such that for each vertex x≠s of G, there are at least k pairwise openly disjoint paths in G from x to s where k∉{3,4,5} if G is directed. Then there exist k spanning trees T ₁, … ,T _k in G directed towards s if G is directed such that for each vertex x≠s of G, the k paths from x to s in T ₁, … ,T _k are pairwise openly disjoint. – The case where G is directed and k∈{3,4,5} remains open.