A note on the half-integral multiflow-problem restricted to minor-closed classes of graphs

Abstract

Seymour (1981) proved that the restriction of the half-integral multiflow-problem to K 5 - free instances is solvable in polynomial time. Middendorf and Pfeiffer (1990) proved the general half-integral multiflow-problem to be NP-complete. Unfortunately, the graphs constructed in their reduction contain arbitrary graphs as minors. We present here a new reduction to prove the NP-completeness of the half-integral multiflow-problem constructing only almost-planar graphs (a graph G is almost-planar if there exists a vertex x ∈ V(G) with G − x planar). This implies that the restriction of the half-integral multiflow- problem to a given minor-closed class of graphs G is NP-complete if F ( G ) (the set of forbidden minors of G ) does not contain an almost-planar graph. In the present note we also address the half-integral directed-multiflow-problem . We prove that even the restriction to directed planar graphs is NP-complete.