Graph Bipartization and Via Minimization.

The vertex- (resp., edge-) deletion graph bipartization problem is the problem of deleting a set of vertices (resp., edges) from a graph so as to make the remaining graph bipartite. In this paper, we first show that the vertex-deletion graph bipartization problem has a solution of size k or less if and only if the edge- deletion graph bipartization problem has a solution of size k or less, when the maximum vertex degree is limited to three. This immediately implies that (1) the vertex-deletion graph bipartization problem is NP-complete for cubic graphs, and (2) the minimum vertex-deletion graph bipartization problem is solvable in polynomial time for planar graphs when the maximum vertex degree is limited to three. We then prove that the vertex- deletion graph bipartization problem is NP-complete for planar graphs when the maximum vertex degree exceeds three. Using this result, we finally show that the via minimization problem, which arises in the design of integrated circuits and printed circuit boards, is NP-complete even when the maximum "junction" degree is limited to four