Paths and Metrics in a Planar Graph with Three or More Holes. I. Metrics

Let G = (VG, EG) be a bipartite planar graph embedded in the euclidean plane and H be a subset of its faces (holes). A. Schrijver proved that if |H| = 2 then there exists a collection C = {m1, ..., mk} of cut metrics on VG such that: (i) m1(e)+ ··· + mk(e) ≤ 1 for any e ∈ EG; and (ii) for any vertices s and t in the boundary of a hole, the value m1(s, t)+ ··· + mk(s, t) is equal to the distance between s and t. This is, in general, not true for |H| = 3. In the present paper one proves that: (*) for |H| = 3, (i)-(ii) hold for some C consisting of cut metrics and 2, 3-metrics (metrics induced by the graph K2, 3); and (**) for |H| =4, (i)-(ii) hold for some C consisting of metrics induced by planar graphs with at most four faces. Using (*), in the sequel to the present paper (Part II) we give a criterion of the existence of edge-disjoint paths connecting certain vertices in a planar graph with three holes, provided that the so-called "parity condition" holds. This extends, in a sense, Okamura′s theorem on edge-disjoint paths in planar graphs with two holes.