Extremal functions for rooted minors

The graph G contains a graph H as a minor if there exist pairwise disjoint sets {S-i subset of V(G)vertical bar i = 1,..., vertical bar V(H)vertical bar} such that for every i, G[S-i] is a connected subgraph and for every edge uv in H, there exists an edge of G with one end in S-u and the other end in S-v. A rooted H minor in G is a minor where each S-i of the minor contains a predetermined x(i) is an element of V(G). We prove that if the constant c is such that every graph on n vertices with cn edges contains an H minor, then every vertical bar V(H)vertical bar-connected graph G with (9c + 26,833 vertical bar V(H)vertical bar)vertical bar V(G)vertical bar edges contains a rooted H minor for every choice of vertices {x(1),...., x(vertical bar V(H)vertical bar)} subset of V(G). The proof methodology is sufficiently robust to find the exact extremal function for an infinite family of rooted bipartite minors previously studied by Jorgensen, Kawarabayashi, and Bohme and Mohar. (C) 2008 Wiley Periodicals, Inc.