Extremal functions for rooted minors

Abstract

AbstractThe graph G contains a graph H as a minor if there exist pairwise disjoint sets {Si ⊆ V(G)|i = 1,…,|V(H)|} such that for every i, G[Si] is a connected subgraph and for every edge uv in H, there exists an edge of G with one end in Su and the other end in Sv. A rooted H minor in G is a minor where each Si of the minor contains a predetermined xi ∈ 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 |V(H)|‐connected graph G with (9c + 26,833|V(H)|)|V(G)| edges contains a rooted H minor for every choice of vertices {x1,…,x|V(H)|} ⊆ 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 Jørgensen, Kawarabayashi, and Böhme and Mohar. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:159–178, 2008