Rooted grid minors

Rooted grid minors

Intuitively, a tangle of large order in a graph is a highly-connected part of the graph, and it is known that if a graph has a tangle of large order then it has a large grid minor. Here we show that for any k, if G has a tangle of large order and Z is a set of vertices of cardinality k that cannot be separated from the tangle by any separation of order less than k, then G has a large grid minor...

Rooted K4-Minors

Let a, b, c, d be four vertices in a graph G. A K4-minor rooted at a, b, c, d consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of G, respectively containing a, b, c, d. We characterise precisely when G contains a K4-minor rooted at a, b, c, d by describing six classes of obstructions, which are the edge-maximal graphs containing no K4-minor rooted at a, b, c, d. The foll...

Extremal functions for rooted minors

The graph G contains a graph H as a minor if there exist pair-wise 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 minor contains a predetermined xi ∈ V (G). We prove that if the constant c is s...

Grid Minors in Damaged Grids

We prove upper and lower bounds on the size of the largest square grid graph that is a subgraph, minor, or shallow minor of a graph in the form of a larger square grid from which a specified number of vertices have been deleted. Our bounds are tight to within constant factors. We also provide less-tight bounds on analogous problems for higher-dimensional grids.

Extremal Functions for Graph Linkages and Rooted Minors