Cops, Robbers, and Threatening Skeletons: Padded Decomposition for Minor-Free Graphs

We prove that any graph excluding K_r as a minor has can be partitioned into clusters of diameter at most Δ while removing at most O(r/Δ) fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove O(r²/Δ) fraction of the edges.

Our result is obtained by a new approach to relate the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops-and-robbers game on excluded-minor graphs can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter O(r) and strong-diameter partitions with padding parameter O(r²) for K_r-free graphs, padding O(k) for graphs with treewidth k, and padding O(log g) for graphs with genus g.