# H-minor free Graph

Recently I cam across a statement which divides graphs into different classes w.r.t. the complexity of problems on them.

planar < bounded genus < H-minor free < general graphs

My question regards the definition of H-minor free graphs. What I don't understand is that apparently H is not fixed.

Is it true that as long as I can be sure that my (large) graph G doesn't contain some! minor, then some problems might be easier to compute on G then on general graphs regardless of what the actual minor is?

Thank you

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Typically in a particular context $H$ is fixed, or is at worst a member of a fixed class of graphs. Take a look at this link, especially the second and third paragraphs, to see some of the useful effects. –  Brian M. Scott Jan 7 '12 at 20:22

Well bounded treewidth graphs are free of "grid minors". Specifically a graph of treewidth k cannot have a $k+1 \times k+1$ grid inside it. On such graphs, it's easier to do various computations –  Suresh Venkat Jan 14 '12 at 1:33