The size of an intertwine

Abstract

An intertwine of two graphs H and H′ is a graph G such that G contains both H and H′ as minors, but no proper minor of G contains both H and H′ as minors. We give an upper bound on the size of an intertwine of two given planar graphs. For two planar graphs H and H′, the bound is triply exponential in O(m ⁵) where m≤max(¦V(H)¦, ¦V(H′)¦). We also give an upper on the size of an intertwine of two given trees T and T′. This bound is exponential in O(m ³ log m) where m≤max(¦V(T)¦, ¦V(T′)¦). Let O ₁ be the set of obstructions for a minor closed family L ₁ and O ₂ the set of obstructions for a minor closed family L ₂. It is a natural to ask the following question: how do we given O ₁ and O ₂ compute the obstructions for L ₁∩L ₂ and L ₁ ∪ L ₂. Both these sets of obstructions are known to be finite, and to obtain the obstructions for L ₁∩ L ₂ is easy. However, to compute the obstructions for L ₁ ∪ L ₂ is hard. Our upper bound enables us to given O ₁ and O ₂ compute a bound on the size of any obstruction for L ₁ ∪ L ₂ whenever L ₁ and L ₂ are families of planar graphs.

Supported by the Swedish Natural Sciences Research Council