The Quest for Unavoidable Sets

Abstract

It is Heinrich Heesch who must be credited with compiling a number of facts about how to search for unavoidable sets of reducible configurations. He provided clues as to which configurations one could most likely ignore. Initially, it was a matter of three “obstructions. ”¹ Even up to now, none of the known methods of reduction have been successful in reducing a configuration whose inner vertices all have at least degree 5 if it “essentially” contains one of the following internal structures:

• 1 An inner vertex with more than three legs.

• 2 An articulation with more than two legs. (This explains an earlier observation on page 166 that configurations with articulations having more than two legs can remain outside of the realm of consideration.)

• 3 A dangling 5-couple, which is a figure consisting of two neighboring inner 5-vertices both of which adjoin the same third distinct inner vertex. The vertices belonging to a dangling 5-couple have exactly three legs each.

Here we use quotation marks instead of italics to point out that is has more to do with a “physical” phenomenon than with precise mathematical definitions and theorems.