We develop a theory of minors for alternating dimaps --- orientably embedded digraphs where, at each vertex, the incident edges (taken in the order given by the embedding) are directed alternately into, and out of, the vertex. We show that they are related by the triality relation of Tutte. They do not commute in general, though do in many circumstances, and we characterise the situations where they do. The relationship with triality is reminiscent of similar relationships for binary functions, due to the author, so we characterise those alternating dimaps which correspond to binary functions. We give a characterisation of alternating dimaps of at most a given genus, using a finite set of excluded minors. We also use the minor operations to define simple Tutte invariants for alternating dimaps and characterise them. We establish a connection with the Tutte polynomial, and pose the problem of characterising universal Tutte-like invariants for alternating dimaps based on these minor operations.
It is interesting to note that this stream of research, first seen in Tutte's 1948 paper @cite_16 , can be traced back to the same source that eventually gave rise to Tutte's work on minor operations and his eponymous polynomial. Historically, the source of both streams was the famous paper on squaring the square'' @cite_10 . The 1948 paper extended the theory to triangulating the triangle'' (where all triangles are equilateral) and introduced triality, among other things. However, this stream has not previously seen the development of minor operations or Tutte-like invariants for alternating dimaps.
@cite_16 @cite_10
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Let T = (T*, T(Delta)) he a spherical latin bitrade. With each a = (a(1), a(2), a(3)) is an element of T* associate a set of linear equations Eq(T, a) of the form b(1) + b(2) = b(3), where b = (b(1), b(2), b(3)) runs through T* a . Assume a(1) = 0 = a(2) and a(3) = 1. Then Eq(T, a) has in rational numbers a unique solution b(i) = (b) over bar (i). Suppose that (b) over bar (i) not equal (c) over bar (i) for all b, c is an element of T* such that (b) over bar (i) not equal c(i) and i is an element of 1, 2, 3 . We prove that then T(Delta) can be interpreted as a dissection of an equilateral triangle. We also consider group modifications of latin bitrades and show that the methods for generating the dissections can be used for a proof that T* can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade T. (C) 2009 Wiley Periodicals, Inc. J Combin Designs 18: 1-24, 2010 We consider the problem of dividing a rectangle into a finite number of non-overlapping squares, no two of which are equal. A dissection of a rectangle R into a finite number n of non-overlapping squares is called a squaring of R of order n; and the n squares are the elements of the dissection. The term “elements” is also used for the lengths of the sides of the elements. If there is more than one element and the elements are all unequal, the squaring is called perfect, and R is a perfect rectangle.