Refined Buneman Trees

2.5 Diagonal quartets
Looking at the definition of a quartet and its associated Buneman score, we
might observe that the quartet q = ab|cd canbeviewedastwodiagonal quartets,
denoted ab||cd and ab||dc. This intuition is not simple, but we can make the
following rewrite to illustrate this point:
β q = 1 2 (min{δ ac + δ bd ,δ ad + δ bc }−(δ ab + δ cd )) (2.2)
{ 1
= min 2 (−δ }
ab + δ bc − δ cd + δ da )
1
2 (−δ (2.3)
ab + δ bd − δ dc + δ ca )
= min{η ab||cd ,η ab||dc } (2.4)
where η ab||cd = 1 2 (δ bc − δ ab + δ ad − δ cd ) is the score of the diagonal quartet
ab||cd.
Looking at Figure 2.5 we can see that the terms η ab||cd and η ab||dc correspond
to a “tour” in one of the two diagonal quartets. In words, we go through either
a → b → c → d → a or a → b → d → c → a, adding or subtracting chords
as appropriate. We might also notice that diagonal quartets are symmetric
on either side of their central edge, i.e. starting out with the diagonal quartet
ab||cd we can swap a ↔ b and c ↔ d or ab ↔ cd to obtain ba||dc or cd||ab, where
η ab||cd = η ba||dc = η cd||ab and all three diagonal quartets still identify the same
quartet ab|cd.
a
c
b
ab|cd
d
a
c
a
c
b
ab||cd
d
b
ab||dc
d
Figure 2.5: The symmetric quartet and its associated asymmetric diagonal quartets.
To keep an ordering on diagonal quartets we shall define the minimum diagonal
of ab|cd: Leta be the smallest (by index in X) of the species a, b, c, d ∈ X.
ab||cd is the minimum diagonal if η ab||cd