Chapter III Duality
III.4 Basic transformation
In a premap M = (Xα,β,P), the deletion of a single edge ex
is called basic deleting an edge and its result is denoted by M −b
ex. The contraction of a link ex is called basic contracting an edge,
and its result is denoted by M •b ex. The two operations are, in all,
called basic subtracting an edge. Similarly, appending a single edge is called basic appending an edge, and splitting a link is called basic splitting an edge. Such two operations are, in all, called basic adding
an edge. Apparently, M +b ex and M ◦b ex are the results of basic
adding an edge ex on M in their own right. Basic subtracting and
basic adding an edge are in all calledbasic transformation. From what we have known above, A premap becomes another premap under basic transformation.
Theorem 3.10 Suppose M′ is a premap obtained by basic transformation from premap M, then M′ is a map if, and only if,
M is a map.
Proof Because a single edge is never a harmonic link, from The- orem 3.4 the theorem holds for basic deleting an edge. Because a link is never a harmonic loop, from Theorem 3.5, the theorem holds for basic contracting an edge. Then, from Theorem 3.7–8, the theorem
holds for basic adding an edge.
Furthermore, for basic transformation, the following conclusion can also be done.
Theorem 3.11 LetM = (X,P) be a map and M∗ = (X∗,P∗), its dual. Then, for any single edge ex in M, (M −b ex)∗ = M∗ •b e∗x
and for a single edge ex not inM, (M +bex)∗ = M∗◦be∗x. Conversely,
for any link ex in M, M •b ex = M∗−b e∗x and for a link ex not in M,
M ◦b ex = M∗ +b e∗x.
Proof Based on the duality between edges as shown in Table 3.1, the statements are meaningful. From Theorem 3.6 and Theorem 3.9, the fist statement is true. In virtue of the duality, the second
III.4 Basic transformation 97
statement is true.
From this theorem, the following two diagrams are seen to be commutative: (Xα,β,P) −−−−−−−−→−bex (Xα,β −Kx,P−bex) ∗xy ∗xy (X∗,P∗) ◦be ∗ x ←−−−−−−−− (X∗ −K∗x,P•∗be∗x) (3.16) and (Xα,β,P) ←−−−−−−−− +bex (Xα,β −Kx,P−bex) ∗xy ∗xy (X∗,P∗) −−−−−−−−→ •be∗x (X∗ −K∗x,P∗ •be∗x) (3.17)
On the basis of basic transformation, an equivalence can be es- tablished for classifying maps in agreement with the classification of surfaces.
III.5 Observations
O3.1 Observe the condition for two permutations Per1and Per2
satisfying (3.1) on the same ground set.
O3.2 Given a tree(e.g., the star of 5 edges), observe how many maps are there for their dual maps all having the tree as under graph. O3.3 Write the dual of map M = (Kx,(x, βx)). If a map has its dual with the same under graph, then it is said to be self-dual for the graph. Observe if M is self-dual for its under graph.
O3.4 Provide a map which is cuttable and its under graph with a cut-vertex.
O3.5 How to distinguish the cuttability of a map and the sep- arability of its under graph?
O3.6 Is the under graph of the dual of a Eulerian map bipar- tite? If yes, explain the why; otherwise, by an example.
O3.7 Is the dual of a preproper map(no double edge) always a preproper map? If yes, explain the why; otherwise, by an example.
O3.8 Is the dual of a proper map(each face formed by a circuit in its under graph) always a proper map? If yes, explain the why; otherwise, by an example.
O3.9 Is the dual of a polygonal map(two face with at most one edge in common) always a polygonal map? If yes, explain the why;
III.6 Exercises 99 otherwise, by an example.
O3.10 Is the under graph of the dual of a map with 3-connected under graph still 3-connected? If yes, explain the why; otherwise, by an example.
III.6 Exercises
E3.1 Prove that the under graph of a map M = (X,P) is a tree if, and only if, its dual M∗ has the following three properties:
(i) M∗ has only one vertex;
(ii) For any x ∈ X, x and γx are in the same orbit of P∗;
(iii) For any y ∈ (x)P∗, there is no subsequence x, y, γx, γy or
x, γy, γx, y in its cycle.
For a mapM = (X(X),P) and S ⊆X, letM[KS] = (KS,P[KS])
where P[KS] is the restriction of P on KS = KS, and M[KS] is said
to be induced on KS from M. Generally speaking, M[KS] is not a
map, but always a premap. A cocircuit of a graph with all of its edges incident to the same vertex is called a proper cocircuit .
E3.2 Let M − ES = M[X − KS], ES = {ex|∀x ∈ S}. Prove
that M − ES, S ⊆ X, is a map if, and only if, there is no proper
cocircuit of G[M] on graph G(M[KS]).
A proper circuit of a map M is such a set C of edges that C∗ =
{e∗|∀e∈ C} is a proper cocircuit of G[M∗].
E3.3 Let H = G[M] and MH be the set of all maps whose
under graphs are H. Prove that if C is a proper circuit of a map M, then it is a proper circuit of all N ∈ MH.
E3.4 Let M • ES = (M∗ −ES∗)∗ where M∗ is the dual of M
and ES∗ = {e∗x|∀x ∈ S}. Prove that M •ES is a map if, and only if,
ES is a proper circuit of M.
E3.5 Prove that a map M is on the sphere if, and only if, each face of its dual M∗ corresponds to a proper cocircuit of M.
If a map has its dual Eulerian, then it is called a dual Eulerian map.
E3.6 Prove that a map is a dual Eulerian map if, and only if, each of its faces is incident with even number of edges.
If a preproper dual Eulerian map has each of its faces partition- able into circuits, then it is called a even assigned map . A map is called bipartite when its under graph is bipartite.
E3.7 Prove that a dual Eulerian map is bipartite if, and only if, it is even assigned.
E3.8 Prove that a quadrangulation is bipartite if, and only if, it is without loop.
For a loopless quadrangulation Q = (Xα,β,Q), from E3.8, its
vertex set V can be partitioned into two subsets V1 and V2 such that
the two ends of each edge are never in the same subset. Such a subset of vertices is called an independent set. Xα(1)1,β1 and X
(2)
α2,β2 stand for the sets of elements in Xα,β incident to, respectively, V1 and V2[Deh1,
Gau1, MuS1].
E3.9 Let σ = βQγ, γ = αβ. Prove that (σx, σQαx), x ∈ Xα,β,
is an angle.
Angles (σx, σQαx) and (x,Qαx) as shown in E3.9 are called an
independent pair . Thus, each face in a quadrangulation has exactly two independent pairs of angles.
E3.10 Let K1 = {1, α1, β1, γ1}, γ1 = α1β1. And, α1 = Q and
β1 = βQγ(= σ as shown in E3.9). Prove
(i) K1 is the Klein group of four elements;
(ii) Xα(1)1,β1 = X
vx∈V1
X
y∈{x}Q
K1y;
(iii) Q1 = (Xα(1)1,β1,Q−11) is a map, where Q1 is the restriction of
Q on Xα(1)1,β1.
Similarly to E3.10, fromV2, another map Q2 can be deduced. Q1
III.7 Researches 101 E3.11 Prove that the two maps in the incident pair of a quad- rangulation are mutually dual.
E3.12 Prove that any planar quadrangulation is loopless. E3.13 Let A and B be the sets of, respective all planar quad- rangulations and all dual pairs of planar maps. Establish a 1–to–1 correspondence between A and B(i.e., bijection).
III.7 Researches
For a map, if the basic deletion of an edge can not be done anymore, then the map is said to be basic deleting edge irreducible. Similarly, if the basic contraction of an edge can not be done on a map anymore, then the map is said to be basic contracting irreducible. R3.1 Given the size, determine the number of self-dual maps as an integral function of the size, or provide a way to list all the self- dual maps of the same size and deduce a relation among the numbers of different sizes.
R3.2 Given the size, determine the number of maps all basic deleting irreducible as an integral function of the size, or provide a way to list all such maps with the same size and deduce a relation among the numbers of different sizes.
R3.3 Given the size, determine the number of maps all basic contracting irreducible as an integral function of the size, or provide a way to list all such maps with the same size and deduce a relation among the numbers of different sizes.
R3.4 For any given graph, determine the number of maps all basic deleting irreducible with the same under graph, or provide a way to list all such maps with the same size and deduce a relation among the numbers of different sizes.
R3.5 For any given map, determine the number of all basic deleting irreducible maps obtained from the map by basic deletion, or
provide a way to list all such maps with the same size and deduce a relation among the numbers of different sizes.
R3.6 For a given graph, determine the number of maps all basic contracting irreducible with the same under graph, or provide a way to list such maps and deduce a relation among the numbers of different sizes.
R3.7 For a given map, determine the number of all basic con- tracting irreducible maps obtained from the map by basic contraction, or provide a way to list such maps and deduce a relation among the numbers of different sizes.
If a map is basic both deleting and contracting irreducible, then it is said to be basic subtracting irreducible.
R3.8 Given the size, determine the number of basic subtracting irreducible maps as an integral function of the size, or provide a way to list all such maps with the same size and deduce a relation among the numbers of different sizes.
R3.9 For a given graph, determine the number of maps all basic subtracting irreducible with the same under graph, or provide a way to list such maps and deduce a relation among the numbers of different sizes.
R3.10 Find a relation between triangulations and quadrangu- lations.
If a map has each of its faces pentagon, then it is called a quin- quangulation. Similarly, the meaning of a hexagonalization.
R3.11 Justify whether or not a triangulation has a spanning quinquangulation or hexagonalization. If do, determine its number.
R3.12 Even assigned conjecture: A bipartite graph without cut-edge has a super map even assigned.