Some general properties

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Figure 2.5.: A construction e₁, . . . , e10of the zigzag graph G = ZZ₅: G_kis drawn in black, G^k in grey and they intersect in the vertices A_k (white circles). These are never more than three, so vw(ZZ₅) ≤ 3 and in fact equality holds.

edges form the graphs G = G⁰⊋ G¹ ⊋ · · · ⊋ G^|E|= ∅ and at each stage k share a set A_k= V (G_k) ∩ V^G^k+1 of active vertices with the so far constructed G_k. The vertex-width bounds the size of these A_k.

Figure 2.5 shows a construction σ of the zigzag graph ZZ₅ with vw(σ) = 3. Obviously there are infinitely many connected graphs G with vw(G) ≤ 3, including all zigzag graphs ZZ_n and the wheels WS_n with n spokes. The aforementioned result is

Theorem 2.4.2 (theorem 118 and corollary 122 of [49]). If vw(G) ≤ 3, then all periods of G are in Z.

This statement means that all coefficients of the ε-expansion of ^ IG Ω (expanding indices a_e = n_e+ εν_e near integers n_e and the dimension D ∈ 2N − 2ε near an even integer) are rational linear combinations of multiple zeta values. By (2.1.19) this property carries over to the Feynman integral Φ(G), up to the Γ-prefactors which introduce the Euler-Mascheroni constant γ_E into the expansion.

2.4.1. Some general properties

Theorem 2.4.3. Every graph G with vw(G) ≤ 3 is planar.

Proof. Since G has vw(G) ≤ r if and only if each of its connected components H ∈ π₀(G) meets vw(H) ≤ r as well, we may restrict to connected G. We can also exclude any parallel edges, self-loops or vertices of valency one (any of these can simply be added without destroying the planarity of an embedding).

We take any construction which achieves vw^e₁, . . . , e_|E|^≤ 3 and inductively assign polar coordinates r : V → N and ϕ: V → ^0,²₃π,⁴₃π^ such that drawing all edges as straight lines yields a planar embedding of G.

Our algorithm iterates over k from 1 to |E|. As illustrated in figure 2.6, at each stage k exactly one of the following cases occurs:

(1) e_k connects v, w ∈ A_k−1

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e_k v w

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(1) (2) (3): v /∈ A_k (3): v ∈ A_k

Figure 2.6.: The proof of theorem 2.4.3 distinguishes the displayed cases to extend the planar embedding of G_k−1 (grey) by the edge e_k= {v, w}. Any forthcoming edges can connect only at the extremal vertices (black dots) on each ray of constant ϕ (dashed).

(2) e_kconnects v, w /∈ A_k−1: Since v, w are incident to at least one further vertex each, we will have A_k−1∪{v, w} ⊆ A˙ _kand therefore |A_k−1| ≤ |A_k|−2 ≤ 1. Hence we can choose ϕ(v) ̸= ϕ(w) both distinct from ϕ(A_k−1) and further set r(v) = r(w) := k.

(3) e_k connects one vertex v ∈ A_k−1 with one vertex w ∈ V (G^k) \ A_k−1: If v /∈ A_k, set ϕ(w) := ϕ(v) and r(w) := k. Otherwise we must have A_k = A_k−1 ∪ {w} (w˙ is incident to at least one further edge, so w ∈ A_k) and from |A_k| ≤ 3 we know

|A_k−1| ≤ 2, so we can choose some ϕ(w) /∈ ϕ(A_k−1) and set r(w) := k.

This construction ensures that for any k, the embedding of G_k with straight lines is contained in the triangle with corners ∆_k = {v_θ}, where v_θ ∈ V_k,θ := V (G_k) ∩ ϕ⁻¹(θ) denotes the farthest vertex r(v_θ) = max r(V_k,θ) of G_k on the ray ϕ = θ. In particular Ak ⊆ ∆_k is a subset of these corners.

By construction all edges lie on the sides of such triangles ∆_k, except for the radial edges in case (3) when v /∈ A_k. None of these can cross and planarity is obvious.

Remark 2.4.4. From this construction it follows that the same sequence of edges gives rise to a construction of the planar dualG of G (relative to this planar embedding) with vw ≤ 3 as well. Note that for 3-connected G, the planar embedding and G are unique [179].

The sets A_kare cuts of G, so the vertex-width vw(G) ≥ κ(G) bounds the connectivity κ(G) := max {n ∈N0: G \ C is connected for all C ⊂ V (G) with |C| = n} . (2.4.3) As mentioned in section 5.2.1, for the computation of Feynman integrals we only need to consider 3-connected simple graphs G, κ(G) ≥ 3. In this case each vertex is at least 3-valent and |A_k| = 3 for all 2 ≤ |E| − 2. Furthermore the first three edges e_σ(1), e_σ(2) and e_σ(3) of any construction σ of G with vw(G) = 3 must either form a triangle or a star: e_σ(1)= {v₁, w} and e_σ(2)= {v₂, w} share one vertex w (otherwise A2= e_σ(1)∪ e˙ _σ(2) has four elements) and if w /∈ e_σ(3), the third edge can only connect v₁ with v₂.

One can therefore test for vw(G) ≤ 3 very efficiently with

Lemma 2.4.5. Given any simple and 3-connected graph G, an algorithm can decide vw(G) = 3 (and if positive provide a construction σ with vw(σ) = 3) in time O (|V | · |E|).

K₅ K_3,3 C O H

Figure 2.7.: The forbidden minors for simple 3-connected graphs G with vw(G) = 3 from theorem 2.4.6 contain the non-planar complete graphs K₅ and K_3,3, as well as three polyhedra: The cube C together with its dual (the octahedron O) and the self-dual heptahedron H.

Proof. Suppose G has a construction σ with vw(σ) = 3 that starts out with a triangle

∆. Then e_σ(4)= {v, w} must connect one of v ∈ ∆ to a new vertex w /∈ ∆, so necessarily v /∈ A₄= {w} ˙∪ ∆ \ {v} and v is 3-valent. Swapping σ(4) with the edge of ∆ that does not touch v yields a construction σ^′ that also achieves vw(σ^′) = 3.

Thus we only need to look for constructions that begin with a star e_σ(i) = {v_i, w}

(1 ≤ i ≤ 3) defined by some three-valent vertex w. Starting from A := {v₁, v₂, v₃} and I := E \ {e₁, e₂, e₃}, repeat the following steps as often as possible:

• Remove any edges from I that connect active vertices: I := I \ {e ∈ I : e ⊆ A}.

• If some v ∈ A is incident to only one edge e = {v, w} ∈ I, remove e from I and replace v by its neighbour w: A := A \ {v} ˙∪ {w}.

If this process ends in I = ∅, the order σ in which edges were removed from I is a construction with vw(σ) = 3. Otherwise, I ̸= ∅ proves that any construction of G starting with the star around w must have vw(σ) ≥ 4.

To implement this test it suffices to scan through the edges e = {v, w} incident to v every time a vertex v is added to A: After deletion of those with w ∈ A, the algorithm is iterated as some one-valent vertex in A is replaced by its neighbour. This procedure requires a time linear in |E| and there are at most |V | initial stars (3-valent vertices w) to check, so the total runtime is in O (|V | · |E|).

Apparently the vertex-width can not decrease when an edge is removed or contracted, so vw(H) ≤ vw(G) for all minors H ⪯ G.²⁰ Hence the theorem of Robertson and Seymour [144] applies: The set of graphs G with vw(G) ≤ 3 can be characterized by a finite set of forbidden minors. For example, the five graphs shown in figure 2.7 each have a vertex-width of four and can thus not appear as a minor of G when vw(G) ≤ 3.

Recently the sufficiency of this condition was proven [21] and we quote

Theorem 2.4.6. A simple, 3-connected graph G has vertex-width vw(G) = 3 if and only if it contains none of {K_3,3, K5, C, O, H} as a minor.

20A minor of a graph G is any graph H = G \ I/K obtained from deletion and contraction of disjoint sets I ˙∪ K ⊆ E(G) of edges.

Note that this result entails an alternative (but non-constructive) proof of theo-rem 2.4.3 via Wagner’s theotheo-rem [173] (K_3,3 and K₅ are the forbidden minors for pla-narity).

Very interestingly, these forbidden minors were originally discovered by Iain Crump in his thesis [71] to describe the seemingly unrelated splitting property as we will mention in the following section.