Pumping Lemmas for Paths

Let P = (s, . . . , t) be a directed path, where each vertex has an input label from Σin. The

tripartition of the vertices ξ(P ) = (D1, D2, D3) is defined as follows:

D1 = Nr−1(s)∪ Nr−1(t),

D2 =

(

N2r−1(s)∪ N2r−1(t))\ D1,

D3 = P \ (D1∪ D2).

See Figure 4.1 for an illustration. More specifically, suppose P = (u1, . . . , uk), and let

i∈ [1, k]. Then we have:

• ui ∈ D1 if and only if i∈ [1, r] ∪ [k − r + 1, k].

• ui ∈ D2 if and only if i∈ [r + 1, 2r] ∪ [k − 2r + 1, k − r].

• ui ∈ D3 if and only if i /∈ [1, 2r] ∪ [k − 2r + 1, k].

Let L: D1 ∪ D2 → Σout assign output labels to D1∪ D2. We say that L is extendible

w.r.t. P if there exists a complete labelingL_⋄ of P such thatL_⋄ agrees withL on D1∪ D2,

and L_⋄ is locally consistent at all vertices in D2∪ D3.

An Equivalence Class. We define an equivalence class ∼ for the directed paths (i.e.,⋆ the set of all non-empty strings in Σ∗_in), as follows.

Consider two directed paths P = (u1, . . . , ux) and P′ = (v1, . . . , vy), and let ξ(P ) = (D1, D2, D3) and ξ(P′) = (D′1, D′2, D3′). Consider the following natural 1-to-1 corre-

spondence ϕ : (D1∪ D2)→ (D′1∪ D′2) defined as ϕ(ui) = vi and ϕ(ux−i+1) = vy−i+1 for each i ∈ [1, 2r]. The 1-to-1 correspondence is well-defined so long as (i) x = y or (ii)

x ≥ 4r and y ≥ 4r. We have P ∼ P⋆ ′ if and only if the following two statements are met:

• Isomorphism: The 1-to-1 correspondence is ϕ well-defined, and for each ui ∈

D1∪ D2, the input label of ui is identical to the input label of ϕ(ui).

• Extendibility: Let L be any assignment of output labels to vertices in D1∪ D2,

and let _L′ be the corresponding output labeling of D′_{1∪ D2}′ under ϕ. Then _{L is} extendible w.r.t. P if and only if _L′ is extendible w.r.t. P′.

Note that for the special case of x _{≤ 4r, we have P ∼ P}⋆ ′ if and only if P is identical to P′.

Define Type(P ) as the equivalence class of P w.r.t. ∼. The following technical lemma⋆ is analogous to Lemma 3.1 in a specialized setting. We only use this lemma to prove the lemmas in Section 4.2.

Lemma 4.1. Let G be a path graph or a cycle graph where all vertices have input labels from Σin. Let P be a directed subpath of G, and let P′ be another directed path such that

Type(P′) = Type(P ). We write ξ(P ) = (D1, D2, D3) and ξ(P′) = (D1′, D′2, D′3). Let L⋄ be

any complete labeling of G such that_L⋄ is locally consistent at all vertices in D2∪ D3. Let

G′ = Replace(G, P, P′) be the graph resulting from replacing P with P′ in G. Then there exists a complete labeling L′_⋄ of G′ such that the following two conditions are met.

1. For each v _{∈ (V (G) \ V (P ))∪(D}1∪D2) and its corresponding v′ ∈ (V (G′)\ V (P′))∪

(D′₁∪ D₂′), we have L_⋄(v) =L′_⋄(v′). Moreover, if v∈ (V (G) \ V (P )) ∪ D1 and L⋄ is

locally consistent at v, then _L′_⋄ is locally consistent at v′. 2. L′_⋄ is locally consistent at all vertices in D₂′ ∪ D′₃.

Proof. The labeling_L′_⋄(v′) of G′ for each v′ _{∈ (V (G}′)\ V (P′))∪ (D′₁∪ D₂′) is chosen “nat- urally” as follows. For each v′ _{∈ V (G}′)\ V (P′), we setL′_⋄(v′) =L_⋄(v) for its corresponding vertex v∈ V (G) \ V (P ). For each v′ ∈ D₁′ ∪ D′₂, we set L′_⋄(v′) = L_⋄(v) for its correspond- ing vertex v ∈ D1 ∪ D2 such that ϕ(v) = v′ in the definition of

⋆

clear that if v _{∈ (V (G) \ V (P )) ∪ D}1 has a locally consistent labeling under L⋄, then its

corresponding vertex v′ _{∈ (V (G}′)\ V (P′))∪ D′₁ also has a locally consistent labeling under

L′

⋄, so Condition 1 holds.

Now, the labeling_L′_⋄ is only undefined for vertices in D′₃. We show that we can complete the labeling in such a way that is locally consistent at all vertices in D₂′ ∪ D′₃. DenoteL as

L⋄ restricted to D1∪D2. Since L⋄ is locally consistent at all vertices in P , the labelingL is

extendible w.r.t. P . Note that if we letL′beL_⋄ restricted to D₁′∪D′₂, then according to the way we defineL′_⋄, the two labeling L′ and L are identical under the 1-to-1 correspondence

ϕ specified in the definition of ∼. That is, for each v⋆ ′ ∈ D₁′ ∪ D′₂, we have L′(v′) =L(v) for its corresponding vertex v _{∈ D}1∪ D2 such that ϕ(v) = v′. Since P

⋆

∼ P′_{, the labeling}

L′ _{must be extendible w.r.t. P}′_{. That is, there is a way to assign L}′

⋄(v′) for each v′ ∈ D3′

such that all vertices in D′_{2∪ D3}′ have locally consistent labelings under_L′_⋄, so Condition 2 holds.

One useful consequence of this lemma is that if we start with a path or a cycle G with a legal labeling, after replacing its subpath P with another one P′ having the same type as

P , then it is always possible to assign output labeling to P′ to get a legal labeling without changing the already-assigned output labels of vertices outside of P′.

Lemma 4.2. Let G be a path graph or a cycle graph where all vertices have input labels from Σin. Let P be a directed subpath of G, and let P′ be another directed path such that

Type(P′) = Type(P ). Let L_⋄ be complete labeling of G that is locally consistent at all vertices in P . Let G′ = Replace(G, P, P′) be the graph resulting from replacing P with P′

in G. Then there exists a legal labeling L′_⋄ of G′ such that the following two conditions are met.

1. For each v ∈ V (G) \ V (P ) and its corresponding v′ ∈ V (G′)\ V (P′), we haveL_⋄(v) =

L′

⋄(v′). Moreover, if L⋄ is locally consistent at v ∈ V (G) \ V (P ), then L′⋄ is locally

consistent at v′.

2. L′_⋄ is locally consistent at all vertices in P′.

Proof. We write ξ(P′) = (D₁′, D′₂, D₃′). Condition 1 in this lemma is implied by Condition 1 in Lemma 4.1. To see that Condition 2 in this lemma holds, notice that in this lemma we additionally require thatL_⋄ is locally consistent at all vertices in P . Therefore, Condition 1 of Lemma 4.1 implies that L′_⋄ is locally consistent at all vertices in D₁′. This observation,

together with Condition 2 of Lemma 4.1, implies that_L′_⋄ is locally consistent at all vertices in P′.

The following lemma is analogous to Theorem 3.4 in a specialized setting. We only use this lemma in Section 4.2.

Lemma 4.3. Let P = (v1, . . . , vk), and let P′ = (v1, . . . , vk−1). Let the input label of vk be

α. Then Type(P ) is a function of α and Type(P′).

Proof. We prove the following stronger statement. Let G be a directed path, and let H be a

directed subpath of G. Suppose H′is another directed path satisfying Type(H) = Type(H′). Let G′ = Replace(G, H, H′) be the result of replacing H with H′ in G. Then we claim that Type(G) = Type(G′). The lemma is a corollary of this claim.

Consider the tripartitions ξ(H) = (B1, B2, B3), ξ(H′) = (B1′, B2′, B3′), ξ(G) =

(D1, D2, D3), and ξ(G′) = (D1′, D′2, D3′). We write B0 = V (G) \ V (H) and B′0 =

V (G′)\ V (H′).

Let ϕ⋆ _{be the natural 1-to-1 correspondence from B}

0∪ B1∪ B2 to B0′ ∪ B1′ ∪ B2′. Note

that D1∪ D2 ⊆ B0∪ B1∪ B2 and D1′ ∪ D2′ ⊆ B0′ ∪ B1′ ∪ B′2. Also, the 1-to-1 correspondence

between D1∪ D2 and D′1∪ D′2 given by ϕ⋆ is exactly the 1-to-1 correspondence ϕ specified

in the requirement of G_{∼ G}⋆ ′.

Let _{L: (D}1∪ D2) → Σout and let L′ be the corresponding output labeling of D′1∪ D2′,

under the 1-to-1 correspondence ϕ. To show that G _{∼ G}⋆ ′, all we need to do is show that

L is extendible w.r.t. G if and only if L′ _{is extendible w.r.t. G}′_{. Since we can also write}

G = Replace(G′, H′, H), it suffices to show just one direction, i.e., if L is extendible then L′ _{is extendible.}

Suppose L is extendible. Then there exists an output labeling L_⋄ of G such that (i) for each v ∈ D1∪ D2, we have L⋄(v) = L(v), and (ii) L⋄ is locally consistent at all vertices

in D2∪ D3. Since D2∪ D3 ⊇ B2 ∪ B3, we can apply Lemma 4.1, which shows that there

exists a complete labeling _L′_⋄ of G′ such that the two conditions in Lemma 4.1 are met. We argue that this implies that_L′ is extendible. We verify that (i)_L′(v′) =L′_⋄(v′) for each

v′ ∈ D′₁∪ D₂′, and (ii) _L′_⋄ is locally consistent at all vertices in D₂′ _{∪ D}′₃.

• Condition 1 of Lemma 4.1 guarantees that _L⋄(v) = L′_⋄(ϕ⋆_{(v)) for each v ∈} (V (G)\ V (H)) ∪ (B1 ∪ B2) = B0 ∪ B1 ∪ B2 and its corresponding vertex ϕ⋆(v) ∈

B′₀ ∪ B′₁ ∪ B′₂. Since D₁′ ∪ D′₂ ⊆ B′₀ ∪ B′₁ ∪ B₂′, we have L′(v′) = L′_⋄(v′) for each

• The fact that _L⋄ is locally consistent at all vertices in D2∪ D3, together with Con-

dition 1 in Lemma 4.1, guarantees that _L′_⋄ is locally consistent at all vertices in (D′₂∪ D′₃)\ B₃′. Condition 2 in Lemma 4.1 guarantees that _L′_⋄ is locally consistent at all vertices in B₂′ _{∪ B3}′. Therefore, _L′_⋄ is locally consistent at all vertices in D₂′ _{∪ D3}′, as required.

The number of types can be upper bounded as follows.

Lemma 4.4. The number of equivalence classes of ∼ (i.e., types) is at most |Σ⋆ in|4r2|Σout|

4r

. Proof. Let P be a directed path, and let ξ(P ) = (D1, D2, D3). Then Type(P ) is determined

by the following information.

• The input labels in D1∪D2. Note that there are at most|Σin|4rpossible input labeling

of D1∪ D2.

• A length-x binary string indicating the extendibility of each possible output labeling of D1∪ D2, where x = |Σout|4r.

Therefore, the number of equivalence classes of _{∼ is at most |Σ}⋆ in|4r2|Σout|

4r

.

Define ℓpumpas the total number of types. Observe that Lemma 4.3 implies that Type(P )

can be computed by a finite automaton whose number of states is the total number of types, which is a constant independent of P . Thus, we have the following two pumping lemmas which allow us to extend the length of a given directed path P while preserving the type of P . The following two lemmas follow from the standard pumping lemma for regular language.

Lemma 4.5. Let P _{∈ Σ}k

in with k ≥ ℓpump. Then P can be decomposed into three substrings

P = x◦ y ◦ z such that (i) |xy| ≤ ℓpump, (ii) |y| ≥ 1, and (iii) for each non-negative integer

i, Type(x◦ yi_{◦ z) = Type(P ).}

Lemma 4.6. For each w ∈ Σ>0

in , there exist two positive integers a and b such that

a + b≤ ℓpump, and Type(wai+b) is invariant for each non-negative integer i.