Graph Isomorphism in Quasipolynomial Time

• p. 84 Acknowledgments, last paragraph

  8.1.1 (Simple quotients of subdirect products)

Judge's note: Viewed crop C111-a (page 84) and crop C112-b (page 60), and checked the contents and Section 8 in the text. Section 8.1 is titled 'Simple quotient of subdirect product' (singular) and has no subsubsections; 8.1.1 is the number of the lemma of the same name. The pointer should read Lemma 8.1.1 or Section 8.1. A reader still finds the place, so minor.

Second check: another model agreed

• p. 5 Sec. 1.1.2, last sentence

  f(n) ≤q1(x)log2 n

• p. 33 Sec. 3.4, Eq. (22)

  T(x) = q1(x)O(log2 x)

Judge's note: Page 5 image shows f(n) ≤ q1(x) with exponent log² n, no O(). The same paragraph writes q(n)^{O(log n)} for the single recursion, and Eq. (22) on page 33 has O(log² x). This is an informal overview sentence, so it is a notational slip rather than a wrong result.

Second check: another model agreed

• p. 7 Sec. 1.2.2, second paragraph

  We shall set this threshold at c log n for some constant c.

• p. 5 Sec. 1.1.2

  When m drops below a threshold ℓ(n) that is polylogarithmic in n

• p. 79 Procedure String-Isomorphism, step 2

  set ℓ= (log n)3

Judge's note: Seen on the images of pages 7 and 5 and the crop of page 79. The three statements describe the same threshold and do not agree. The introduction is an informal overview, and the remark near Theorem 8.5.1 says a bound of the form c log n0 would suffice for the group theory (for k), so either form could be defended; it is an inconsistency, not an error in the algorithm.

Second check: another model agreed

• p. 46 Lemma 7.4.1

  2/3 ≤ β ≤ 1 be a threshold parameter

• p. 45 Theorem 7.3.1

  let 2/3 ≤ β < 1 be a threshold parameter

• p. 10 Sec. 1.3.3, after Theorem 1.3.3

  the result would remain valid for any constant 0.5 < α < 1

• p. 46 Theorem 7.3.3

  Let 3/4 ≤α < 1 be a threshold parameter.

Judge's note: Page 46 image: Theorem 7.3.1 (stated p. 45) has β < 1, Lemma 7.4.1 has β ≤ 1; with β = 1 the lemma is vacuous, so this is harmless. Page 52 image: "with α := 2/3" for a lemma whose parameter is β. Both are minor notation inconsistencies. The third point (p. 10: "the result would remain valid for any constant 0.5 < α < 1" against Theorems 7.3.1 and 7.3.3, stated for ≥ 2/3 and ≥ 3/4) is an informal remark about Theorem 1.3.3 that may be true by adjusting the proofs; the later theorems do not state that range, but this is not shown to be an error.

Second check: another model agreed

• p. 19 Proof of Lemma 2.5.9, definition of Nj(x)

  Let Nj(x) = {y ∈ C2 | c(x, y) = j}.

• p. 19 Proof of Lemma 2.5.10

  Therefore, M(x, y) = ⋃{Nj(x) | j ∈ L}

Only the authors can say what is right here.

Judge's note: Checked page 19. Lemma 2.5.9's proof defines Nj(x) only for y ∈ C2. In Lemma 2.5.10 the colours j ∈ L satisfy Rj ⊆ C1×C1, so under the printed definition every Nj(x) with j ∈ L is empty, and the equality with the nonempty set M(x,y) (it contains x) fails as printed. The proof evidently means the generalised set {z | c(x,z)=j}, whose size is also dj, and with that reading the size count and the lemma still hold. So the step is a notation slip and the stated result stands (S3).

Second check: another model agreed

• p. 32 Sec. 3.4

  C1 is an absolute constant. For x < 2 we set T(x) = Ttrans(x) = 1. For x ≥ C0 (an absolute constant)

Only the authors can say what is right here.

Judge's note: Read all of Sec. 3.4 on pages 32-33: C1 appears only in its declaration and in no recurrence; the threshold is called C0 and introduced again as an absolute constant. Dangling or misnamed constant; no effect on the results.

Second check: another model agreed

• p. 39 Proof of Observation 6.1.2, Case 1

  is an α-coloring (since α ≥1/2), and therefore

Only the authors can say what is right here.

Judge's note: p.39. An α-coloring means every colour class has size ≤ α|Γ| (defined in the text, extract line 1956). With |W| > αn the class W breaks this. The intended conclusion (an α′-coloring) holds directly: |W| < α′n and |Ω\W| ≤ (1−α)n ≤ α′n. The step is false as printed; the result stands.

Second check: another model agreed

• p. 42 Proposition 6.2.3 hypotheses

  Let 3 ≤k ≤n/4.

• p. 54 Section 7.11

  Assume now m > 2ℓ.

• p. 58 Section 7.11.2, Case B1

  Ωi ←Γi, n ←m, k ←ℓ.

• p. 58 Case B1

  Ωi ←Γi, n ←m, k ←ℓ

Only the authors can say what is right here.

Judge's note: Checked p.42 (Prop. 6.2.3 hypothesis 3 ≤ k ≤ n/4), p.54 ("Assume now m > 2ℓ"), and pp.57–58 (Case B: no further assumption on m; p.58 assigns n ← m, k ← ℓ). For 2ℓ < m < 4ℓ the stated hypothesis fails. Whether the proposition actually needs k ≤ n/4 cannot be settled here; the authors should state the threshold or weaken the hypothesis.

Second check: another model agreed

• p. 46 Theorem 7.3.2

  let 2/3 ≤ α < 1 be a threshold parameter

• p. 48 Procedure Bipartite Split-or-Johnson, Input

  a threshold parameter 3/4 ≤ α < 1

• p. 49 Procedure step 7 comment

  The symmetry defect of X on V1 is ≥ 1/3 > 1 − α

• p. 52 Section 7.9 (II)

  Recursively apply Theorem 7.3.2 to X′ with threshold parameter α = 2/3.

Only the authors can say what is right here.

Judge's note: Checked p.46 (Thm 7.3.2: 2/3 ≤ α; Lemma 7.4.1 returns α ≥ 2/3), p.48 (procedure input 3/4 ≤ α < 1; Sec. 7.5 says it proves Thm 7.3.2), p.49 (step 7 comment) and p.52 (call at α = 2/3). The ranges disagree. At α = 2/3, 1/3 > 1 − α is false, though ≥ 1 − α, which is what the hypothesis needs, still holds. Only the authors can say whether the procedure is meant for α ≥ 2/3.

Second check: another model agreed

• p. 50 Lemma 7.7.1

  at only additive polynomial cost (no multiplicative cost) a canonical bipartite graph

• p. 50 Procedure ImprimitiveCase, step 2(i)

  This involves choosing i at a multiplicative cost of m.

• p. 50 Lemma 7.7.1 statement

  return, at only additive polynomial cost (no multiplicative cost) a canonical bipartite graph

Only the authors can say what is right here.

Judge's note: p.50. The lemma covers both subcases. Subcase 2(i) incurs multiplicative cost m ≤ n2/2; subcase 2(ii) (contraction) does not. Polynomial cost, so the bounds are unaffected, but the lemma statement does not match the procedure.

Second check: another model agreed

• p. 52 Section 7.8, Case 3b

  Let t = ⌈(7/4) log2 n1⌉. So t ≤(3/4)d1.

Only the authors can say what is right here.

Judge's note: pp.51–52 and Lemma 2.6.8 (extract line 1334). With n1 = 7000 (log2 ≈ 12.773), d1 = 30 > 29.80 is in Case 3b, but t = 23 > 22.5. The step is false as printed. Lemma 2.6.8's proof only uses |S∩E| ≥ (3/4)d ≥ t with integer |S∩E|, so t ≤ ⌈(3/4)d1⌉ suffices, and that does follow. The result stands.

Second check: another model agreed

• p. 54 Section 7.11, situation paragraph

  and therefore t < 2 log n2/ log m

Only the authors can say what is right here.

Judge's note: p.54. The Johnson scheme only requires m ≥ 2t+1 (extract line 1196). For m = 65, t = 32, log2 C(65,32) ≈ 61.66 and 2·61.66/log2 65 ≈ 20.5 < 32. The paper's own Lemma 7.11.1 proof (p.56) uses C(a,b) > a^{b/2} only for b ≤ √a. Under the later assumption m > 2ℓ we have t ≤ log2 n2 < √m, so the uses downstream hold.

Second check: another model agreed

• p. 67 Corollary 9.0.10, statement

  by recursively calling ≤ 4k instances

• p. 67 Proof of Corollary 9.0.10

  calling 3k instances of window size n/k

• p. 67 Proof of Corollary 9.0.10, last sentence

  reducing G-isomorphism to two instances of G1-isomorphism

• p. 67 Corollary 9.0.10 statement and proof

  recursively calling ≤4k instances

Only the authors can say what is right here.

Judge's note: p.67. The A(Γ) case gives 3 N-instances × k orbits = 3k. The S(Γ) case reduces to two G1 instances, each handled as the A(Γ) case, so 6k > 4k. Cor. 9.0.12 on the same page states ≤ 6k. Only a constant is affected.

Second check: another model agreed

• p. 74 Theorem 10.2.1 statement

  nO(1) instances of size ≤2n/3

• p. 74 Proof, item 1

  each N-orbit on Ωhas size ≤4n/5

Only the authors can say what is right here.

Judge's note: Page image p. 74 shows both texts. Cor. 5.2.2 (p. 30 text) gives a max(2/3, α)-coloring, which for the 4/5-partition of step 1 is 4/5, not 2/3. Only branch 2b1 (via Cor. 9.0.13, p. 68, "maximum size ≤2n/3") yields the 2n/3 bound. I found no statement that reconciles the two: Sec. 11.2 treats classes above 2n/3 separately, but for Procedure Align, not for step 1. Step 1 could be read as the first outcome (a 4/5-partition of Γ found after individualisation) processed at once, and a 4n/5 bound does not change the quasipolynomial analysis, so this is a mismatch between the statement and one branch of the proof, not a wrong result. Only the author can say which reading is intended.

Second check: another model agreed

• p. 79 Master Algorithm, step 4, t=1 case

  G = A(Ω); isomorphism only depends on the multiplicity of each letter

Judge's note: Checked page 79. For G = A(Ω), if every letter occurs exactly once, two strings with the same multiplicities can differ by an odd permutation, and then they are not A(Ω)-isomorphic. Equal multiplicities are enough only when some letter repeats, or when G = S(Ω). The case is still trivially solvable in polynomial time once a parity check is added, so this is a wording slip in an algorithm comment (S3).

Second check: another model agreed

• p. 1 Footnote *

  CCF-7443327 (2014-current)

Only the authors can say what is right here.

Judge's note: Rests on outside knowledge: that NSF award numbers of this era start with the two-digit fiscal year (so a 2014 award would not start with 74), and the reviewer's recollection of the real award number. The page image does print CCF-7443327; whether it is wrong, and what the right number is, only the author can say.

Second check: none (judge only)

• p. 1 Footnote *

  (2010-2014), and CCF-0830370 (2008–2010)

Judge's note: Seen on the page 1 image: the first two ranges have a short hyphen, the third a longer en dash. Purely typographic.

Second check: none (judge only)

• p. 6 Sec. 1.2.1, last paragraph

  consist of those σ ∈G that satisfy

Judge's note: Read on the page 6 image.

Second check: none (judge only)

• p. 7 Sec. 1.2.2

  we are able to infer global automorophisms

• p. 7 Sec. 1.2.2, penultimate paragraph

  global automorophisms

Judge's note: Read on the page 7 image; the parenthesis in the same sentence has the correct spelling.

Second check: none (judge only)

• p. 7 Sec. 1.2.2

  One of our main technical result says

Judge's note: Read on the page 7 image.

Second check: none (judge only)

• p. 8 Sec. 1.3.1, first sentence

  our main group theoric tool

Judge's note: Read on the page 8 crop; the section heading directly above has "group-theoretic".

Second check: none (judge only)

• p. 9 Theorem 1.3.1(b)

  (Affected Orbits Lemma)

• p. 63 Corollary 8.3.7

  Corollary 8.3.7 (Affected Orbit Lemma)

Judge's note: Page 9 image has the plural; the page 63 crop has the singular. The text layer also shows the plural again in Section 10 ("By Cor. 8.3.7 (“Affected Orbits Lemma”)"). Cosmetic naming inconsistency.

Second check: none (judge only)

• p. 13 Sec. 2.2, before Theorem 2.2.3

  We quote if from [DiM, Thm. 5.4A].

• p. 13 before Theorem 2.2.3

  We quote if from [DiM, Thm. 5.4A]

Judge's note: Read on the page 13 crop.

Second check: none (judge only) · checked by hand

• p. 14 Observation 2.3.6

  is an t-ary

• p. 15 Proof of Observation 2.4.9

  such that the 3-cyle (x, y, z)

• p. 44 Proof of Corollary 7.1.3

  contrary assumption.

• p. 48 Procedure Bipartite Split-or-Johnson

  Output: Output: item (a) or (b) of Theorem 7.3.2.

Judge's note: Each of the six read on its full page image and present as quoted. All cosmetic. C017 and C019 repeat two of these items.

Second check: none (judge only)

• p. 17 Definition of binary configuration, item (ii)

  (∀i)(Ri ⊆ diag(Ω) or Ri ∩ diag(Ω) = ∅

Judge's note: Page 17 image: item (ii) ends with ∅ and no closing bracket, while item (iii) directly below is balanced.

Second check: none (judge only)

• p. 17 after item (iii)

  We write j = i− if R−i = Rj).

Judge's note: Page 17 image: the parenthetical "(This is the binary case ... Section 2.3.)" is already closed before this sentence, so the final ")" has no opening partner.

Second check: none (judge only)

• p. 17 Proof of Observation 2.5.2

  c(z′, y′) = i.

Judge's note: Page 17 image: the sentence ends "= i." with no closing bracket.

Second check: none (judge only)

• p. 20 Proof of Lemma 2.5.15, Necessity

  So J ∩ W = ∅ if j ≠ i and J ∩ W = ∅ if j ≠ i and J ∩ W = {y} if j = i

• p. 20 Proof of Lemma 2.5.15, Necessity

  J ∩ W = ∅ if j ̸= i and J ∩ W = ∅ if j ̸= i

Judge's note: Page 20 image: the repetition is in the typeset text across a line break, not an extraction artefact. The Sufficiency part states the clause once.

Second check: none (judge only)

• p. 21 Proof of Lemma 2.5.18

  a contradition, proving item (a)

• p. 21

  hence x, y are strong twins, a contradition, proving item (a)

Judge's note: Page 21 image shows "contradition"; the word is spelled correctly elsewhere in the same proof.

Second check: none (judge only)

• p. 23 Johnson scheme definition

  t-subsets of an k-set Γ

Judge's note: Page 23 image shows "an k-set Γ".

Second check: none (judge only)

• p. 29 before Eq. (17)

  The following straighforward identity

Judge's note: Page 29 image shows "straighforward" (missing t).

Second check: none (judge only)

• p. 30 Proof of Observation 3.1.10

  the restiction of the elements of S to ∆

• p. 30 Proof of Observation 3.1.10

  the restiction of σ to ∆belongs to

Judge's note: Page 30 image shows "restiction" in both places.

Second check: none (judge only)

• p. 32 Sec. 3.4

  assuming x is sufficently large

Judge's note: Page 32 image shows "sufficently" (the same paragraph has "sufficiently large x" spelled correctly a few lines above).

Second check: none (judge only)

• p. 36 Proposition 5.1.3

  if the partition was admissibe

• p. 36

  so if the partition was admissibe, it remains admissible

Judge's note: Page 36 image shows "admissibe" followed by the correct "admissible" in the same sentence.

Second check: none (judge only)

• p. 36 Observation 5.1.4

  (at leat least two blocks

• p. 36

  (at leat least two blocks, the blocks have size ≥2)

Judge's note: Page 36 image shows "at leat least two blocks" exactly as quoted: a misspelling plus a duplicated word.

Second check: none (judge only)

• p. 36 category ColoredPartitions

  The morphisms are the bijection that preserve color

Judge's note: Page 36 image, paragraph after Definition 5.1.1: singular "bijection" with plural verbs "are" and "preserve".

Second check: none (judge only)

• p. 38 Proof of Prop. 5.2.3

  if 1 ≤ s ≤ mi − 1 and 1 ≤ t − s ≤ m2 − 1

Judge's note: Page 38 image: the subscript is clearly i in "1 ≤ s ≤ m_i − 1" both after the definition of a_s and again in "under the conditions 1 ≤ s ≤ m_i − 1 and 1 ≤ t − s ≤ m_2 − 1". With a_s = C(m_1,s)C(m_2,t−s) and t_1 = s, t_2 = t − s, the paired condition must be on m_1; the index i is not bound there. A subscript slip that a reader corrects easily and that does not affect the argument.

Second check: none (judge only)

• p. 40 Proof of Theorem 6.1.5

  otherwise we succed on line 02

Judge's note: Page 40 image, proof of Theorem 6.1.5, second paragraph: "succed" is printed.

Second check: none (judge only)

• p. 40 after Theorem 6.1.5

  then the configurations X∗(S) is not a clique configuration

Judge's note: Page 40 image, paragraph after Theorem 6.1.5: plural noun with singular verb, as quoted.

Second check: none (judge only)

• p. 41 Proof of Claim 6.1.8, Case 2

  So if X∗(Q) cannot be a clique; Case 2 cannot occur.

Judge's note: Page 41 image, end of Case 2: printed as quoted. Intended sense is evidently "So X*(Q) cannot be a clique; Case 2 cannot occur."

Second check: none (judge only)

• p. 41 Claim 6.1.8, Case 1

  |C \ Q| > αn > n/2 This shows that Case 1 cannot occur.

Judge's note: Page 41 image, end of Case 1: the line ends with "> n/2" with no punctuation and the next line starts a new sentence "This shows...".

Second check: none (judge only)

• p. 42 Proposition 6.2.3

  we can either refute isomorphism is X1 and X2 find a pair of canonical

• p. 42 Proposition 6.2.3

  refute isomorphism is X1 and X2 find a pair of canonical k-ary relational structures

Judge's note: Page 42 image shows the wording as quoted: "is" in place of "of" and the "or" of the either/or missing. The intended meaning is recoverable from the proof ("exit with rejection") and from the application on page 58, so this is a copy-edit error in a proposition statement, not a change of content.

Second check: none (judge only)

• p. 42 Sec. 6.2, first paragraph

  (the case of Johnson graphs) and and Case 2c

• p. 47 Proof of Lemma 7.4.1

  is between 1/2 and and α

• p. 69 Definition 10.1.4

  be a permutation group and and ϕ

• p. 42

  (the case of Johnson graphs) and and Case 2c of the proo

Judge's note: All three confirmed on the page images (42, 47, 69). Page 47 also prints "apply Theorem 7.3.2 to X to obtain either obtain a canonical colored α-partition", a further duplicated word in the next paragraph.

Second check: code check

• p. 42 Sec. 6.2

  namely, an objects Xi(L)

Judge's note: Page 42 image shows "namely, an objects X_i(L) for every L".

Second check: none (judge only)

• p. 42 Proof of Prop. 6.2.3

  the set of odered k-tuples

• p. 42

  denote the set of odered k-tuples of distinct elements

Judge's note: Page 42 image, proof of Prop. 6.2.3: "odered" is printed.

Second check: none (judge only)

• p. 43 End of Sec. 6.2

  This completes the proof of

Judge's note: Page 43 image: the sentence ends after "of", followed only by the end-of-proof box and then the Section 7 heading. Nothing is lost to extraction.

Second check: none (judge only)

• p. 43 Sec. 7 introduction

  Canonical auxiliary structres are constructed

Judge's note: Page 43 image, Sec. 7 introduction, shows "structres".

Second check: none (judge only)

• p. 45 Proof of Proposition 7.2.6

  we may assume the density of Y is |E|/(|V1||V2|) ≤

Judge's note: Page 45 image: the proposition is stated for X = (V1, V2; E) and the proof refers to "the density of Y". The same proof also prints "Let x ∈ Vi be adjacent to y", where x ∈ S is needed for the next sentence to follow. The "if S" slip in the same proof is covered by C067.

Second check: none (judge only)

• p. 45 Sec. 7.2

  “twin-or-equal” realtion is an equivalence relation

• p. 45

  twin-or-equal” realtion is an equivalence relation

Judge's note: Page 45 image, first line, shows "realtion".

Second check: none (judge only)

• p. 45 Proof of Prop. 7.2.6

  y is adjacent to all vertices if S

Judge's note: Page 45 image shows "all vertices if S".

Second check: none (judge only)

• p. 46 Lemma 7.4.1 (ii)

  by computig a threshold

Judge's note: Page 46 image shows "computig".

Second check: none (judge only)

• p. 47 Proof of Lemma 7.4.1

  between 1/2 and and α

• p. 47

  ect of X in C2 is between 1/2 and and α then we have a ca

Judge's note: Page 47 image shows "and and" within a single line, so it is not a line-break artefact.

Second check: code check

• p. 47 Proof of Lemma 7.4.1

  to obtain either obtain a canonical colored α-partition

Judge's note: Page 47 image shows "to obtain either obtain".

Second check: none (judge only)

• p. 48 Procedure Bipartite Split-or-Johnson

  Output: item (a) or (b) of Theorem 7.3.2.

Judge's note: Page 48 image: "Output:" appears in the label column and again at the start of the description on the same line. It is in the page, not an extraction artefact.

Second check: none (judge only)

• p. 49 Procedure Bipartite Split-or-Johnson, step 7

  dure selects j ∈ {1, 2}. Update X and Xi to their induced subconfigurations on V1 ∪ Ci

Judge's note: Page 49 image shows the text as quoted. Reduce-Part2-by-Color (p. 47) outputs the index j of the selected part, so the update is to V1 ∪ Cj; i in the same sentence is the running index of X_i and of "the Ci are nonempty". An index slip with an evident fix.

Second check: none (judge only)

• p. 49 Remark 7.5.1

  could be helpful7.)

Judge's note: Checked the full page image of PDF page 49. The text reads "helpful7.). We only land in case (iv)", so there is a full stop inside the parenthesis after the footnote mark and another after the parenthesis. The inner one is superfluous.

Second check: none (judge only)

• p. 51 Sec. 7.8, Case 1

  V1 can be indentified with

• p. 51

  V1 can be indentified with V1

Judge's note: Checked the page image of PDF page 51; the misspelling "indentified" is printed under Case 1.

Second check: none (judge only)

• p. 52 Sec. 7.9, item (II)

  ular bipartite graph X′′ = (V′1, V′2; E′)

• p. 52 Sec. 7.9, item (II)

  Recursively apply Theorem 7.3.2 to X′ with threshold parameter α = 2/3.

• p. 52 Sec. 7.9 (II)

  semireg-

• p. 52 Sec. 7.9 (II)

  Recursively apply Theorem 7.3.2 to X′

Judge's note: p.52 image: X′′ is introduced and X′ is used in the next sentence. Evidently the same graph.

Second check: none (judge only)

• p. 52 Sec. 7.9, item (II)

  Johnson scheme on a subset W of V2 of size

• p. 52 Sec. 7.9, item (II)

  to the canonical 2-coloring (W2, V2 \ W2) of V2

• p. 52 Sec. 7.9, item (II)

  with W := V′1 ∪ V′2 ⊆ V2

Judge's note: p.52 image confirms that the symbol W is used for two different sets and that W2 is never defined. Notational inconsistency; the meaning is recoverable.

Second check: none (judge only)

• p. 57 Case A, Sec. 7.11.2

  its counterpart airising from the other input

Judge's note: Checked the page image of PDF page 57; "airising" is printed in the parenthetical under Case A.

Second check: none (judge only)

• p. 57 Sec. 7.11.2

  let X[L] denote the subgraph of X induced by

• p. 57 Sec. 7.11.2

  Determine the isomorphisms among all the X(L) for

Judge's note: Checked page images of PDF pages 57 and 58. X[L] is defined and used in the Case A and Case B headings ("Not all the X[L] are isomorphic", "All the X[L] are isomorphic"), while X(L) is used in the running text ("isomorphism type of X(L)", "Aut(X(L))", "Since all the X(L) are isomorphic"). Both plainly denote the same induced subgraph, so the meaning is not in doubt; it is a notational inconsistency only.

Second check: none (judge only)

• p. 88 Reference [Me]

  Ulrich Meierfrankefeld: Non-finitary locally finite simple groups

• p. 60 Lemma 8.1.3

  Lemma 8.1.3 (Meierfrankenfeld)

• p. 88 Reference [Me]

  Ulrich Meierfrankefeld

• p. 88 Reference [McP]

  Practical Graph Isomoprhism, II

Judge's note: Viewed page 88 and crop C112-b (page 60). The reference entry lacks the second 'n' that the lemma heading has, and it is also the only entry on the page set in sans-serif instead of small capitals. The 'Isomoprhism' misspelling that one reviewer note adds is handled in C114.

Second check: none (judge only)

• p. 61 footnote 8

  (see. e. g., [HoS])

Judge's note: Checked the crop of PDF page 61. The footnote reads "adopted in computational group theory (see. e. g., [HoS])." with a full stop directly after "see".

Second check: none (judge only)

• p. 65

  namely, the idenity.

Judge's note: Checked the crop of PDF page 65; the last line of the remark prints "idenity".

Second check: none (judge only)

• p. 66 Before Proposition 8.5.4

  Finally, and algorithmic observation.

• p. 66 Definition 9.0.7

  Note that ay intersection of subcosets

Judge's note: Checked the page image of PDF page 66; the first line reads "Finally, and algorithmic observation." The second typo one reviewer listed here ("ay intersection") is real too and is recorded under C089.

Second check: none (judge only)

• p. 66 Def. 9.0.7

  Note that ay intersection of subcosets

• p. 66

  (Note that ay intersection of subcosets of G

Judge's note: Checked the page image of PDF page 66; Definition 9.0.7 prints "Note that ay intersection of subcosets".

Second check: none (judge only)

• p. 67 Proof of Cor. 9.0.12

  We claim that if If IsoG(x, y)ϕ

Judge's note: Page image p. 67 shows "if If" as typeset. Fix: delete one "if".

Second check: none (judge only)

• p. 68 Corollary 9.0.13 statement

  a giant representation. where |Γ| = m

Judge's note: Page image p. 68 shows the full stop followed by lower-case "where". The parallel Corollaries 9.0.10 and 9.0.12 on p. 67 have a comma here.

Second check: none (judge only)

• p. 69 Definition 10.1.4

  permutation group and and ϕ

• p. 69

  ≤S(Ω) be a permutation group and and ϕ : G →S(Γ) a homom

Judge's note: Page image p. 69 shows "and and" as typeset.

Second check: code check

• p. 72 Notation 10.1.9

  x^W : |omgea → (Σ ∪ {∗})

• p. 72 Notation 10.1.9

  xW : |omgea →(Σ∪{∗})

Judge's note: Page image p. 72 shows the literal italic text "|omgea" where the domain Ω belongs (the next line uses "u ∈ Ω \ W"). It is in the typeset PDF, not an extraction artefact.

Second check: none (judge only)

• p. 73 before Prop. 10.1.13

  the minial degree of a permutation group

Judge's note: Page image p. 73 shows "minial degree" in italics. Prop. 10.1.13 just below spells it "minimal degree".

Second check: none (judge only)

• p. 76 Sec. 11 introduction

  correspondigly came at a multiplicative cost

Judge's note: Page image p. 76 shows "correspondigly".

Second check: none (judge only)

• p. 76 Sec. 11.1

  A common features of the categories

Judge's note: Page image p. 76 shows "A common features ... is that". Fix: "A common feature".

Second check: none (judge only)

• p. 77 Procedure Align, step 4

  is the disjoin union of the ∆j

• p. 77

  Γ is the disjoin union of the ∆j

Judge's note: Page image p. 77 shows "disjoin union".

Second check: none (judge only)

• p. 77 Sec. 11.2

  We are looking a function T(x)

Judge's note: Page image p. 77 shows "We are looking a function T(x)".

Second check: none (judge only)

• p. 82 Sec. 13.3, second paragraph

  Cai, Furer,

• p. 86 Reference [CaiFI]

  Martin F¨urer

Judge's note: Viewed both crops. Page 82 shows 'Furer' with a plain u; the reference list on page 86 shows 'Fürer' with the umlaut. The same page 82 sets 'Lovász' with its accent, so the missing umlaut is a slip.

Second check: none (judge only)

• p. 88 Reference [McP]

  Practical Graph Isomoprhism, II.

• p. 85 Reference [BaBS]

  Polynomial-Time Theory of Mtrix

Judge's note: Viewed crop C113-b (page 85): the title reads 'Mtrix Groups'. This cluster is kept for the [BaBS] misspelling; the 'Isomoprhism' misspelling in [McP] is C114. A reviewer's side remark that [Mat] has no year is also true on page 88 ('Info. Proc. Lett. 8 pp. 131–132.').

Second check: none (judge only)

• p. 88 Reference [McP]

  Practical Graph Isomoprhism, II.

• p. 88 reference [McP]

  Practical Graph Isomoprhism, II

Judge's note: Viewed page 88: the [McP] title reads 'Isomoprhism' with the letters r and p transposed.

Second check: none (judge only)

• p. 88 Reference [OWWZ]

  Symp. Disr. Alg. (SODA’14)

• p. 88

  Proc. 25th ACM–SIAM Symp. Disr. Alg. (SODA’14)

Judge's note: Viewed page 88: the [OWWZ] entry reads 'Symp. Disr. Alg.', an abbreviation of 'Discrete' missing its 'c'.

Second check: none (judge only)