Supplementary information Efficient Enumeration of Monocyclic Chemical Graphs with Given Path Frequencies

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1 Supplementay infomation Efficient Enumeation of Monocyclic Chemical Gaphs with Given Path Fequencies Masaki Suzuki, Hioshi Nagamochi Gaduate School of Infomatics, Kyoto Univesity {m Tatsuya Akutsu Bioinfomatics Cente, Institute fo Chemical Reseach, Kyoto Univesity S1 Poof of Lemmas and Theoems S1.1 Poof of Lemma 1 Poof: Case I. C contains the oot (centoid) of T : Since the centoid is in C, the 1-augmented tee G = T + xy has no heavy edge incident to C and we check if π(g) = T accoding to the definition of paent in Case 1. Then we compute σ (v) fo all vetices v V (T ) and σ (e) fo all simple edges e in C, as discussed in the definition of paent π in Case 1. If xy E (C) then T + xy is a child of T ; othewise T + xy is not a child of T. To compute σ, we need to know the signatue σ(t w ) of each tee T w with w N(v) and v V (T ). Since C contains the oot of T, all these tees T w appea as subtees of T ooted at w and we know thei signatue σ(t w ) fom the codes δ(τ) and M(τ) on the labeling dfs(τ). With an adequate data stuctue, we can compute σ(t w ) of each tee T w in O(1) time and testing whethe T = π(t +xy) can be done in O( V (C) 2 ) time due to veifying that c (xy) is the lexicogaphically maximum code among c (e) fo at most V (C) simple edges e in C, whose length ae 2 V (C) 1. Case II. C does not contain the oot (centoid) of T : Since the centoid is not in C, G = T +xy has a heavy edge v w incident to a vetex v in C and we check if π(g) = T accoding to the definition of paent in Case 2. In Case 2, only one of the two edges in C incident to v is emoved fom G to define its paent π(g). Hence G = T + xy can be a child of T only when xy is incident to v, i.e., one of x and y (say x) is equal to v and the othe y is a descendant of v = x (othewise we conclude that T + xy is not a child of T ). If the othe edge xz in C incident to v = x is not a simple edge, then T + xy is a child of T. When xz is a simple edge, we compute σ (xy) and σ (xz) as discussed in the definition of paent π in Case 2. Then T + xy is a child of T if and only if σ (xy) is not smalle than σ (xz). We can compute σ in the simila manne of Case I. We hee emak that afte the fist stage, we have not computed the signatue σ of any tee T w with w N(v ) since none of such tee is a subtee ooted at w in the tee T ooted at its centoid. Howeve, such tees ae ignoed in computing σ in Case 2. We can compute σ (xy) and σ (xz) in O( V (C) ) time and testing whethe T = π(t + xy) can be accomplished in O( V (C) ) time due to compaing codes c (xy) and c (xz), whose length ae 2 V (C) 1. The above pocedue is descibed as follows. Algoithm ChildTest Input: A ooted left-heavy multi-tee T and a pai (x, y) of nonadjacent vetices in V (T ). 1

2 Output: yes if T + xy is a child; no othewise. 1: if The cycle C in T + xy contains the oot (centoid) of T then /* Case I */ 2: Compute σ (v) fo all vetices v V (T ) and σ (e) fo all simple edges e in C, as in the definition of paent π in Case 1; 3: Retun yes if xy E (C) o no othewise 4: else /* Case II: Now G = T + xy has a heavy edge v w incident to a vetex v in C */ 5: if xy is not incident to v then etun no 6: else 7: if the othe edge xz in C incident to v = x is not a simple edge then etun yes 8: else /* xz is a simple edge */ 9: Compute σ (xy) and σ (xz) as in the definition of paent π in Case 2; 10: if σ (xy) is not smalle than σ (xz) then etun yes 11: else etun no 12: end if 13: end if 14: end if 15: end if S1.2 Poof of Theoem 2 Poof: Fo a tee T and two vetices x, y V (T ), let P (x, y) denote the unique path in T that connects x and y. Fo a vetex in the unique cycle C in a 1-augmented tee G, let G u denote the component containing u that appeas afte emoving the two edges in C incident to u. Hence G u is a subtee of G, and we egad it as a subtee ooted at u. Let σ (u; C) denote the signatue of G u ; σ (u; C) = σ (v; C) if and only if G u G v. When G is obtained fom a tee T by adding an edge uv, the unique cycle in G is denoted by C uv. lca(u, v) denote the least common ancesto of u and v, i.e., the highest vetex in P (u, v) (in a tee T ooted at a bicentoid c T =, the least common ancesto lca(x, y) of two vetices such that x V (T ) and y V (T ) is defined to be the edge c T = ). In a ooted tee, a pai of vetices x and y is called compaable if one of them is an ancesto of the othe. Let dfs be the depth-fist seach numbe obtained by tavesing T stating fom, whee = c T o the left endpoint of c T =. To pove this theoem, we deive a contadiction assuming that thee ae pais of nonadjacent vetices {u, v}, {u, v } V (T ) such that T + uv T + u v but T + uv T + u v. Among such pais, we choose {u, v}, {u, v } V (T ) so that dfs(u) + dfs(v) + dfs(u ) + dfs(v ) is minimized. Assume without loss of geneality that dfs(u) < dfs(v), dfs(u ) < dfs(v ) and dfs(u) dfs(u ). Note that the assumption implies that vetex lca(u, u ) has no two childen a and a in T such that u V (T a ), u V (T a ) and T a T a, (1) whee T + ûv T + u v holds fo the vetex û V (T a ) with ξ(û) = u in the ooted-isomophism ξ fom T a to T a. This is because T + uv T + ûv if and only if T + uv T + u v (by T + ûv T + u v ) and theeby if (1) holds, then (u, v) and (û, v ) would have a smalle sum of dfs than that by (u, v) and (u, v ). Similaly lca(v, v ) has no two childen a and a in T such that v V (T a ), v V (T a ) and T a T a. We fist pove that T + uv T + u v implies that lca(u, v) = lca(u, v ) = c T (note that this implies that u, u V (T ) and v, v V (T ) when c T = ). Assume indiectly that lca(u, v) c T (when c T = ) and lca(u, v) {, } (when c T = ). Then the vetex a = lca(u, v) has a unique code σ (a; C uv ) among all vetices in C uv, since the 2

3 subtee G a of G = T + uv contains c T and has moe than n/2 vetices. Hence ψ(a) is a vetex b V (C u v ) which has the same code σ (a; C uv ) = σ (b; C u v ), indicating that the subtee G b of G = T + u v must contain c T. Moeove σ (a; C uv ) = σ (b; C u v ) means that ψ maps c T = (esp., c T = ) in T + uv to c T = (esp., c T = ) in T + u v, i.e., T + uv T + u v, a contadiction to the assumption on (u, v) and (u, v ). Hence lca(u, v) = c T (when c T = ) and lca(u, v) {, } (when c T = ). Analogously, lca(u, v ) = c T o lca(u, v ) {, } (when c T = ). When c T =, this poves that lca(u, v) = lca(u, v ) = = c T. Conside the case of c T = and lca(u, v) lca(u, v ), i.e., lca(u, v) = o lca(u, v ) = (since dfs(u) dfs(u ) and dfs() < dfs( )). Assume lca(u, v) = (the case of lca(u, v ) = can be teated analogously). Then vetex = lca(u, v) still has a unique code σ (; C uv ) among all vetices in C uv, whee G contains a subtee (ooted at ) with at least n/2 vetices, and ψ() must be vetex V (C u v ) which can have the same code σ (; C uv ) = σ ( ; C u v ). Then lca(u, v ) = holds (othewise G cannot contain a subtee with at least n/2 vetices). This means that the subtees T and T of T ae ooted-isomophic, and ψ() = and ψ( ) =, i.e., T + uv T + u v, a contadiction to the assumption on (u, v) and (u, v ). This completes the poof of popety that lca(u, v) = lca(u, v ) = c T. Now lca(u, u ) and lca(v, v ) ae both on the cycles C uv and C u v. In what follows, we show that thee ae ooted-isomophic subtees T a and T b of T, whee a and b ae childen of unicentoid c T = o a = and b = of bicentoid c T =, such that u, u V (T a ), v, v V (T b ) and the ooted-isomophism ξ fom T a to T b satisfies ξ(u) = v and ξ(u ) = v. (2) Note that this implies that T + uv and T + u v admit a ooted-isomophism, which contadicts the assumption on (u, v) and (u, v ). Visiting C uv (esp., C u v ) in the clockwise ode means to tavese the cycle stating fom lca(u, u ), visiting the ancestos of lca(u, u ) and then visiting the est of vetices in the cycle. Let σ (C uv ) = (σ (t 1 ; C uv ), σ (t 2 ; C uv ),..., σ (t c ; C uv )) be the sequence of codes of vetices in C uv that appea fom the fist vetex t 1 = lca(u, u ) to the last vetex t c (c = V (C uv ) = V (C u v ) ) in the clockwise ode. Let σ (C u v ) = (σ (t 1; C u v ), σ (t 2; C u v ),..., σ (t c; C u v )) be the sequence of codes of vetices in C u v that appea fom the fist vetex t 1 = lca(u, u ) to the last vetex t c in the clockwise ode. The isomophism ψ fom G = T + uv to G = T + u v maps σ (C uv ) to σ (C u v ) in one of the following ways: (i) otate σ (C uv ) by some amount 0 s < c in the clockwise ode to get σ (C uv ); o (ii) flip σ (C uv ) at t 1 and then otate the flipped sequence by some amount 0 s < c in the clockwise ode to get σ (C uv ). The isomophism ψ in the fome is denoted by ψ = (1, s) and the latte by ψ = ( 1, s). We distinguish the thee cases: Case 1. Each of (u, u ) and (v, v ) is compaable in T ; Case 2. Exactly one of (u, u ) and (v, v ) is compaable in T ; and Case 3. Each of (u, u ) and (v, v ) is incompaable in T. Case 1. Each of (u, u ) and (v, v ) is compaable in T : By lca(u, v) = lca(u, v ) = c T and dfs(u) dfs(u ), we see that u is an ancesto of u and v is an ancesto of v. In this case, the unique path P (u, v) of T connecting u and v contains vetices u, u, v and v in this ode. Denote the sequence of vetices in P (u, v) by u 0 (= u ), u 1,..., u h (= u),..., u L h (= v ),..., u L (= v), whee 1-augmented tee T + uv (esp., T = u v ) has a unique cycle C uv = (u h, u h+1,..., u L ) (esp., C u v = (u h, u h+1,..., u L h, u 0, u 1,..., u h 1 )) and it holds V (C uv ) = V (C u v ) = L h + 1. (See Figue S1.) We fist pove that σ (u h ; C uv ) = σ (u L h ; C u v ) holds. Since σ (u h ; C uv ) is the signatue of subtee G uh of 1-augmented tee G = T + uv, it cannot be equal to σ (u i ; C u v ) fo any i = 0, 1,..., h (othewise G uh would be a pope subtee of itself fo i < h o the degee of u h in T + uv would be equal to that of u h in T + u v fo i = h). Hence ψ(u h ) {u h+1, u h+2,..., u L h }. Note that σ (x; C uv ) = σ (x; C u v ) fo all x {u h+1, u h+2,..., u L h 1 } = V (C uv ) V (C u v ) {u h, u L h }. Hence ψ(ψ(u h )) {u h+1, u h+2,..., u L h } and thee is an intege p 1 such that ψ p (u h ) = u L h since ψ gives a bijection between V (C uv ) and V (C u v ). This poves that 3

4 Figue S1: Case 1 (Each of (u, u ) and (v, v ) is compaable in T ). σ (u h ; C uv ) = σ (u L h ; C u v ), i.e., the subtee G u h of 1-augmented tee G = T + uv is ootedisomophic to the subtee G u L h of 1-augmented tee G = T + u v. We futhe pove that σ (u L j ; C uv ) = σ (u j ; C u v ) holds fo all j = 0, 1,..., h 1. To deive a contadiction, assume that thee is an intege k h 1 such that σ (u L k ; C uv ) σ (u k ; C u v ) and σ (u L j ; C uv ) = σ (u j ; C u v ) holds fo all j = k+1, k+2,..., h. (See Figue S2.) Conside ψ(u L k ). Note that ψ(u L k ) u L h, since G ul k of G is a pope subtee of T ul k. If ψ(u L k ) = u p {u k+1, u k+2,..., u h }, then σ (u p ; C u v ) = σ (u L p ; C uv ) holds by assumption on k and we egads that u L k is futhe mapped to u L p V (C uv ) to apply ψ to u L p. Moe fomally we conside a mapping ξ such that ξ(x) = ψ(x) fo x {u h+1, u h+2,..., u L h 1 } and ξ(u p ) = ψ(u L p ) fo u p {u k+1, u k+2,..., u h }. Hence by epeatedly mapping with ξ as long as it is mapped to a vetex in {u k+1, u k+2,..., u h } {u h+1, u h+2,..., u L h 1 }, we see that thee is a vetex u q with 0 q < k such that σ (u L k ; C uv ) = σ (u q ; C u v ) (ecall that σ (u L k ; C uv ) σ (u k ; C u v )). This means that the subtee G u L k of 1-augmented tee G = T + uv is adjacent to a pope descendant of u k in T, which contadicts that G uh G u L h (since σ (u L j ; C uv ) = σ (u j ; C u v ) holds fo all j = k, k + 1,..., h 1). Theefoe σ (u L j ; C uv ) = σ (u j ; C u v ) holds fo all j = 0, 1,..., h. (3) We now distinguish two cases: (i) ψ = (1, s); and (ii) ψ = ( 1, s). (i) ψ = (1, s): As we have obseved that thee is an intege p 1 such that ψ p (u h ) = u L h. Let a 0 = u h, a i = ψ i (u h ) fo i = 1, 2,..., p, whee vetices a 0 = u h, a 1,..., a p = u L h belong to V (C uv ) V (C u v ), but they may not appea along C uv in this ode. In paticula, σ (a i ; C uv ) = σ (a i ; C u v ) holds fo all 1 i p 1. Let {u l 0, u l1,..., u lp } = {a 0, a 1,..., a p }, whee h = l 0 < l 1 < < l p = L h. Let S 1, S 2,..., S p be the sequences of vetices between two consecutive vetices u li and u li+1 ; i.e., S i = (u li 1 +1, u li 1 +2,..., u li 1) fo each i = 1, 2,..., p. Let S 1 = (u 0, u 1,..., u l1 1, u l1 ) and S p = (u lp 1, u lp 1+1,..., u L ). Let σ (S i ) denote the sequence of codes σ (x) of vetices x along S i (note that σ (x; C uv ) = σ (x; C u v ) fo each of such x). Let σ (S 1) denote the sequence of codes σ (x; C u v ) of vetices x along S 1 and σ (S p) the sequence of codes σ (x; C uv ) of vetices x along S p. (See Figue S3.) We will show that (a) each of S i, S 1 and S p is symmetic in the sense that it is identical with its evese sequence; and (b) σ (S 1) = σ (S p) and fo each S i = (u j,..., u j ), thee is a sequence S i in its symmetic position (i.e., S i = (u L j,..., u L j )). Note that conditions (a) and (b) indicate that T has two subtees T a and T b satisfying condition (2), and hence T + uv and T + u v admit 4

5 Figue S2: Case 1 (Each of (u, u ) and (v, v ) is compaable in T ). Figue S3: Case (i) ψ = (1; s). 5

6 a ooted-isomophism, which contadicts the assumption on (u, v) and (u, v ). Since ψ = (1, s) maps a vetex u k in C uv to the vetex u k+s (when k +s L) o u k+s L (when k + s > L), we see that each sequence S i (o S p) in G is entiely mapped to anothe sequence S j fo some j (o S 1) in G. Let us tace this stating fom sequence S p in C uv, whee we call S p a long sequence. By ψ = (1, s), sequence S p is mapped to some sequence S i1 in C u v. Then sequence S i1 in C uv is mapped to some sequence S i2 in C u v. In this way, we have a seies of sequences S i1, S i2,..., S ik such that S ij is mapped to S ij+1 until S ik = S 1 holds fo some k. We call S i1, S i2,..., S ik 1 long sequences. Note that S ik = S 1 is not in C uv and S 1 is a subsequence of S 1. Let us continue to tace how S 1 will be mapped by ψ = (1, s). The sequence S 1 will be mapped to some sequence S ik+1 in C u v. Similaly with long sequences S i 1, S i2,..., S ik 1, we have a seies of sequences S ik+1, S ik+2,..., S ip = S p such that S ij is mapped to S ij+1, whee we call S ik+1, S ik+2,..., S ip = S p shot sequences. Fist we pove popety (a). Let g = S 1 /h and d = S 1 hg, whee S 1 = hg+d and d < h. By (3) and σ (S ik ) = σ (S 1), the subsequence A = (σ (u 0 ; C u v ), σ (u 1 ; C u v ),..., σ (u h 1 ; C u v )) of the fist h elements of σ (S 1) is symmetic with the subsequence of the last h elements of σ (S 1). Note that σ (S 1 ) is obtained fom σ (S 1) by deleting the last h elements. Since σ (S 1 ) = σ (S p ) is equal to the sequence obtained fom σ (S 1) by deleting the fist h elements, which implies that σ (S 1) is g 1 epetitions of A, finishing with the sequence B = (σ (u 0 ; C u v ), σ (u 1 ; C u v ),..., σ (u d 1 ; C u v )) of the fist d element of σ (S 1) followed by the evese of sequence A (ecall that the subsequence A of the fist h elements of σ (S 1) is symmetic with the subsequence of the last h elements of σ (S 1)). Then each of A and B is symmetic to itself, and σ (S 1) is epesented by an altenating sequence B, A, B,..., A, B of them. This poves that σ (S) of a shot sequence S is symmetic and σ (S) of a long sequence S is symmetic. Next we pove popety (b). Since ψ is a otational mapping with a fixed shift s, the positions of vetices in U = {u l0, u l1,..., u lp } = {a 0, a 1,..., a p } ae symmetic; i.e., u li U if and only u L li U. We fist flip the positions of all vetices u i in P (u, v), i.e., conside the mapping fom u i to u L i. Let S p denote the long sequence obtained fom S p by flipping the positions of the vetices in S p. Then conside the evese ψ 1 of mapping ψ, and tace how the sequence S p will be mapped to othe sequences in the flipped positions. Let S i 1, S i 2,..., S i k 1 be the esulting seies of long sequences. We have anothe intepetation of the seies; S i 1, S i 2,..., S i k 1 is the evese of seies S i1, S i2,..., S ik 1 obtained in the above. Since the position of S i 1, S i 2,..., S i k 1 is flipped, the positions of S i1, S i2,..., S ik 1 fom a symmetic distibution ove the path P (u h, u L h ) in T. The est of shot sequences also admits symmetic positions, poving popety (b). (ii) ψ = ( 1, s): We fist pove that ψ(u h ) = u L h. As we have obseved, ψ(u h ) {u 0, u 1,..., u h }. Let ψ(u h ) = u k with h + 1 k L h 1. (See Figue S4.) Recall that σ (C u v ) is obtained by flipping σ (C uv ) at t 1 = lca(u, u ) = u h and then otating the flipped sequence by some amount 0 s < c = V (C uv ) in the clockwise ode. We see that s = k h, since the position of u h will not change by the flipping and will change to u k by the otation by s. Hence ψ maps the code σ (u k, C uv ) of u k (u h+s ) to that of a vetex in G by fist picking the position of its flipped vetex u L+1 s and by shifting it by s = k h; i.e., ψ(u k ) = u (L+1 s)+s, which is vetex u h in C u v. Thus ψ(ψ(u h)) = u h and σ (u h, C uv ) = σ (u h, C u v ). This, howeve, contadicts that the degee of u h in G is lage than that of u h in G. Hence ψ(u h ) = u L h. Analogously we have ψ(u L h ) = u h. This means that σ (C u v ) and σ (C uv ) ae symmetic with espect to the axis passing though the centoid c T. Theefoe T has two subtees T a and T b satisfying condition 2, and hence T + uv and T + u v admit a ooted-isomophism, which contadicts the assumption on (u, v) and (u, v ). Case 2. Exactly one of (u, u ) and (v, v ) is compaable in T : Assume that neithe of u and u is an ancesto of the othe and v is an ancesto of v (the othe case can be teated symmetically). Denote the sequence of vetices in path P (u, v ) in T by u 0 (= u), u 1,..., u h (= lca(u, u )),..., u L l (= v),..., u L (= v ), and that in P (u, lca(u, u )) by u 0 (= u ), u 1,..., u h (= u h ), whee h h = l 0. (See Figue S5.) In this case, we pove that σ (u h ; C uv ) = σ (u h ; C u v ), which indicates that the two childen a = u h 1 and a = u h 1 of lca(u, u ) satisfy the condition (1), a contadiction to the assumption on choice of (u, v) and (v, v ). 6

7 Figue S4: Case (ii) ψ = ( 1; s). Figue S5: (a) x < h in Case 2 (Exactly one of (u, u ) and (v, v ) is compaable in T ). 7

8 Figue S6: (b) x = h in Case 2 (Exactly one of (u, u ) and (v, v ) is compaable in T ). Note that ψ(u L l ) {u L l, u L l+1,..., u L } holds by the same eason obseved in Case 1. Since σ (z; C uv ) = σ (z; C u v ) fo all z {u h+1, u h+2,..., u L l 1 }, thee is an intege p 1 such that ψ p (u L l ) = u x {u 0, u 1,..., u h }. Similaly, thee ae integes q and y such that ψ q (u h ) = u y {u L l, u L l+1,..., u L }. By the assumption of σ (u h ; C uv ) σ (u h ; C u v ), we have ψ q (u h ) u h. We distinguish two subcases. (a) x < h : In this case, the subtee T ul l of T is ooted-isomophic to the subtee G u of x G = T + u v, and the subtee G uh of G = T + uv popely contains a subtee ooted-isomophic to G u T u x L l. (See Figue S5.) This means that ψ q (u h ) = u y {u L l, u L l+1,..., u L } cannot hold, since othewise G u y of G = T + u v would popely contain a subtee ooted-isomophic to T ul l. Hence case (a) cannot occu. (b) x = h : In this case, the degee of u h in G (also in G) is equal to that degee of u L l in G = T + uv, which is lage than that of u L l in G = T + u v. This means that σ (u h ; C uv ) = σ (u L l ; C u v ) and y L l. (See Figue S6.) Now G u h G u y and G ul l G u h. Since y L l, G ul l (and hence G u h ) popely contains a subtee T ooted-isomophic to G u y ( G uh ). Moeove such a subtee T cannot appea as a subtee of T t of any othe child t of u h than t = u h 1 in G u h (othewise T would contain a subtee ooted-isomophic to T t ). This means that two subtees T uh 1 and T ul l+1 of T ae ooted-isomophic. We futhe pove that fo each i = 1, 2,..., l, it holds σ (u h i ; C uv ) = σ (u L l+i ; C u v ), i.e., G u h i G u L l+i. (4) Assume that thee is an intege 1 k l such that σ (u h i ; C uv ) = σ (u L l+i ; C u v ) fo all i = 1, 2,..., k 1 and σ (u h k ; C uv ) σ (u L l+k ; C u v ). Since G u h i G u L l+i (1 i < k), we see fom G uh k G u L l+k that u h k has a child w such that the subtee T w in T is ooted-isomophic to the subtee T ul l+k+1 in T. (See Figue S7.) This means that σ (u h k ; C uv ) σ (z; C u v ) fo all z {u L l+k+1, u L l+k+2,..., u L }. Let us tace how u h k in G = T + uv will be mapped to vetices in G = T + u v by ψ. Also σ (u h k ; C uv ) σ (u L l ; C u v ) since the subtee G u h k of G = T + uv is a pope subtee of T ul l. As we have obseved, if ψ(u h k ) = t {u h+1, u h+2,... u L l 1 }, whee σ (t; C uv ) = σ (t; C u v ) fo all t {u h+1, u h+2,... u L l 1 }, then we continue to conside ψ(t) of t V (C uv ). Similaly, if ψ(u h k ) = u L l+i {u L l+1, u L l+2,... u L l+k 1 }, whee σ (u h i ; C uv ) = σ (u L l+i ; C u v ) 8

9 Figue S7: Case 2 (Exactly one of (u, u ) and (v, v ) is compaable in T ). (1 i < k), then we continue to conside ψ(u h i ) of u h i V (C uv ). Since ψ is a bijection, by epeating this we see that u h k will be mapped to a vetex z {u L l+k+1, u L l+k+2,..., u L }, which contadicts that σ (u h k ; C uv ) σ (z; C u v ) fo all z {u L l+k+1, u L l+k+2,..., u L }. This poves (4). Now let us tace how u h in G = T + uv will be mapped to vetices in G = T + u v. Note that σ (u h ; C uv ) σ (u L l ; C u v ), as aleady obseved. Also σ (u h ; C uv ) σ (u i ; C u v ) fo all i = 0, 1,..., h since G uh cannot be ooted-isomophic to its pope subtee (ecall that σ (u h ; C uv ) σ (u h ; C u v ) is aleady assumed in Case 2). As we have obseved, if ψ(u h) = t {u h+1, u h+2,... u L l 1 }, whee σ (t; C uv ) = σ (t; C u v ) fo all t {u h+1, u h+2,... u L l 1 }, then we continue to conside ψ(t) of t V (C uv ). Similaly, if ψ(u h ) = u L l+i {u L l+1, u L l+2,... u L }, whee σ (u h i ; C uv ) = σ (u L l+i ; C u v ) (1 i l) by (4), then we continue to conside ψ(u h i) of u h i V (C uv ). Since ψ is a bijection, this can be epeated infinitely many time, contadicting the finiteness of gaphs. This poves that (b) cannot occu eithe. Case 3. Each of (u, u ) and (v, v ) is incompaable in T : Denote the sequence of vetices in the path P (u, v) by u 0 (= u), u 1,..., u h (= lca(u, u )),..., u L l (= lca(v, v )),..., u L (= v), and those in P (u, v ) by u 0 (= u ), u 1,..., u h (= lca(u, u )),..., u L l (= lca(v, v )),..., u L (= v ), whee h + l = h + l. (See Figue S8.) We fist claim that σ (u h ; C uv ) σ (t ; C u v ) fo all t {u 0, u 1,..., u h } {u L l +1, u L l +2,..., u L }. (5) By assumption on (u, v) and (u, v ), we know that σ (u h ; C uv ) σ (u h ; C u v ), whee u h = u h = lca(u, u ) (othewise two childen a = u h 1 and a = u h 1 in T of lca(u, u ) would satisfy the condition (1)). Clealy σ (u h ; C uv ) σ (t; C u v ) fo all t {u 0, u 1,..., u h 1 }, since othewise the subtee G uh of G = T + uv would be a pope subtee of itself. Hence to pove (5) by deiving a contadiction, it suffices to assume that σ (u h ; C uv ) = σ (z; C u v ) fo some z {u L l +1, u L l +2,..., u L }. Hence the subtee G z of G = T + u v is ooted-isomophic to the subtee G uh of G = T +uv. Let us tace how u L l = lca(v, v ) is mapped to vetices in C u v. As we have obseved, if ψ(u L l) = t {u h+1, u h+2,... u L l 1 }, whee σ (t; C uv ) = σ (t; C u v ) fo all t {u h+1, u h+2,... u L l 1 }, then we continue to conside ψ(t) of t := ψ(t). Hence u L l will be mapped to a vetex t {u 0, u 1,..., u h } {u L l, u L l +1,..., u L }, whee σ (u L l ; C uv ) = σ (t ; C u v ) holds. Since G u L l popely contains a subtee ootedisomophic to G uh, the vetex t cannot be in {u 0, u 1,..., u h 1 }. Similaly if t {u L l +1, u L l +2, 9

10 Figue S8: Case 3 (Each of (u, u ) and (v, v ) is incompaable in T )...., u L }, then the subtee G t of G = T + u v would be a pope subtee of itself G ul l G t, a contadiction. Theefoe thee ae only two cases: (a) t = lca(v, v ) = u L l = u L l ; and (b) t = lca(u, u ) = u h = u h. (a) t = lca(v, v ) = u L l = u L l : Let us tace how vetex u L l (= lca(v, v ) = u L l ) in C u v will be mapped to vetices in C uv by ψ 1. By assumption on (u, v) and (u, v ), we know that σ (u L l ; C uv ) σ (u L l ; C u v ) (othewise two childen a = u L l+1 and a = u L l +1 in T of lca(v, v ) would satisfy the condition (1)). Also σ (t; C uv ) σ (u L l ; C u v ) fo all t {u L l+1, u L l+2,..., u L }, since othewise G u L l would be a pope subtee of itself. We see that σ (u h ; C uv ) σ (u L l ; C u v ) since G u L l popely contains a subtee ooted-isomophic to G uh. Hence thee is a vetex u x with 0 x h 1 such that σ (u x ; C uv ) = σ (u L l ; C u v ). Next conside how vetex u h (= lca(u, u ) = u h ) in C u v will be mapped to vetices in C uv by ψ 1. Again by assumption on (u, v) and (u, v ), we know that σ (u h ; C uv) σ (u h ; C u v ). Also σ (t; C uv ) σ (u h ; C u v ) fo all t {u 0, u 1,..., u h 1 }, since othewise G u would be a pope h subtee of itself. We see that σ (u h ; C uv ) σ (u h ; C u v ) since u x V (G u ) means that G h u h popely contains a subtee ooted-isomophic to G uh. Hence the emaining possibility is that σ (u L l ; C uv ) = σ (u h ; C u v ). Howeve, this is also impossible because G u popely contains a h subtee ooted-isomophic to G ul l G u L l. Theefoe case (a) cannot occu. (b) t = lca(u, u ) = u h = u h : Now G u L l G u h, which popely contains a subtee ootedisomophic to G z G uh by assumption. Let us tace how vetex u h (= lca(u, u ) = u h ) in C u v will be mapped to vetices in C uv by ψ 1. Now σ (u h ; C uv ) σ (u h ; C u v ). Also σ (u i ; C uv ) σ (u h ; C u v ) fo all i = 0, 1,..., h 1 since othewise G u L l would be a pope subtee of itself. By assumption on (u, v) and (u, v ), σ (u h ; C uv ) σ (u h ; C u v ). Hence the emaining possibility is that thee is a vetex u y {u L l+1, u L l+2,..., u L } such that σ (u y ; C uv ) = σ (u h ; C u v ). Now conside how vetex u L l (= lca(v, v ) = u L l ) in C u v will be mapped to vetices in C uv by ψ 1. Since ψ is a bijection, thee is a vetex w {u 0, u 1,..., u h } {u L l, u L l+1,..., u L } such that σ (w; C uv ) = σ (u L l ; C u v ). Note that the subtee G u has the vetex u y and L l popely contains a subtee ooted-isomophic to G u. Clealy σ (u i ; C uv ) σ (u h L l ; C u v ) fo all i = L l+1, L l+2,..., L since othewise G u would be a pope subtee of itself. Hence w L l {u 0, u 1,..., u h } {u L l }. By assumption on (u, v) and (u, v ), σ (u L l ; C uv ) σ (u L l ; C u v ). 10

11 Figue S9: Sequences A, A, B and B. Also σ (u i ; C uv ) σ (u L l ; C u v ) fo all i = 0, 1,..., h 1 since G u popely contains a L l subtee ooted-isomophic to G u h. Finally σ (u h ; C uv ) σ (u L l ; C u v ) since G u popely L l contains a subtee ooted-isomophic to G uh. Theefoe such a vetex w cannot exist and (b) does not occu eithe, poving (5). Symmetically we also obtain σ (t ; C u v ) σ (u L l ; C uv ) fo all t {u 0, u 1,..., u h } {u L l +1, u L l +2,..., u L }, and σ (u h ; C u v ) σ (t; C uv ) σ (u L l ; C u v ) fo all t {u 0, u 1,..., u h } {u L l+1, u L l+2,..., u L }. Then (5) implies σ (u h ; C uv ) = σ (u L l ; C u v ), by epeatedly taking t := ψ(t ) as long as t {u h+1, u h+2,... u L l 1 }, whee σ (t ; C uv ) = σ (t ; C u v ) fo all t {u h+1, u h+2,... u L l 1 }. Analogously we also have σ (u L l ; C uv ) = σ (u h ; C u v ). Hence we have the following popeties. σ (u h ; C uv ) = σ (u L l ; C u v ), σ (u L l ; C uv ) = σ (u h ; C u v ), σ (u h ; C uv ) σ (t ; C u v ) σ (u L l ; C uv ) fo all t {u 0, u 1,..., u h } {u L l +1, u L l +2,..., u L }, and σ (u h ; C u v ) σ (t; C uv ) σ (u L l ; C u v ) fo all t {u 0, u 1,..., u h } {u L l+1, u L l+2,..., u L }. (6) In (6), σ (u h ; C uv ) = σ (u L l ; C u v ) and σ (u L l ; C uv ) = σ (u h ; C u v ) mean that G u h G u L l and G ul l G u h. Note that G uh (esp., G u h ) is obtained fom T uh by emoving the subtee T uh 1 (esp., T u h 1 ), while G u L l (esp., G u L l ) is obtained fom T ul l by emoving the subtee T ul l+1 (esp., T u L l +1 ). (See Figue S9.) Since T u h 1 T u h 1 and T u L l+1 T u L l +1 by assumption, we see that T u h 1 T u L l+1 and T uh 1 T u L l +1. (7) We distinguish two subcases: (i) ψ = ( 1, s); and (ii) ψ = (1, s). (i) ψ = ( 1; s): In this case, we can pove that ψ(lca(u, u )) = lca(v, v ) and ψ(lca(v, v )) = lca(u, u ) in the same manne of Case 1. This means that σ (C u v ) and σ (C uv ) ae symmetic with espect to the axis passing though the centoid c T. Theefoe T has two subtees T a and T b satisfying condition (2), and hence T + uv and T + u v admit a ooted-isomophism, which contadicts the assumption on (u, v) and (u, v ). 11

12 (ii) ψ = (1, s): In this case, the cyclic sequence σ (C uv ) will match with σ (C u v ) afte taking lca(u, u ) as the common stat position and otating σ (C uv ) by s in the clockwise ode. By V (C uv ) = V (C u v ), we have h + l = h + l and L h l = L h l. By (6), vetex lca(u, u ) = u h = u h (esp., lca(v, v ) = u L l = u L l ) in C uv will neve be mapped to any vetex u i with i {0, 1,..., h } {L l +1, L l +2,..., L} afte epeating the otation by any numbe of times. This means that s L h l = L h l (esp., h + l = h + l < s). Let A be the subsequence of σ (C uv ) fom u L l+1 to u h 1 ; i.e., A = (σ (u L l+1 ; C uv ), σ (u L l+2 ; C uv ),..., σ (u L ; C uv ), σ (u 0 ; C uv ), σ (u 1 ; C uv ),..., σ (u h 1 ; C uv )), whee the length A of A is h + l 2 (< s). Let B be the subsequence of σ (C u v ) fom u h+1 to u 2h+l 1 ; i.e., B = (σ (u h+1 ; C u v ), σ (u h+2 ; C u v ),..., σ (u 2h+l 1 ; C u v )), whee A = B. (See Figue S9.) Since vetex u L l will match with u h by epeating the otation of σ (C uv ) by s some numbe of times, we see that A matches with B afte epeating the otation by some numbe of times. Let B be the subsequence of σ (C uv ) fom u h+1 to u 2h+l 1 ; i.e., B = (σ (u h+1 ; C uv ), σ (u h+2 ; C uv ),..., σ (u 2h+l 1 ; C uv )), whee B is equal to B since {u h+1, u h+2,..., u 2h+l 1 } V (C uv ) V (C u v ) {u h, u L l }. Let A be the subsequence of σ (C u v ) fom u L l +1 to u h 1 ; i.e., A = (σ (u L l +1; C u v ), σ (u L l +2; C u v ),..., σ (u L; C u v ), σ (u 0; C u v ), σ (u 1; C u v ),..., σ (u h 1; C u v )), Similaly A matches with B (= B). Theefoe we have A = A. Note that h h since othewise A = A would imply that a = u h 1 and a = u h 1 in T satisfy (1). Assume h < h (the othe case of h > h can be teated analogously). Since A = A, we know that σ (u h i ; C uv ) = σ (u h i ; C u v ) fo i = 1, 2,..., h (< h ). (8) Symmetically we also see that σ (u L l+i ; C uv ) = σ (u L l +i ; C u v ) fo i = 1, 2,..., l (< l). (9) Recall that T u T h u 1 L l+1 and T uh 1 T u by (7). Fom this, (8) and (9), we hee show L l +1 that σ (u h i ; C uv ) = σ (u h i ; C u v ) = σ (u L l+i ; C uv ) = σ (u L l +i ; C u v ) fo i = 1, 2,..., min{h, l }. (10) To pove this, we indiectly assume that thee is an index j ( min{h, l }) such that σ (u h j ; C uv ) σ (u L l +j ; C u v ) (i.e., G u h j G u ) and σ (u h i ; C uv ) = σ (u L l L l +i ; C u v ) fo all i = +j 1, 2,..., j 1, whee the latte indicates that T uh j T u L l +j and T u h j T u L l+j since (7) holds. (See Figue S10.) This and G uh j G u mean that T uh j 1 T u L l +j L l. +j+1 Let k be the numbe of subtees T t ooted at a child t of u L l +j such that T t T u L l, whee +j+1 k 1 (possibly t = u L l +j+1 ). Since T u h j T u L l but T u +j h j 1 T u, we see that L l +j+1 the numbe of subtees T x ooted at a child x ( u h j 1 ) of u h j such that T x T u L l is +j+1 also k. Since σ (u h j ; C uv ) = σ (u h j ; C u v ), we see that the numbe of subtees T y ooted at a child y ( u h j 1 ) of u h j such that T y T u L l +j+1 is also k. By T u h j T u L l+j, we see that 12

13 Figue S10: T + uv and T + u v satisfying equation (10). the numbe of subtees T z ooted at a child z ( u L l+j+1 ) of u L l+j such that T z T u L l +j+1 is also k (note that T ul l+j+1 T u L l ). Since σ (u +j+1 L l+j ; C uv ) = σ (u L l +j ; C u v ), this means that the numbe of subtees T w ooted at a child w ( u L l +j+1 ) of u L l +j such that T w T u is also k. This contadicts that the numbe of such subtees must be k 1, since L l +j+1 T u L l is counted in the oiginal k. This poves (10). This also implies that h = l, h = l and +j+1 T u T h u h 1 L h +h+1 (since h l means T uh j 1 T u fo j = min{h, l }, which leads L l +j+1 to a contadiction in the same manne). We next pove that subsequence C = (σ (u L h +h+1; C uv ), σ (u L h +h+2; C uv ),..., σ (u L ; C uv )) of A is symmetic (i.e., it matches with its evese sequence). Note that C is equal to subsequence C = (σ (u 0; C u v ), σ (u 1; C u v ),..., σ (u h h 1; C u v )) of A, since each of C and C coesponds to the same subsequence of A = A. (See Figue S11.) Note that C passes though tee T ul h +h+1 ( T u ) fom u h L h +h+1 to u h 1 L, while C (= C) passes though tee T u ( T h u h 1 L h +h+1 ) fom u 0 to u h h 1. If σ (u L h+h +1; C uv ) σ (u L ; C uv ) (= σ (u h h 1 ; C u v )) then T u L h+h +2 T u and u h h 2 h h 1 has some othe child t ( u h h 2 ) such that T t T ul h+h +2. In this case, the code σ (u L ; C uv ) (= σ (u h h 1 ; C u v )) of vetex u L means that thee is a child x of u L such that T x T ul h+h +2, which is a contadiction (since T x is now a pope subtee of T ul h+h +2 ). Hence σ (u L h+h +1; C uv ) = σ (u L ; C uv ). Moe geneally assume that thee is an index k ( h h 2 ) such that σ (u L h+h +1+k; C uv ) σ (u L k ; C uv ) and σ (u L h+h +1+i; C uv ) = σ (u L i ; C uv ) fo all i = 0, 1,..., k 1. By the second condition, we have T ul h+h +k T u. By the fist condition, T h u h k L h+h +k+1 T u holds h h k 1 and vetex u h h k has some othe child t ( u h h k 1 ) such that T t T ul h+h +k+1. In this case, the code σ (u L k ; C uv ) (= σ (u h h k ; C u v )) of vetex u L k means that thee is a child x of u L k such that T x T ul h+h +k+1, which is a contadiction (since T x is now a pope subtee of T ul h+h +k+1 ). Fo indices k (> h h 2 ), we can apply the same agument vetices fom u h h 1 to u 0 to get a contadiction. This poves that C (= C ) is symmetic. 13

14 Figue S11: Sequences C and C. By applying a simila agument in Case 1(i), we see that T has two subtees T a and T b satisfying condition (2), and hence T + uv and T + u v admit a ooted-isomophism, which contadicts the assumption on (u, v) and (u, v ). This completes a poof of this theoem. S1.3 Poof of Lemma 3 Poof: Fo a vetex u in the unique cycle C in a 1-augmented tee G, let G u denote the component containing u that appeas afte emoving the two edges in C incident to u. Hence G u is a subtee of G, and we egad it as a subtee ooted at u. Let σ (u; C) denote the signatue of G u ; σ (u; C) = σ (v; C) if and only if G u G v. When G is obtained fom a tee T by adding an edge uv, the unique cycle in G is denoted by C uv. In a ooted tee, a pai of vetices x and y is called compaable if one of them is an ancesto of the othe. Let dfs be the depth-fist seach numbe obtained by tavesing T stating fom, whee = c T o the left endpoint of c T =. Intuitively admissible edges ae defined so that each admissible pai (u, v) is lexicogaphically smalle than any pai (u, v ) such that T + uv T + u v. To pove the lemma, it suffices to show that (I) Fo any nonadmissible pai of nonadjacent vetices u, v V (T ), thee is an admissible pai (u, v ) such that T + u v T + uv; and (II) Fo two admissible pais (u, v) and (u, v ), if T + uv T + u v then u = u and v = v. (I) Let (u, v) be a nonadmissible pai of nonadjacent vetices with dfs(u) < dfs(v). Hence (u, v) does not satisfy one of conditions (1)-(3) in the definition of admissible pais. Let G = T + uv. To show (I), we assume that (u, v) is lexicogaphically minimum among all nonadmissible pais (x, y) such that T + xy G. (i) (u, v) does not satisfy condition (1), i.e., thee is a vetex w V (P (lca(u, v), )) {} such that copy(w) = 1 (possibly u = w when u = lca(u, v)): Then the subtee T w ooted at w is ooted-isomophic to the subtee T x ooted at the left sibling x = left(w) of w, implying that T x contains a pai (u, v ) such that T + u v is ooted-isomophic to T + uv. Then (u, v ) must be admissible, since othewise it would contadict the choice of (u, v). (ii) (u, v) does not satisfy condition (2), i.e., thee is a vetex w with copy(w) = 1 such that (a) w V (P (u, gua(u, v)) {lca(u, v), gua(v, u)}; o (b) w V (P (v, gua(v, u)) {lca(u, v), gua(v, u)}: Fist conside case (a), whee u lca(u, v) (since othewise u = lca(u, v), gua(u, v) = gua(v, u), 14

15 and V (P (u, gua(u, v)) {lca(u, v), gua(v, u)} = ). As in (i), T w is ooted-isomophic to T x ooted at the left sibling x = left(w) of w, and T x contains a vetex u such that T + u v T + uv. Then (u, v) must be admissible by the lexicogaphical minimality in the choice of (u, v). In (b) (possibly u = lca(u, v)), T w T x fo the left sibling x = left(w) of w, and T x contains a vetex v such that T + uv T + uv. Then (u, v ) must be admissible by the lexicogaphical minimality of (u, v). (iii) (u, v) with copy(gua(v, u)) = 1 does not satisfy condition (3); i.e., (a) u = lca(u, v), which violates condition (3-i) gua(u, v) = left(gua(v, u)); (b) u lca(u, v) and gua(u, v) left(gua(v, u)); o (c) u lca(u, v), gua(u, v) = left(gua(v, u)), and dfs(v) < dfs(û) fo the copy û of vetex u in T gua(v,u). In (a) and (b), vetex x = left(gua(v, u)) satisfies T x T gua(v,u) and T x contains a vetex v ( v) such that T + uv T + uv. Then (u, v ) must be admissible by the lexicogaphical minimality of (u, v). In (c), thee is a vetex ˆv V (T gua(u,v) ) such that T + ˆvû T + uv. Since dfs(ˆv) = dfs(v) V (T gua(u,v) ) and dfs(u) = dfs(û) V (T gua(u,v) ) hold, dfs(v) < dfs(û) implies dfs(ˆv) < dfs(u). Then (ˆv, û) must be admissible by the lexicogaphical minimality of (u, v). (II) Let (u, v) and (u, v ) be two admissible pais such that T +uv T +u v. Assume without loss of geneality that (u, v) is lexicogaphically smalle than (u, v ), i.e., dfs(u) dfs(u ), and if dfs(u) = dfs(u ) then dfs(v) dfs(v ). We show that u = u and v = v. We fist claim lca(u, v) = lca(u, v ). (11) Note that if lca(u, v) is not the unicentoid c T = then among the codes in σ (C uv ) the vetex lca(u, v) V (C uv ) has the unique code σ (lca(u, v); C uv ), since only subtee G lca(u,v) of 1-augmented tee G = T + uv contains at least n/2 + 1 vetices. Similaly, if lca(u, v ) is not the unicentoid c T = the vetex lca(u, v ) V (C u v ) has the unique code σ (lca(u, v ); C u v ). If lca(u, v) = = lca(u, v ) then we ae done. The emaining case is that lca(u, v) o lca(u, v ). In this case, we assume that lca(u, v) lca(u, v ) to deive a contadiction. Without loss geneality, we assume that lca(u, v). Then lca(u, v ) also holds, since othewise fo any vetex t in C u v (such as t = lca(u, v )), σ (C u v ) cannot contain a code σ (t; C u v ) that indicates that the subgaph G t of G = T + u v ooted at t has at least n/2 vetices. This and T + uv T + u v mean that σ (lca(u, v); C uv ) = σ (lca(u, v ); C u v ), i.e., the subtees G lca(u,v) and G lca(u,v ) ae ooted-isomophic. This implies that T gua(x,y) T gua(y,x) fo two vetices x = lca(u, v) and y = lca(u, v ) and copy(gua(x, y)) = 1 o copy(gua(y, x)) = 1, contadicting that both (u, v) and (u, v ) satisfy the condition (1) in the definition of admissible pais. This poves (11). Now gua(u, v), gua(v, u), gua(u, v ) and gua(v, u ) ae childen of lca(u, v) = lca(u, v ) in T. We next claim u lca(u, v) = lca(u, v ) u (12) Assume that u = lca(u, v) o u = lca(u, v ), say u = lca(u, v) (the case of u = lca(u, v ) can be handled symmetically). In this case, u = lca(u, v ) also holds by lca(u, v) = lca(u, v ), since othewise the degee of lca(u, v) in G = T + uv would be lage than that of it in G = T + u v, contadicting that σ (lca(u, v); C uv ) = σ (lca(u, v ); C u v ) holds by T + uv T + u v. Now u = lca(u, v) = lca(u, v ) = u. Assume v v (othewise we ae done with u = u and v = v ), whee dfs(v) < dfs(v ) is assumed without loss of geneality. Clealy v is not an ancesto of v. Then by σ (lca(u, v); C uv ) = σ (lca(u, v ); C u v ), we have T gua(v,u) T gua(v,u ) and hence copy(gua(v, u )) = 1. Howeve, in this case, (u, v ) with u = lca(u, v ) does not satisfy condition (3-i) in the definition of admissible pais, a contadiction. This poves (12). Thidly we claim gua(u, v) = gua(u, v ) and gua(v, u) = gua(v, u ). (13) 15

16 Figue S12: Thee cases (c-i), (c-ii) and (c-iii). Let a = gua(u, v), a = gua(u, v ), b = gua(v, u) and b = gua(v, u ). (a) a a and b = b (whee dfs(u) < dfs(u ) is assumed without loss of geneality): In the isomophism ψ fo G = T + uv G = T + u v, lca(u, v) in G coesponds to lca(u, v ) in G, and σ (lca(u, v); C uv ) = σ (lca(u, v ); C u v ) implies that T a T a and copy(a ) = copy(gua(u, v )) = 1. Hence (u, v ) does not satisfy the condition (2) in the definition of admissible pais, a contadiction. (b) a = a and b b (whee dfs(v) < dfs(v ) is assumed without loss of geneality): As in (a), we see that T b T b and copy(b ) = copy(gua(v, u )) = 1. Hence (u, v ) does not satisfy the condition (3-i) in the definition of admissible pais, a contadiction. (c) a a and b b (whee dfs(u) < dfs(u ) is assumed without loss of geneality): Thee ae thee cases: (c-i) dfs(u) < dfs(v) < dfs(u ) < dfs(v ); (c-ii) dfs(u) < dfs(u ) < dfs(v ) < dfs(v); and (c-iii) dfs(u) < dfs(u ) < dfs(v) < dfs(v ). (See Figue S12.) In (c-i), the subtees T 1 = T a, T 2 = T b, T 3 = T a and T 4 = T b appea in this ode fom left to ight in T, whee T i T j (i < j) implies T i T i T j fo any i < i < j since T is a left-heavy tee. Hence the isomophism ψ fo G = T + uv G = T + u v implies T a T a and T b T b. Hence we have T a T b T a T b in (c-i). With a simila agument fom (c-i), we have T a T a and T b T b in (c-ii); and T a T a and T b T b in (c-iii). In any of these thee cases, copy(gua(u, v )) = 1 and (u, v ) does not satisfy the condition (2) in the definition of admissible pais, a contadiction. This poves that a = a and b = b ; i.e., (13). In a ooted-isomophism ψ between G = T +uv and G = T +u v, vetex lca(u, v) = lca(u, v ) in G coesponds to itself in G, meaning that thee ae only two possible ooted-isomophisms ψ: (i) one peseves the diection of cycles, i.e., ψ(gua(u, v)) = gua(u, v); and (ii) the othe evese the diection, i.e., ψ(gua(u, v)) = gua(v, u). We fist conside the ooted-isomophism ψ in (i). When u is an ancesto of u, the degee deg(u; G) of u in G = T + uv is lage than the degee deg(u; G ) of u in G = T + u v, and it cannot hold that σ (u; C uv ) = σ (u; C u v ), a contadiction to ψ. When neithe of u and u is an ancesto of the othe (whee dfs(u) < dfs(u ) is assumed without loss of geneality), we have σ (lca(u, u ); C uv ) = σ (lca(u, u ); C u v ), implying that T gua(u,u ) T gua(u,u) and copy(gua(u, u)) = 1. (See Figue S13.) This contadicts that (u, v ) satisfies the condition (2) in the definition of admissible pais. Finally conside the case whee neithe of v and v is an ancesto 16

17 Figue S13: Neithe of u and u is an ancesto of the othe. of the othe (whee dfs(v) < dfs(v ) is assumed without loss of geneality). As in the above case, we have T gua(v,v ) T gua(v,v) and copy(gua(v, v)) = 1. This contadicts that (u, v ) satisfies the condition (2) in the definition of admissible pais. In what follows, we conside the ooted-isomophism ψ in (ii). Since we know that G = T + uv and G = T + u v ae ooted-isomophic, we have ψ(gua(u, v)) = gua(v, u) and ψ(gua(v, u)) = gua(u, v). We distinguish the thee cases: Case 1. Each of (u, u ) and (v, v ) is compaable in T ; Case 2. Exactly one of (u, u ) and (v, v ) is compaable in T ; and Case 3. Each of (u, u ) and (v, v ) is incompaable in T. Case 1. Each of (u, u ) and (v, v ) is compaable in T : By dfs(u) dfs(u ), u is an ancesto of u and v is an ancesto of v: In this case, the unique path P (u, v) of T connecting u and v contains vetices u, u, v and v in this ode. Denote the sequence of vetices in P (u, v) by u 0 (= u ), u 1,..., u h (= u),..., u g (= lca(u, v)),..., u L h (= v ),..., u L (= v), whee 1-augmented tee T + uv (esp., T = u v ) has a unique cycle C uv = (u h, u h+1,..., u L ) (esp., C u v = (u h, u h+1,..., u L h, u 0, u 1,..., u h 1 )) and it holds V (C uv ) = V (C u v ) = L h + 1. (a) g h = L h g (the depth of u in T is equal to that of v in T ): Since σ (u i ; C uv ) = σ (u L i ; C u v ) fo all i = h, h+1,..., g, we see that T gua(u,u ) T gua(v,v ) and copy(gua(v, u )) = 1. (See Figue S14.) This contadicts that (u, v ) satisfies the condition (3-ii) in the definition of admissible pais. (b) g h L h g: Conside the case of g h < L h g, i.e., the depth of u in T is smalle than that of v in T (the othe case can be teated analogously). (See Figue S15.) In this case, u h (esp., u 2g h ) in G = T + uv is mapped to u 2g h (esp., u h ) in G = T + u v by ψ, and we have σ (u h ; C uv ) = σ (u 2g h ; C u v ) and σ (u 2g h ; C uv ) = σ (u h ; C u v ). By noting that u 2g h V (C uv ), σ (u 2g h ; C uv ) = σ (u 2g h ; C u v ), indicating that σ (u h ; C uv ) = σ (u h ; C u v ). This, howeve, is impossible because the degee of u h in G is lage than that in G. Case 2. Exactly one of (u, u ) and (v, v ) is compaable in T : Assume that neithe of u and u is an ancesto of the othe and v is an ancesto of v (the othe case can be teated symmetically). Denote the sequence of vetices in path P (u, v ) in T by u 0 (= u), u 1,..., u h (= lca(u, u )),..., u g (= lca(u, v)),..., u L l (= v),..., u L (= v ), and that in P (u, lca(u, u )) by u 0 (= u ), u 1,..., u h (= u h). (a) v v (whee dfs(v) < dfs(v ) is assumed without loss of geneality) and g h > L l g (the depth of u h in T is lage than that of v in T ): (See Figue S16.) In this case, u L l (esp., u 2g L+l ) in G = T + uv is mapped to u 2g L+l (esp., u L l ) in G = T + u v by ψ, and we 17

18 Figue S14: (a) in Case 1. Figue S15: (b) in Case 1. 18

19 Figue S16: (a) in Case 2. have σ (u L l ; C uv ) = σ (u 2g L+l ; C u v ) and σ (u 2g L+l ; C uv ) = σ (u L l ; C u v ). By noting that u 2g L+l V (C uv ), σ (u 2g L+l ; C uv ) = σ (u 2g L+l ; C u v ), indicating that σ (u L l ; C uv ) = σ (u L l ; C u v ). This, howeve, is impossible because the degee of u L l in G is lage than that in G. (b) v v (whee dfs(v) < dfs(v ) is assumed without loss of geneality) and g h L l g (the depth of u h in T is not smalle than that of v in T ): (See Figue S17.) In this case, u h (esp., u 2g h ) in G = T + uv is mapped to u 2g h (esp., u h ) in G = T + u v by ψ, and we have σ (u h ; C uv ) = σ (u 2g h ; C u v ) and σ (u 2g h ; C uv ) = σ (u h ; C u v ). By noting that u 2g h V (C uv ), it holds σ (u 2g h ; C uv ) = σ (u 2g h ; C u v ), indicating that σ (u h ; C uv ) = σ (u h ; C u v ). This, howeve, means that T gua(u,u ) T gua(u,u) and copy(gua(u, u)) = 1, and that (u, v ) does not satisfy the condition (2) in the definition of admissible pais. (c) v = v : Let us tace how vetex u h in G will be mapped to vetices in G = T + u v by ψ. Since the code σ (u h ; C uv ) epesents the set T of subtees T t ooted at childen t ( u h 1 ) of u h, vetex u h cannot be mapped to any vetex in {u h 1, u h 2,..., u 0} (since these vetices ae contained in T u T ). Hence ψ(u h) {u h 1 h, u h+1,..., u L l (= v = v )}. Since σ (u i ; C uv ) = σ (u i ; C u v ) fo these vetices u i {u h+1,..., u L l (= v = v )}, thee is an intege j 1 such that ψ j (u h ) = u h. This, howeve, means that T gua(u,u ) T gua(u,u) and copy(gua(u, u)) = 1, and that (u, v ) does not satisfy the condition (2) in the definition of admissible pais. Case 3. Each of (u, u ) and (v, v ) is incompaable in T : Assume that u u and dfs(u) < dfs(u ) without loss of geneality. Denote the sequence of vetices in the path P (u, v) by u 0 (= u), u 1,..., u h (= lca(u, u )),..., u g (= lca(u, v)),..., u L l (= lca(v, v )),..., u L (= v), and those in P (u, v ) by u 0 (= u ), u 1,..., u h (= lca(u, u )),..., u g (= u g = lca(u, v)),..., u L l (= lca(v, v )),..., u L (= v ), whee h + l = h + l. (a) g h L l g: Conside the case of g h < L l g, i.e., the depth of lca(u, u ) in T is lage than that of lca(v, v ) in T (the othe case can be teated analogously). As in Case 2(b), u h (esp., u 2g h ) in G = T + uv is mapped to u 2g h (esp., u h ) in G = T + u v by ψ, and we have σ (u h ; C uv ) = σ (u 2g h ; C u v ) and σ (u 2g h ; C uv ) = σ (u h ; C u v ). By noting that u 2g h V (C uv ), σ (u 2g h ; C uv ) = σ (u 2g h ; C u v ), indicating that σ (u h ; C uv ) = σ (u h ; C u v ). This meas that T gua(u,u ) T gua(u,u) and copy(gua(u, u)) = 1, and that (u, v ) does not satisfy the condition (2) in the definition of admissible pais. (b) g h = L l g (the depth of lca(u, u ) in T is equal to that of lca(v, v ) in T ): Let 19

20 Figue S17: (b) in Case 2. T uh (esp., T ul l ) denote the set of subtees T t of T ooted at a child t of u h (esp., u L l ). Let a = gua(u, u ), a = gua(u, u), b = gua(v, v ) and b = gua(v, v), whee T a T a and T b T b (othewise (u, v) o (u, v ) would not satisfy the condition (2) in the definition of admissible pais). (See Figue S18.) Let A be the subsequence of σ (C uv ) fom b = u L l+1 to a = u h 1, i.e., A = (σ (u L l+1 ; C uv ), σ (u L l+2 ; C uv ),..., u L, u 0,..., σ (u h 1 ; C uv )), and let A be the subsequence of the evesal of σ (C u v ) fom a = u h 1 to b = u L l +1, i.e., A = (σ (u h 1 ; C u v ), σ (u h 2 ; C u v ),..., u 0, u L,..., σ (u L l +1 ; C u v )). Then A = A by the isomophism ψ. Since u h (esp., u 2g h ) in G = T +uv is mapped to u L l (esp., u h ) in G = T +u v by ψ, we have σ (u h ; C uv ) = σ (u L l ; C u v ) and σ (u L l ; C uv ) = σ (u h ; C u v ). This means that the set T uh {T a } of subtees is ooted-isomophic to T ul l {T b }, while T uh {T a } of subtees is ooted-isomophic to T ul l {T b }. Since T a T a and T b T b, this implies that T a T b, T b T a, and T gua(u,v) T gua(v,u). Hence by assumption of dfs(u) < dfs(u ), we have dfs(u) < dfs(u ) < dfs(v ) < dfs(v). Let ˆv V (T a ) be the symmetic copy of v V (T b ) and û V (T b ) be the symmetic copy of u V (T a ). Let  be the subsequence of the evesal of σ (Cˆvû ) fom a = u h 1 to b = u L l+1. Then T + uv T + ˆvû and  = A = A. Now we show that u = ˆv and v = û, which indicates that (u, v ) does not satisfy the condition (3-ii) in the definition of admissible pais. Assume u ˆv (the case of v û can be teated analogously). (See Figue S19.) Note that u, ˆv V (T a ). When one of u and ˆv is an ancesto of the othe in T a (say ˆv is an ancesto of u ), ˆv V (Cˆvû ) V (C u v ) holds but σ (ˆv; Cˆvû ) σ (ˆv; C u v ) (since the degee of ˆv in T + ˆvû is lage than that in T + u v ), contadicting  = A. Hence the emaining case is that neithe of u and ˆv is an ancesto of the othe. Let t 1 = gua(u, ˆv) and t 2 = gua(ˆv, u ), which ae childen of lca(u, ˆv). Then by T + u v T + ˆvû, we have σ (gua(u, ˆv); C u v ) = σ (gua(ˆv, u ); Cˆvû ) by  = A, which implies that T t1 T t2. If dfs(ˆv) < dfs(u ) then copy(gua(u, ˆv)) = 1 holds, indicating that (u, v ) does not satisfy the condition (2) in the definition of admissible pais. On the othe hand (dfs(u ) < dfs(ˆv)) copy(gua(ˆv, u )) = 1 holds. Then the symmetic copy of w V (T b ) of gua(ˆv, u ) also satisfies copy(w) = 1, indicating that (u, v ) does not satisfy the condition (2) in the definition of admissible pais. 20