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The Complexity of the Evolution of Graph Labelings Geir Agnarsson Raymond Greenlaw Sanpawat Kantabutra.

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1 The Complexity of the Evolution of Graph Labelings Geir Agnarsson Raymond Greenlaw Sanpawat Kantabutra

2 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 2 Outline Introduction Preliminaries and Problem Definitions Relating Vertex and Edge Relabeling Tight Bounds for the Relabeling Problem Relabeling with Privileged Labels Open Problems References

3 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 3 Outline Introduction Preliminaries and Problem Definitions Relating Vertex and Edge Relabeling Tight Bounds for the Relabeling Problem Relabeling with Privileged Labels Open Problems References

4 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 4 Introduction Graph labeling is a well-studied subject in computer science and mathematics. Problem that has widespread applications, including in many other disciplines. Graph Relabeling Problem is a variant of graph labeling. See next slide for example.

5 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 5 Introduction Two configurations, a “start” and “goal” are given. Transform the “start” to the “goal” configuration with certain conditions on the movement of labels.

6 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 6 Introduction Presentation of new results and the extension of existing results; in particular, –NC 1 reduce the Vertex Relabeling problem to the Edge Relabeling Problem and vice versa. –Provide upper and lower bounds on the complexity of the Vertex and Edge Relabeling Problems. –Provide precise characterizations of when instances of relabeling with privileged labels are solvable.

7 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 7 Introduction A number of puzzles can be viewed as relabeled graphs. One of the most famous is the 15-Puzzle which consists of 15 tiles on a 4 x 4 board with one position empty. Graph labeling has been studied (and continues to have ongoing research) in the context of cartography, codings, colorings, and rankings.

8 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 8 Introduction Applications in bioinformatics, networks, and VLSI; also appear in unexpected places. Graph Relabeling Problem can model a wormhole routing in processor networks. Well-known Pancake Flipping Problem can be modeled as a special case of our problem.

9 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 9 Outline Introduction Preliminaries and Problem Definitions Relating Vertex and Edge Relabeling Tight Bounds for the Relabeling Problem Relabeling with Privileged Labels Open Problems References

10 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 10 Preliminaries and Problem Definitions Let S V, S E N = {1,2,…}. A labeling L V of V is a mapping L V : V  S V. A labeling L E of E is a mapping L E : E  S E. S V = {1,2,…,n} and S E = {1,2,…,m}. Graphs are associated with labelings using angle bracket notation. A mutation or flip function f maps triples to triples.

11 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 11 Preliminaries and Problem Definitions A consecutive vertex mutation function is defined where f maps a pair to a pair with the following conditions: –L V = L′ V, except on two vertices u and w. –{u, w} E. –L V (u) = L′ V (w) and L V (w) = L′ V (u). –S V = {1,2,…,n}. –f is a bijection.

12 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 12 Preliminaries and Problem Definitions Vertex Relabeling Problem Instance: A graph G, labelings L V and L′ V, and t N. Question: Can labeling L V evolve into L′ V in t or fewer vertex mutations?

13 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 13 Preliminaries and Problem Definitions Edge Relabeling Problem Instance: A graph G, labelings L E and L′ E, and t N. Question: Can labeling L E evolve into L′ E in t or fewer edge mutations?

14 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 14 Outline Introduction Preliminaries and Problem Definitions Relating Vertex and Edge Relabeling Tight Bounds for the Relabeling Problem Relabeling with Privileged Labels Open Problems References

15 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 15 Relating Vertex and Edge Relabeling Theorem (Vertex/Edge Relabeling Related) Vertex Relabeling Problem is NC 1 many-one reducible to the Edge Relabeling Problem, and the Edge Relabeling Problem is NC 1 many-one reducible to the Vertex Relabeling Problem.

16 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 16 Outline Introduction Preliminaries and Problem Definitions Relating Vertex and Edge Relabeling Tight Bounds for the Relabeling Problem Relabeling with Privileged Labels Open Problems References

17 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 17 Tight Bounds for the Relabeling Problem Theorem (Vertex Relabeling Upper Bound) Let G = (V,E) be a graph, L V and L′ V vertex labelings, and t = n(n — 1)/2, then the answer to the Vertex Relabeling Problem is YES. That is, any labeled graph can evolve into any other labeled graph in at most n(n — 1)/2 mutations.

18 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 18 Tight Bounds for the Relabeling Problem Theorem (Vertex Relabeling Upper Bound) Proof: –Consider the number of mutations required to change an arbitrary labeling L V into an arbitrary labeling L′ V. –Construct a spanning tree T of G. Let p 1 p 2 … p n be vertex numbers (not labels) that denote the Prüfer code order when the leaves of T are deleted during the process of constructing a Prüfer code. –Note, p j {v i | 1 ≤ i ≤ n} for 1 ≤ j ≤ n.

19 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 19 Tight Bounds for the Relabeling Problem Theorem (Vertex Relabeling Upper Bound) Proof (cont.): –Not interested in the actual Prüfer code itself but rather just the leaf elimination order. –Mutate labels from L V into their positions in L′ V in the order specified by the p i ’s and along the path in the spanning tree from their starting position in L V to their final position in L′ V.

20 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 20 Tight Bounds for the Relabeling Problem Theorem (Vertex Relabeling Upper Bound) Proof (cont.): –To move L′ V (p 1 ) from the initial labeling to its final position can take at most n — 1 mutations. Note, p 1 is an initial leaf in T, and T contains at most n – 1 edges. –To move L′ (p 2 ) from the initial labeling to its final position, we need at most n — 2 mutations, since L′ (p 1 ) is already in its rightful place.

21 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 21 Tight Bounds for the Relabeling Problem Theorem (Vertex Relabeling Upper Bound) Proof (cont.): –After k iterations, where all of the labels L′ V (p 1 ) through and including L′ (p k ) are in their correct places, then, to move L′ V (p (k+1) ) to its correct place, we need at most n – k – 1 mutations, since the remaining spanning tree induced by the vertex set, V(T) — {p i | 1 ≤ i ≤ k} has at most n — k — 1 edges.

22 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 22 Tight Bounds for the Relabeling Problem Theorem (Vertex Relabeling Upper Bound) Proof (cont.): –No mutations are performed in locations of the tree that have already been completed. –The total number of mutations is therefore at most n(n — 1)/2. □

23 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 23 Tight Bounds for the Relabeling Problem Corollary (Edge Relabeling Upper Bound) Let G = (V,E) be a graph, L E and L′ E edge labelings, and t = m(m — 1)/2, then the answer to the Edge Relabeling Problem is YES. That is, any labeled graph can evolve into any other labeled graph in at most m(m — 1)/2 mutations.

24 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 24 Tight Bounds for the Relabeling Problem Theorem (Lower Bounds for Relabeling Graphs) [Muir 1882] There is a graph G = (V,E), labelings L V and L′ V, and t = (n(n — 1)/2) — 1 such that the Vertex Relabeling Problem has an answer of NO. That is, there exist two labelings that require n(n — 1)/2 mutations to evolve one into the other. There is a graph H = (V′,E′), labelings L E′ and L′ E′, and t = (m(m — 1)/2) — 1 such that the Edge Relabeling Problem has an answer of NO.

25 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 25 Tight Bounds for the Relabeling Problem Theorem (Lower Bounds for Relabeling Graphs) Proof: We provide two vertex labelings on a path that require n(n — 1)/2 mutations to be evolved from one to the other. For convenience, a labeling is represented by a permutation of {1,2,…,n}. There exists a permutation, namely, n(n — 1)…1, where at least n(n — 1)/2 mutations are needed to obtain it from the permutation 1 2 … n.

26 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 26 Tight Bounds for the Relabeling Problem Theorem (Lower Bounds for Relabeling Graphs) Proof (cont.): If we use fewer than n(n — 1)/2 mutations on 1 2 … n, we can not end up with n(n — 1)…1. View the permutation a 1 a 2 …a n as a string s. Continued…

27 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 27 Tight Bounds for the Relabeling Problem Theorem (Lower Bounds for Relabeling Graphs) Attach a unique parameter p(s) to each string by letting p(s) be the number of occurrences a i and a j, where a i > a j among all i a j }|. p(s) is the number of inversions of the permutation.

28 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 28 Tight Bounds for the Relabeling Problem Theorem (Lower Bounds for Relabeling Graphs) Proof (cont.): There are exactly n(n — 1)/2 pairs i < j if i and j are among 1,2,…,n, so for every such strings s, we clearly have 0 ≤ p(s) ≤ n(n — 1)/2. Each mutation reduces or increases the value of p(. ) by exactly one. If s′ is the string obtained from s, then |p(s′)–p(s)| = 1, the absolute value of p(s′)–p(s) equals 1.

29 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 29 Tight Bounds for the Relabeling Problem Theorem (Lower Bounds for Relabeling Graphs) Proof (cont.): Since p(1 2…n) = 0 and p(n(n — 1)…1) = n(n — 1)/2, we need at least n(n — 1)/2 mutations to obtain n(n — 1)…1 from 1 2…n. In fact, we can use exactly n(n — 1)/2 mutations to obtain n(n — 1)…1 from 1 2…n. This shows there are graphs and vertex labelings that require n(n — 1)/2 mutations. □

30 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 30 Tight Bounds for the Relabeling Problem Theorem (Tight Bound on Path Relabeling Complexity) [Muir 1882] Let P n be the path on n vertices, L V and L′ V vertex labelings, and t N. Then the answer to the Vertex Relabeling Problem is YES if and only if t ≥ p(L V, L′ V ). Proof: For the sake of simplicity, assume we are to evolve s into s′.

31 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 31 Tight Bounds for the Relabeling Problem Theorem (Tight Bound on Path Relabeling Complexity) Proof (cont.): A mutate sequence is a sequence of strings with s 0 = s, s m = s′, and where s i+1 is obtained from s i by a single mutation, 0 ≤ i ≤ m — 1. In this case, we have for an arbitrary labeling s = a 1 a 2 … a n that the parameter p(. ) satisfies

32 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 32 Tight Bounds for the Relabeling Problem Theorem (Tight Bound on Path Relabeling Complexity) Proof (cont.): We now argue by using induction on n that p(s) mutations suffice to evolve s into s′. This is clearly true for n = 2. Assume this assertion is true for length (n — 1)-strings, and let s = a 1 a 2 …a n be such that n = a i, for a fixed i, 1 ≤ i ≤ n. In this case, we have p(s) = n — i + p( ŝ ), where ŝ = a 1 …a i-1 a i+1 …a n.

33 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 33 Tight Bounds for the Relabeling Problem Theorem (Tight Bound on Path Relabeling Complexity) Proof (cont.): Clearly in s we can move n = a i to the rightmost position by precisely n — i mutations. By induction, we can obtain 1 2 … (n — 1) from ŝ by p(ŝ) mutations. Hence, we are able to evolve s into s′ using p(s) mutations.

34 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 34 Tight Bounds for the Relabeling Problem Theorem (Tight Bound on Path Relabeling Complexity) Proof (cont.): If we have two vertex labelings L V and L′ V of the vertices of the path P n, we can define the corresponding relative parameter p(L V, L′ V ), as p(s), where s is the unique permutation obtained from L V by renaming the labels in L′ V from left-to- right as 1,2,…,n. We note that p(L V, L′ V ) = p(L′ V, L V ). □

35 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 35 Tight Bounds for the Relabeling Problem Theorem (Tight Bound on Path Relabeling Complexity) Let P n be the path on n vertices, L V and L′ V vertex labelings, and t N. Then we can evolve the labeling L V into L′ V using t mutations if and only if t = p(L V, L′ V ) + 2k for some nonnegative integer k.

36 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 36 Tight Bounds for the Relabeling Problem Theorem (Tight Bound on Path Relabeling Complexity) Proof: By the previous theorem we can always evolve L V into L′ V using the minimum of p(L V, L′ V ) mutations. Repeating the last mutation 2k times is not going to alter L′ V, since repeating a fixed mutation an even number of times corresponds to the identity (or neutral) relabeling. For any nonnegative integer k one can always evolve L V into L′ V using t = p(L V, L′ V ) + 2k mutations.

37 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 37 Tight Bounds for the Relabeling Problem Theorem (Tight Bound on Path Relabeling Complexity) Proof (cont.): We show that t — p(L V, L′ V ) must be even. Assume L V is given by the string s = a 1 a 2 …a n and L′ V by the string s′ = 1 2 … n. Now, let and be two mutation sequences with s 0 = s′ 0 = s and s m = s′ m′ = s′.

38 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 38 Tight Bounds for the Relabeling Problem Theorem (Tight Bound on Path Relabeling Complexity) Proof (cont.): Since p(s 0 ) = p(s′ 0 ) = p(s) and p(s m ) = p(s′ m′ ) = 0, we have and (cont. next slide)

39 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 39 Tight Bounds for the Relabeling Problem Theorem (Tight Bound on Path Relabeling Complexity) Proof (cont.): where = |{i {1,…,m} : p(s i ) – p(s i+1 ) = 1}|, = |{i {1,…,m} : p(s i ) – p(s i+1 ) = -1}|, = |{i {1,…,m′ } : p(s′ i ) – p(s′ i+1 ) = 1}|, and = |{i {1,…,m′ } : p(s′ i ) – p(s′ i+1 ) = -1}|. We have. m = and m′ = and we obtain…

40 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 40 Tight Bounds for the Relabeling Problem Theorem (Tight Bound on Path Relabeling Complexity) Proof (cont.): m′ — m =, and thus m and m′ must have the same parity. This shows that if L V is evolved into L′ V in exactly t mutations, then t — p(s) must be even. □ This theorem re-establishes a known result in the theory of permutations that each permutation has unique parity.

41 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 41 Outline Introduction Preliminaries and Problem Definitions Relating Vertex and Edge Relabeling Tight Bounds for the Relabeling Problem Relabeling with Privileged Labels Open Problems References

42 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 42 Relabeling with Privileged Labels Definition (Vertex Relabeling with Privileged Labels Problem) Instance: A graph G, labelings L V and L′ V, a nonempty set S {1,2,…,n} of privileged labels, and t N. Question: Can labeling L V evolve into L′ V in less than t restricted vertex mutations?

43 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 43 Relabeling with Privileged Labels Definition (Edge Relabeling with Privileged Labels Problem) Instance: A graph G, labelings L E and L′ E, a nonempty set S {1,2,…,n} of privileged labels, and t N. Question: Can labeling L E evolve into L′ E in less than t restricted edge mutations?

44 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 44 Relabeling with Privileged Labels Theorem (General Unsolvability, Privileged Labels) Among all connected vertex labeled graphs on n vertices with k privileged labels where k {0,1,…,n — 2}, the Vertex Relabeling with Privileged Labels Problem is, in general, unsolvable. Among all connected edge labeled graphs on m edges with k privileged labels where k {0,1,…,m — 2}, the Edge Relabeling with Privileged Labels Problem is, in general, unsolvable.

45 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 45 Relabeling with Privileged Labels Definition ((n x n)-Puzzle Problem) Instance: Two n x n board configurations B 1 and B 2, and k N. Question: Is there a sequence of at most k moves that transforms B 1 into B 2 ?

46 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 46 Relabeling with Privileged Labels Theorem (Intractability, Privileged Labels) The Vertex Graph Relabeling with Privileged Labels Problem is NP-complete. Proof: Reduce the (n x n)-Puzzle Problem to the Vertex Graph Relabeling with Privileged Labels Problem by taking G as an n x n mesh, L V corresponding to B 1, L' V corresponding to B 2, S = {n 2 } corresponding to the blank space, and t = k.

47 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 47 Relabeling with Privileged Labels Theorem (Intractability, Privileged Labels) Proof (cont.): The instance of the (n x n)- Puzzle Problem is YES if and only if the answer to the constructed instance of the Vertex Graph Relabeling with Privileged Labels Problem is also YES.

48 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 48 Relabeling with Privileged Labels Theorem (2-Connected Unsolvability, Privileged Labels) Among all 2-connected vertex labeled graphs on n vertices with k privileged labels where k {0,1,…,n — 3}, the Vertex Relabeling with Privileged Labels Problem is, in general, unsolvable. Among all 2- connected edge labeled graphs on m edges with k privileged labels where k {0,1,…, m — 3}, the Edge Relabeling with Privileged Labels Problem is, in general, unsolvable.

49 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 49 Relabeling with Privileged Labels Theorem (2-Connected Unsolvability, Privileged Labels) Proof: Let n ≥ 3 and consider two vertex labelings L V and L′ V of the cycle C n, where we have precisely k privileged labels p 1,…,p k, where k {0,1,…,n — 3}. For a fixed planar embedding of C n, assume the labelings are given cyclically in clockwise order: L V : (p 1,…,p k,1,2,3,…,n – k), and L′ V : (p 1,…,p k,2,1,3,…,n – k).

50 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 50 Relabeling with Privileged Labels Claims If a simple graph is neither a path nor a cycle, then it has a spanning tree that is not a path (and hence contains a vertex of degree of at least three). Among vertex labeled trees, which are not paths, with exactly two non-privileged labels, any two labels can be swapped using restricted mutations.

51 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 51 Relabeling with Privileged Labels Lemma Among vertex labeled trees, which are not paths, with exactly two non-privileged labels, the Vertex Relabeling with Privileged Labels Problem is solvable and in P.

52 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 52 Relabeling with Privileged Labels Theorem (Vertex Solvability, Two Privileged Labels) Among all connected vertex labeled graphs G on n ≥ 4 vertices with all but two vertex labels privileged, the Vertex Relabeling with Privileged Labels Problem is solvable if and only if G is not a path.

53 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 53 Relabeling with Privileged Labels Theorem (Edge Solvability, Two Privileged Labels) Among all connected edge labeled graphs G on n ≥ 4 edges with all but two edge labels privileged, the Edge Relabeling with Privileged Labels Problem is solvable if and only if G is not a path.

54 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 54 Outline Introduction Preliminaries and Problem Definitions Relating Vertex and Edge Relabeling Tight Bounds for the Relabeling Problem Relabeling with Privileged Labels Open Problems References

55 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 55 Open Problems Study other types of mutation functions where, for example, labels along an entire path are mutated, or where labels can be reused. In the parallel setting, compute the sequence of mutations required for the evolution of one labeling into another. The parallel time for computing the sequence could be much smaller than the sequential time to execute the mutation sequence.

56 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 56 Open Problems For various classes of graphs determine the probability of one labelings evolving naturally into another. Such an evolution of a labeling could be used to model mutation periods. Study the properties of the graphs of all labelings. In this graph all labelings of a given graph are vertices and two vertices are connected if they are one mutation apart. Other conditions for edge placement may also be worthwhile to examine.

57 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 57 Open Problems Determine if there is a version of the Edge Relabeling with Privileged Labels Problem that is NP-complete. Define the cost of a mutation sequence to be the sum of the weights on all edges that are mutated. Determine mutation sequences that minimize the cost of evolving one labeling into another. Explore other cost functions.

58 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 58 Outline Introduction Preliminaries and Problem Definitions Relating Vertex and Edge Relabeling Tight Bounds for the Relabeling Problem Relabeling with Privileged Labels Open Problems References

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60 The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 60 References Gardner, M. Mathematical Puzzles of Sam Loyd, Dover Publications, 1959. Gates, W. H. and Papdimitriou, C. H. Bounds for sorting by reversal. Discrete Mathematics, 27:47-57, 1979. Grace, T. Graceful, Harmonious, and Sequential Graphs. Ph.D. Thesis, University of Illinois at Chicago Circle, 1982. Grace, T. On sequential labelings of graphs. Journal of Graph Theory, 7:195-201, 1983. Greenlaw, R., Halldórsson, M., and Petreschi, R. On Computing Prüfer Codes and Their Corresponding Trees Optimally in Parallel (Extended Abstract). Proceedings of Journées de l'Informatique Messine (JIM 2000), Université de Metz, France, Laboratoire d'Informatique Théorique et Appliqueé, editor D. Kratsch, pages 125-130, 2000. Greenlaw, R., Hoover, H. J., and Ruzzo, W. L. Limits to Parallel Computation: P- Completeness Theory, Oxford University Press, 1995. Hungerford, Thomas W. Algebra, Graduate Texts in Mathematics GTM—73, Springer Verlag, 1987. Kakoulis, K. G. and Tollis, I. G. On the complexity of the edge label placement problem. Computational Geometry, 18(1):1-17, 2001. Kantabutra, S. The complexity of label relocation problems on graphs. Proceedings of the 8th Asian Symposium on Computer Mathematics, National University of Singapore, Singapore, December 2007.

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