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The Complexity of the Evolution of Graph Labelings. Geir Agnarsson Raymond Greenlaw Sanpawat Kantabutra. Outline. Introduction Preliminaries and Problem Definitions Relating Vertex and Edge Relabeling Tight Bounds for the Relabeling Problem Relabeling with Privileged Labels Open Problems

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the complexity of the evolution of graph labelings

The Complexity of the Evolution of Graph Labelings

Geir Agnarsson

Raymond Greenlaw

Sanpawat Kantabutra

outline
Outline
  • Introduction
  • Preliminaries and Problem Definitions
  • Relating Vertex and Edge Relabeling
  • Tight Bounds for the Relabeling Problem
  • Relabeling with Privileged Labels
  • Open Problems
  • References
outline3
Outline
  • Introduction
  • Preliminaries and Problem Definitions
  • Relating Vertex and Edge Relabeling
  • Tight Bounds for the Relabeling Problem
  • Relabeling with Privileged Labels
  • Open Problems
  • References
introduction
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.
introduction5
Introduction
  • Two configurations, a “start” and “goal” are given. Transform the “start” to the “goal” configuration with certain conditions on the movement of labels.
introduction6
Introduction
  • Presentation of new results and the extension of existing results; in particular,
    • NC1 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.
introduction7
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.
introduction8
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.
outline9
Outline
  • Introduction
  • Preliminaries and Problem Definitions
  • Relating Vertex and Edge Relabeling
  • Tight Bounds for the Relabeling Problem
  • Relabeling with Privileged Labels
  • Open Problems
  • References
preliminaries and problem definitions
Preliminaries and Problem Definitions
  • Let SV, SEN = {1,2,…}.
  • A labeling LV of V is a mapping LV: V  SV.
  • A labeling LE of E is a mapping LE: ESE.
  • SV = {1,2,…,n} and SE = {1,2,…,m}.
  • Graphs are associated with labelings using angle bracket notation.
  • A mutation or flip function f maps triples <G, LV,LE> to triples <G, L′V,L′E>.
preliminaries and problem definitions11
Preliminaries and Problem Definitions
  • A consecutive vertex mutation function is defined where f maps a pair <G, LV> to a pair <G, L′V> with the following conditions:
    • LV = L′V, except on two vertices u and w.
    • {u, w} E.
    • LV(u) = L′V(w) and LV(w) = L′V(u).
    • SV = {1,2,…,n}.
    • f is a bijection.
preliminaries and problem definitions12
Preliminaries and Problem Definitions

Vertex Relabeling Problem

  • Instance: A graph G, labelings LV and L′V, and tN.
  • Question: Can labeling LV evolve into L′V in t or fewer vertex mutations?
preliminaries and problem definitions13
Preliminaries and Problem Definitions

Edge Relabeling Problem

  • Instance: A graph G, labelings LE and L′E, and tN.
  • Question: Can labeling LE evolve into L′E in t or fewer edge mutations?
outline14
Outline
  • Introduction
  • Preliminaries and Problem Definitions
  • Relating Vertex and Edge Relabeling
  • Tight Bounds for the Relabeling Problem
  • Relabeling with Privileged Labels
  • Open Problems
  • References
relating vertex and edge relabeling
Relating Vertex and Edge Relabeling

Theorem (Vertex/Edge Relabeling Related)

  • Vertex Relabeling Problem is NC1 many-one reducible to the Edge Relabeling Problem, and the Edge Relabeling Problem is NC1 many-one reducible to the Vertex Relabeling Problem.
outline16
Outline
  • Introduction
  • Preliminaries and Problem Definitions
  • Relating Vertex and Edge Relabeling
  • Tight Bounds for the Relabeling Problem
  • Relabeling with Privileged Labels
  • Open Problems
  • References
tight bounds for the relabeling problem
Tight Bounds for the Relabeling Problem

Theorem (Vertex Relabeling Upper Bound)

  • Let G = (V,E) be a graph, LV 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.
tight bounds for the relabeling problem18
Tight Bounds for the Relabeling Problem

Theorem (Vertex Relabeling Upper Bound)

  • Proof:
    • Consider the number of mutations required to change an arbitrary labeling LV into an arbitrary labeling L′V.
    • Construct a spanning tree T of G. Let p1p2 … pn 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, pj {vi | 1 ≤i≤n} for 1 ≤j≤n.
tight bounds for the relabeling problem19
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 LV into their positions in L′V in the order specified by the pi’s and along the path in the spanning tree from their starting position in LV to their final position in L′V .
tight bounds for the relabeling problem20
Tight Bounds for the Relabeling Problem

Theorem (Vertex Relabeling Upper Bound)

  • Proof (cont.):
    • To move L′V(p1) from the initial labeling to its final position can take at most n — 1 mutations. Note, p1 is an initial leaf in T, and T contains at most n – 1 edges.
    • To move L′ (p2) from the initial labeling to its final position, we need at most n — 2 mutations, since L′ (p1) is already in its rightful place.
tight bounds for the relabeling problem21
Tight Bounds for the Relabeling Problem

Theorem (Vertex Relabeling Upper Bound)

  • Proof (cont.):
    • After k iterations, where all of the labels L′V(p1) through and including L′ (pk) 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) — {pi | 1 ≤i≤k} has at most n — k — 1 edges.
tight bounds for the relabeling problem22
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. □
tight bounds for the relabeling problem23
Tight Bounds for the Relabeling Problem

Corollary (Edge Relabeling Upper Bound)

  • Let G = (V,E) be a graph, LE 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.
tight bounds for the relabeling problem24
Tight Bounds for the Relabeling Problem

Theorem (Lower Bounds for Relabeling Graphs) [Muir 1882]

  • There is a graph G = (V,E), labelings LV 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 LE′ and L′E′, and t = (m(m— 1)/2) — 1 such that the Edge Relabeling Problem has an answer of NO.
tight bounds for the relabeling problem25
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.
tight bounds for the relabeling problem26
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 a1a2…an as a string s.
  • Continued…
tight bounds for the relabeling problem27
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 ai and aj, where ai > aj among all i < j, so p(s) = |{{i,j} : 1 ≤i<j≤n and ai > aj}|.
  • p(s) is the number of inversions of the permutation.
tight bounds for the relabeling problem28
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.
tight bounds for the relabeling problem29
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.

tight bounds for the relabeling problem30
Tight Bounds for the Relabeling Problem

Theorem (Tight Bound on Path Relabeling Complexity) [Muir 1882]

  • Let Pn be the path on n vertices, LV and L′V vertex labelings, and tN. Then the answer to the Vertex Relabeling Problem is YES if and only if t ≥ p(LV,L′V).
  • Proof: For the sake of simplicity, assume we are to evolve s into s′.
tight bounds for the relabeling problem31
Tight Bounds for the Relabeling Problem

Theorem (Tight Bound on Path Relabeling Complexity)

  • Proof (cont.): A mutate sequence is a sequence of strings with s0 = s, sm = s′, and where si+1 is obtained from si by a single mutation, 0 ≤i≤m — 1.
  • In this case, we have for an arbitrary labeling s = a1a2 … an that the parameter p(.) satisfies
tight bounds for the relabeling problem32
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 = a1a2…an be such that n = ai, for a fixed i, 1 ≤i≤n. In this case, we have p(s) = n — i + p(ŝ), where ŝ = a1…ai-1ai+1…an.
tight bounds for the relabeling problem33
Tight Bounds for the Relabeling Problem

Theorem (Tight Bound on Path Relabeling Complexity)

  • Proof (cont.): Clearly in s we can move n = ai 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.
tight bounds for the relabeling problem34
Tight Bounds for the Relabeling Problem

Theorem (Tight Bound on Path Relabeling Complexity)

  • Proof (cont.): If we have two vertex labelings LV and L′V of the vertices of the path Pn, we can define the corresponding relative parameter p(LV,L′V), as p(s), where s is the unique permutation obtained from LV by renaming the labels in L′V from left-to-right as 1,2,…,n. We note that p(LV,L′V) = p(L′V,LV). □
tight bounds for the relabeling problem35
Tight Bounds for the Relabeling Problem

Theorem (Tight Bound on Path Relabeling Complexity)

  • Let Pn be the path on n vertices, LV and L′V vertex labelings, and tN. Then we can evolve the labeling LV into L′V using t mutations if and only if t = p(LV,L′V) + 2k for some nonnegative integer k.
tight bounds for the relabeling problem36
Tight Bounds for the Relabeling Problem

Theorem (Tight Bound on Path Relabeling Complexity)

  • Proof: By the previous theorem we can always evolve LV into L′V using the minimum of p(LV,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 LV into L′V using t = p(LV,L′V) + 2k mutations.
tight bounds for the relabeling problem37
Tight Bounds for the Relabeling Problem

Theorem (Tight Bound on Path Relabeling Complexity)

  • Proof (cont.): We show that t — p(LV,L′V) must be even. Assume LV is given by the string s = a1a2…an and L′V by the string s′ = 1 2 … n.
  • Now, let and be two mutation sequences with s0 = s′0 = s and sm = s′m′ = s′.
tight bounds for the relabeling problem38
Tight Bounds for the Relabeling Problem

Theorem (Tight Bound on Path Relabeling Complexity)

  • Proof (cont.): Since p(s0) = p(s′0) = p(s) and p(sm) = p(s′m′) = 0, we have

and (cont. next slide)

tight bounds for the relabeling problem39
Tight Bounds for the Relabeling Problem

Theorem (Tight Bound on Path Relabeling Complexity)

  • Proof (cont.): where
  • = |{i {1,…,m} : p(si) – p(si+1) = 1}|,
  • = |{i {1,…,m} : p(si) – p(si+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…
tight bounds for the relabeling problem40
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 LVis evolved intoL′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.
outline41
Outline
  • Introduction
  • Preliminaries and Problem Definitions
  • Relating Vertex and Edge Relabeling
  • Tight Bounds for the Relabeling Problem
  • Relabeling with Privileged Labels
  • Open Problems
  • References
relabeling with privileged labels
Relabeling with Privileged Labels

Definition (Vertex Relabeling with Privileged Labels Problem)

  • Instance: A graph G, labelings LV and L′V, a nonempty set S {1,2,…,n} of privileged labels, and tN.
  • Question: Can labeling LV evolve into L′V in less than t restricted vertex mutations?
relabeling with privileged labels43
Relabeling with Privileged Labels

Definition (Edge Relabeling with Privileged Labels Problem)

  • Instance: A graph G, labelings LE and L′E, a nonempty set S {1,2,…,n} of privileged labels, and tN.
  • Question: Can labeling LE evolve into L′E in less than t restricted edge mutations?
relabeling with privileged labels44
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.
relabeling with privileged labels45
Relabeling with Privileged Labels

Definition ((n x n)-Puzzle Problem)

  • Instance: Two n x n board configurations B1 and B2, and k N.
  • Question: Is there a sequence of at most k moves that transforms B1 into B2?
relabeling with privileged labels46
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, LV corresponding to B1, L'V corresponding to B2, S = {n2} corresponding to the blank space, and t = k.
relabeling with privileged labels47
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.
relabeling with privileged labels48
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.
relabeling with privileged labels49
Relabeling with Privileged Labels

Theorem (2-Connected Unsolvability, Privileged Labels)

  • Proof: Let n ≥ 3 and consider two vertex labelings LV and L′V of the cycle Cn, where we have precisely k privileged labels p1,…,pk, where k {0,1,…,n — 3}. For a fixed planar embedding of Cn, assume the labelings are given cyclically in clockwise order: LV : (p1,…,pk,1,2,3,…,n – k), and L′V : (p1,…,pk,2,1,3,…,n – k).
relabeling with privileged labels50
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.
relabeling with privileged labels51
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.
relabeling with privileged labels52
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.
relabeling with privileged labels53
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.
outline54
Outline
  • Introduction
  • Preliminaries and Problem Definitions
  • Relating Vertex and Edge Relabeling
  • Tight Bounds for the Relabeling Problem
  • Relabeling with Privileged Labels
  • Open Problems
  • References
open problems
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.
open problems56
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.
open problems57
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.
outline58
Outline
  • Introduction
  • Preliminaries and Problem Definitions
  • Relating Vertex and Edge Relabeling
  • Tight Bounds for the Relabeling Problem
  • Relabeling with Privileged Labels
  • Open Problems
  • References
references
References
  • Agnarsson, G. and Greenlaw, R. Graph Theory: Modeling, Applications, and Algorithms. Prentice Hall, 2007.
  • Agnarsson, G., Greenlaw, R., and Kantabutra, S. The Complexity of the Graph Relabeling Problem. Manuscript, 2008.
  • Appel, K. and Haken, W. Every Map is Four Colorable. Providence, RI: American Mathematical Society, first edition, 1989.
  • Chang, G. J., Hsu, D. F., and Rogers, D. G. Additive variations on a graceful theme: some results on harmonious and other related graphs. Congressus Numerantium, 32:181-197, 1981.
  • Cormen, T. H., Leiserson, C. E., Rivest, R. L., and Stein, C. Introduction to Algorithms, second edition, MIT Press, 2001.
  • Even, S. and Tarjan, R. E. A combinatorial problem which is complete in polynomial space. Journal of the Association for Computing Machinery, 23:710-719, 1976.
  • Fraenkel, A. S., Garey, M. R., Johnson, D. S., Schaefer, T., and Yesha, Y. The complexity of Checkers on an N x N board—Preliminary Report, Proceedings of the 19th Annual Symposium on the Foundations of Computer Science, IEEE Computer Society, Long Beach, CA, pages 55-64, 1978.
  • Gallian, J. A dynamic survey of graph labeling (10th edition). Electronic Journal of Combinatorics, 14:1-180, 2007.
references60
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.
references61
References
  • Kato, T. and Imai, H. The NP-completeness of the character placement problem of 2 or 3 degrees of freedom. Record of Joint Conference of Electrical and Electronic Engineers in Kyushu (in Japanese), pages 11-18, 1988.
  • Knuth, Donald E. The Art of Computer Programming, Volume 3, second edition, Addison Wesley, 1998.
  • Lam, T. W. and Yue, F. L. Optimal Edge Ranking of Trees in Linear Time. Algorithmica, 30(1): 12-33, 2001.
  • Lee, S. M., Lee, A. N-T., Sun, H., and Wen, I. On the integer-magic spectra of graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 42:77-86, 2002.
  • Lee, S. M. and Wong, H. On the integer spectra of the power of paths. Journal of Combinatorial Math and Combinatorial Computing, 42:187-194, 2002.
  • Lichtenstein, D. and Sipser, M. GO is Pspace hard. Proceedings of the 19th Annual Symposium on the Foundations of Computer Science, IEEE Computer Society, Long Beach, CA, pages 48-54, 1978.
  • Marks, J. and Shieber, S. The computational complexity of cartographic label placement. Technical Reports TR-05-91. Advanced Research in Computing Technology, Harvard University, 1991.
  • Moon, J. W. Counting Labelled Trees Canadian Mathematical Monographs William Clowes and Sons, Limited, 1970.
references62
References
  • Muir, T. A treatise on the theory of determinants. McMillan and Co., London, 1882.
  • Muir, T. A treatise on the theory of determinants. Revised and enlarged by W. H. Metzler. Dover Publications Inc. New York, 1960.
  • Ratner, D. and Warmuth, M. The (n2-1)-puzzle and related relocation problems. Journal of Symbolic Computation, 10:111-137, 1990.
  • Rosa, A. On certain valuations of the vertices of a graph. Theory of Graphs (International Symposium, Rome, July 1966). Gordon and Breach, NY and Dunod Paris, pages 349-355, 1967.
  • Schaefer, T. J. Complexity of some two-person perfect-information games. Journal of Computer and System Science, 16:185-225, 1978.
  • Sedláček, J. Problem 27, in Theory of Graphs and Its Applications. Proc. Symposium Smolenice, pages 163-167, 1963.
  • Slocum, J. and Sonneveld, D. The 15 puzzle: how it drove the world crazy. The puzzle that started the craze of 1880. How America's greatest puzzle designer, Sam Loyd, fooled everyone for 115 years. Beverly Hills, CA, Slocum Puzzle Foundation, 2006.
  • Stewart, B. M. Magic graphs. Canadian Journal of Mathematics, 18:1031-1059, 1966.
  • Stewart, B. M. Supermagic complete graphs. Canadian Journal of Mathematics, 19:427-438, 1967.
references63
References
  • Storey, W. E. Notes on the 15 Puzzle 2. American Journal of Mathematics, 2(4):399-404, 1879.
  • Wilkinson, B. and Allen, M. Parallel Programming: Techniques and Applications Using Networked Workstations and Parallel Computers, Prentice Hall, 1999.
  • Wilson, R. M. Graph puzzles, homotopy, and the alternating group. Journal of Combinatorial Theory B, 16:86-96, 1974.