Abstract
An L(2,1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that |f(x) − f(y)| ≥ 2 if x and y are adjacent and |f(x) − f(y)| ≥ 1 if x and y are at distance 2, for all x and y in V(G). A k-L(2,1)-labeling is an L(2,1)-labeling f:V(G)→{0,...,k}, and the L(2,1)-labeling problem asks the minimum k, which we denote by λ(G), among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2, and tree is one of very few classes for which the problem is polynomially solvable. The running time of the best known algorithm for trees had been O(Δ4.5 n) for more than a decade, and an O( min {n 1.75,Δ1.5 n})-time algorithm has appeared recently, where Δ is the maximum degree of T and n = |V(T)|, however, it has been open if it is solvable in linear time. In this paper, we finally settle this problem for L(2,1)-labeling of trees by establishing a linear time algorithm.
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References
Bodlaender, H.L., Kloks, T., Tan, R.B., van Leeuwen, J.: Approximations for λ-coloring of graphs. The Computer Journal 47, 193–204 (2004)
Calamoneri, T.: The L(h,k)-labelling problem: A survey and annotated bibliography. The Computer Journal 49, 585–608 (2006), http://www.dsi.uniroma1.it/~calamo/PDF-FILES/survey.pdf (January 13, 2009)
Chang, G.J., Ke, W.-T., Kuo, D., Liu, D.D.-F., Yeh, R.K.: On L(d,1)-labeling of graphs. Discr. Math. 220, 57–66 (2000)
Chang, G.J., Kuo, D.: The L(2,1)-labeling problem on graphs. SIAM J. Discr. Math. 9, 309–316 (1996)
Fiala, J., Golovach, P.A., Kratochvíl, J.: Distance constrained labelings of graphs of bounded treewidth. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 360–372. Springer, Heidelberg (2005)
Fiala, J., Golovach, P.A., Kratochvíl, J.: Distance constrained labelings of trees. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 125–135. Springer, Heidelberg (2008)
Fiala, J., Golovach, P.A., Kratochvíl, J.: Computational complexity of the distance constrained labeling problem for trees (Extended abstract). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 294–305. Springer, Heidelberg (2008)
Fiala, J., Kloks, T., Kratochvíl, J.: Fixed-parameter complexity of λ-labelings. Discr. Appl. Math. 113, 59–72 (2001)
Goldberg, A.V., Rao, S.: Beyond the flow decomposition barrier. J. ACM 45, 783–797 (1998)
Griggs, J.R., Yeh, R.K.: Labelling graphs with a condition at distance 2. SIAM J. Disc. Math. 5, 586–595 (1992)
Hasunuma, T., Ishii, T., Ono, H., Uno, Y.: An O(n 1.75) algorithm for L(2,1)-labeling of trees. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 185–197. Springer, Heidelberg (2008)
Hasunuma, T., Ishii, T., Ono, H., Uno, Y.: A linear time algorithm for L(2,1)-labeling of trees. CoRR abs/0810.0906 (2008)
Havet, F., Reed, B., Sereni, J.-S.: L(2,1)-labelling of graphs. In: Proc. 19th SIAM-SODA, pp. 621–630 (2008)
Hopcroft, J.E., Karp, R.M.: An n5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2, 225–231 (1973)
Kratochvíl, J., Kratsch, D., Liedloff, M.: Exact algorithms for L(2,1)-labeling of graphs. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 513–524. Springer, Heidelberg (2007)
Wang, W.-F.: The L(2,1)-labelling of trees. Discr. Appl. Math. 154, 598–603 (2006)
Yeh, R.K.: A survey on labeling graphs with a condition at distance two. Discr. Math. 306, 1217–1231 (2006)
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Hasunuma, T., Ishii, T., Ono, H., Uno, Y. (2009). A Linear Time Algorithm for L(2,1)-Labeling of Trees. In: Fiat, A., Sanders, P. (eds) Algorithms - ESA 2009. ESA 2009. Lecture Notes in Computer Science, vol 5757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04128-0_4
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DOI: https://doi.org/10.1007/978-3-642-04128-0_4
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