Abstract
An H(p,q)-labeling of a graph G is a vertex mapping f:V G →V H such that the distance between f(u) and f(v) (measured in the graph H) is at least p if the vertices u and v are adjacent in G, and the distance is at least q if u and v are at distance two in G. This notion generalizes the notions of L(p,q)- and C(p,q)-labelings of graphs studied as particular graph models of the Frequency Assignment Problem. We study the computational complexity of the problem of deciding the existence of such a labeling when the graphs G and H come from restricted graph classes. In this way we extend known results for linear and cyclic labelings of trees, with a hope that our results would help to open a new angle of view on the still open problem of L(p,q)-labeling of trees for fixed p > q > 1 (i.e., when G is a tree and H is a path).
We present a polynomial time algorithm for H(p,1)-labeling of trees for arbitrary H. We show that the H(p,q)-labeling problem is NP-complete when the graph G is a star. As the main result we prove NP-completeness for H(p,q)-labeling of trees when H is a symmetric q-caterpillar.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Calamoneri, T.: The L(h,k)-labeling problem: A survey and annotated bibliography. Computer Journal 49(5), 585–608 (2006)
Chang, G.J., Ke, W.-T., Liu, D.D.-F., Yeh, R.K.: On l(d,1)-labellings of graphs. Discrete Mathematics 3(1), 57–66 (2000)
Chang, G.J., Kuo, D.: The L(2,1)-labeling problem on graphs. SIAM Journal of Discrete Mathematics 9(2), 309–316 (1996)
Courcelle, B.: The monadic second-order logic of graphs. I: Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)
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., Kloks, T., Kratochvíl, J.: Fixed-Parameter Complexity of λ-Labelings. In: Widmayer, P., Neyer, G., Eidenbenz, S. (eds.) WG 1999. LNCS, vol. 1665, Springer, Heidelberg (1999)
Fiala, J., Kratochvíl, J.: Complexity of Partial Covers of Graphs. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 537–549. Springer, Heidelberg (2001)
Fiala, J., Kratochvíl, J.: Partial covers of graphs. Discussiones Mathematicae Graph Theory 22, 89–99 (2002)
Fiala, J., Kratochvíl, J., Kloks, T.: Fixed-parameter complexity of λ-labelings. Discrete Applied Mathematics 113(1), 59–72 (2001)
Fiala, J., Kratochvíl, J., Pór, A.: On the computational complexity of partial covers of theta graphs. Electronic Notes in Discrete Mathematics 19, 79–85 (2005)
Fiala, J., Kratochvíl, J., Proskurowski, A.: Distance Constrained Labeling of Precolored Trees. In: Restivo, A., Ronchi Della Rocca, S., Roversi, L. (eds.) ICTCS 2001. LNCS, vol. 2202, pp. 285–292. Springer, Heidelberg (2001)
Fiala, J., Kratochvíl, J., Proskurowski, A.: Systems of distant representatives. Discrete Applied Mathematics 145(2), 306–316 (2005)
Garey, M.R., Johnson, D.S.: Computers and Intractability. W. H. Freeman and Co., New York (1979)
Golovach, P.A.: Systems of pair of q-distant representatives and graph colorings (in Russian). Zap. nau. sem. POMI 293, 5–25 (2002)
Griggs, J.R., Yeh, R.K.: Labelling graphs with a condition at distance 2. SIAM Journal of Discrete Mathematics 5(4), 586–595 (1992)
Harary, F.: Graph theory. Addison-Wesley Series in Mathematics IX (1969)
Leese, R.A.: A fresh look at channel assignment in uniform networks. In: EMC 1997 Symposium, Zurich, pp. 127–130 (1997)
Leese, R.A.: Radio spectrum: a raw material for the telecommunications industry. In: 10th Conference of the European Consortium for Mathematics in Industry, Goteborg (1998)
Leese, R.A., Noble, S.D.: Cyclic labellings with constraints at two distances. Electr. J. Comb. 11(1) (2004)
Liu, D.D.-F., Zhu, X.: Circulant distant two labeling and circular chromatic number. Ars Combinatoria 69, 177–183 (2003)
Matoušek, J., Nešetřil, J.: Invitation to Discrete Mathematics. Oxford University Press, Oxford (1998)
McDiarmid, C.: Discrete mathematics and radio channel assignment. In: Recent advances in algorithms and combinatorics. CMS Books Math./Ouvrages Math. SMC, vol. 11, pp. 27–63. Springer, Heidelberg (2003)
Yeh, R.K.: A survey on labeling graphs with a condition at distance two. Discrete Mathematics 306(12), 1217–1231 (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fiala, J., Golovach, P.A., Kratochvíl, J. (2008). Distance Constrained Labelings of Trees. In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2008. Lecture Notes in Computer Science, vol 4978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79228-4_11
Download citation
DOI: https://doi.org/10.1007/978-3-540-79228-4_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79227-7
Online ISBN: 978-3-540-79228-4
eBook Packages: Computer ScienceComputer Science (R0)