Abstract
Minus domination in graphs is a variant of domination where the vertices must be labeled -1,0,+1 such that the sum of labels in each N[v] is positive. (As usual, N[v] means the set containing v together with its neighbors.) The minus domination number γ − is the minimum total sum of labels that can be achieved. In this paper we prove linear lower bounds for γ − in graphs either with Δ ≤ 3, or with Δ ≤ 4 but without vertices of degree 2. The central section is concerned with complexity results for Δ ≤ 4: We show that computing γ − is NP-hard and MAX SNP-hard there, but that γ − can be approximated in linear time within some constant factor. Finally, our approach also applies to signed domination (where the labels are -1,+1 only) in small-degree graphs.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berman, P., Fujito, T.: On approximation properties of the independent set problem for degree 3 graphs. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 449–460. Springer, Heidelberg (1995)
Chang, G.J., Pandu Rangan, C., Coorg, S.R.: Weighted independent perfect domination on cocomparability graphs. In: Ng, K.W., Balasubramanian, N.V., Raghavan, P., Chin, F.Y.L. (eds.) ISAAC 1993. LNCS, vol. 762, pp. 506–515. Springer, Heidelberg (1993)
Dunbar, J., Goddard, W., Hedetniemi, S.T., Henning, M.A., McRae, A.: The algorithmic complexity of minus domination in graphs. Discrete Applied Math. 68, 73–84 (1996)
Dunbar, J., Hedetniemi, S.T., Henning, M.A., McRae, A.: Minus domination in regular graphs. Discrete Math. 149, 311–312 (1996)
Dunbar, J., Hedetniemi, S.T., Henning, M.A., McRae, A.: Minus domination in graphs. Discrete Math. (to submitted)
Dunbar, J., Hedetniemi, S.T., Henning, M.A., Slater, P.J.: Signed domination in graphs. Graph Theory, Combinatorics and Applications, 311–322 (1995)
Favaron, O.: Signed domination in regular graphs. Discrete Math. 158, 287–293 (1996)
Garey, M.R., Johnson, D.S.: Computers and Intractability - a Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Hell, P., Kirkpatrick, D.G.: Algorithms for degree constrained graph factors of minimum deficiency. J. Algorithms 14, 115–138 (1993)
Natanzon, A., Shamir, R., Sharon, R.: A polynomial approximation algorithm for the minimum fill-in problem. In: 30th ACM STOC 1998, pp. 41–47 (1998)
Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comp. System Sc. 43, 425–440 (1991)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Damaschke, P. (1998). Minus Domination in Small-Degree Graphs. In: Hromkovič, J., Sýkora, O. (eds) Graph-Theoretic Concepts in Computer Science. WG 1998. Lecture Notes in Computer Science, vol 1517. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10692760_2
Download citation
DOI: https://doi.org/10.1007/10692760_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65195-6
Online ISBN: 978-3-540-49494-2
eBook Packages: Springer Book Archive