Abstract
A set D ⊆ V is called a dominating set of G = (V,E) if |N G [v] ∩ D| ≥ 1 for all v ∈ V. The Minimum Domination problem is to find a dominating set of minimum cardinality of the input graph. In this paper, we study the Minimum Domination problem for star-convex bipartite graphs, circular-convex bipartite graphs and triad-convex bipartite graphs. It is known that the Minimum Domination Problem for a graph with n vertices can be approximated within ln n. However, we show that for any ε > 0, the Minimum Domination problem does not admit a (1 − ε)ln n-approximation algorithm for star-convex bipartite graphs with n vertices unless NP ⊆ DTIME(n O(loglogn)). On the positive side, we propose polynomial time algorithms for computing a minimum dominating set of circular-convex bipartite graphs and triad-convex bipartite graphs, by polynomially reducing the Minimum Domination problem for these graph classes to the Minimum Domination problem for convex bipartite graphs.
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References
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in graphs: Advanced topics, vol. 209. Marcel Dekker Inc., New York (1998)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs, vol. 208. Marcel Dekker Inc., New York (1998)
Garey, M.R., Johnson, D.S.: Computers and Interactability: a guide to the theory of NP-completeness. W.H. Freeman and Co., San Francisco (1979)
Bertossi, A.A.: Dominating sets for split and bipartite graphs. Inform. Process. Lett. 19, 37–40 (1984)
Müller, H., Brandstädt, A.: The NP-completeness of steeiner tree and dominating set for chordal bipartite graphs. Theoret. Comput. Sci. 53, 257–265 (1987)
Damaschke, P., Müller, H., Kratsch, D.: Domination in convex and chordal bipartite graphs. Inform. Process. Lett. 36, 231–236 (1990)
Liang, Y.D., Blum, N.: Circular convex bipartite graphs: maximum matching and hamiltonial circuits. Inform. Process. Lett. 56, 215–219 (1995)
Lu, M., Liu, T., Xu, K.: Independent domination: Reductions from circular- and triad-convex bipartite graphs to convex bipartite graphs. In: Fellows, M., Tan, X., Zhu, B. (eds.) FAW-AAIM 2013. LNCS, vol. 7924, pp. 142–152. Springer, Heidelberg (2013)
Lu, Z., Liu, T., Xu, K.: Tractable connected domination for restricted bipartite graphs (Extended abstract). In: Du, D.-Z., Zhang, G. (eds.) COCOON 2013. LNCS, vol. 7936, pp. 721–728. Springer, Heidelberg (2013)
Liu, T., Lu, M., Lu, Z., Xu, K.: Circular convex bipartite graphs: Feedback vertex sets. Theoret. Comput. Sci. (2014), doi: 10.1016/j.tcs.2014.05.001
Song, Y., Liu, T., Xu, K.: Independent domination on tree convex bipartite graphs. In: Snoeyink, J., Lu, P., Su, K., Wang, L. (eds.) AAIM 2012 and FAW 2012. LNCS, vol. 7285, pp. 129–138. Springer, Heidelberg (2012)
Liu, W.J.T., Wang, C., Xu, K.: Feedback vertex sets on restricted bipartite graphs. Theoret. Comput. Sci. 507, 41–51 (2013)
Zhang, Y., Bao, F.S.: A review of tree convex sets test. Computational Intelligence 28, 358–372 (2012)
Bang-Jensen, J., Huang, J., MacGillivray, G., Yeo, A.: Domination in convex bipartite and convex-round graphs. Recent Trends in Computational Math. and its Applications. International Journal of Mathematical Sciences 5 (2006)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)
Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45, 634–652 (1998)
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Pandey, A., Panda, B.S. (2015). Domination in Some Subclasses of Bipartite Graphs. In: Ganguly, S., Krishnamurti, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2015. Lecture Notes in Computer Science, vol 8959. Springer, Cham. https://doi.org/10.1007/978-3-319-14974-5_17
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DOI: https://doi.org/10.1007/978-3-319-14974-5_17
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