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Algorithmic Aspects of Domination in Graphs

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Handbook of Combinatorial Optimization

Abstract

Domination in graph theory has many applications in the real world such as location problems. A dominating set of a graph G = (V, E) is a subset D of V such that every vertex not in D is adjacent to at least one vertex in D. The domination problem is to determine the domination number \(\gamma (G)\) of a graph G that is the minimum size of a dominating set of G. Although many theoretic theorems for domination and its variations have been established for a long time, the first algorithmic result on this topic was given by Cockayne, Goodman, and Hedetniemi in 1975. They gave a linear-time algorithm for the domination problem in trees by using a labeling method. On the other hand, at about the same time, Garey and John constructed the first (unpublished) proof that the domination problem is NP-complete. Since then, many algorithmic results are studied for variations of the domination problem in different classes of graphs. This chapter is to survey the development on this line during the past 36 years. Polynomial-time algorithms using labeling method, dynamic programming method, and primal–dual method are surveyed on trees, interval graphs, strongly chordal graphs, permutation graphs, cocomparability graphs, and distance-hereditary graphs. NP-completeness results on domination are also discussed.

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  1. Department of Mathematics, National Taiwan University, 10617, Taipei, Taiwan

    Gerard Jennhwa Chang

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  1. Department of Industrial and Systems Eng, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Computer Science, University of Texas, Dallas, Richardson, Texas, USA

    Ding-Zhu Du

  3. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

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Chang, G.J. (2013). Algorithmic Aspects of Domination in Graphs. In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_26

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