CDS in Planar Graphs

Ding-Zhu Du and Peng-Jun Wan
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Ding-Zhu Du: University of Texas, Dallas
Peng-Jun Wan: Illinois Institute of Technology

Chapter Chapter 12 in Connected Dominating Set: Theory and Applications, 2013, pp 183-191 from Springer

Abstract: Abstract Although Min-CDS in general graphs is hard to approximate, the restriction to certain special graph classes admits much better approximation results. Min-CDS in planar graphs remains NP-hard even for planar graphs that are regular of degree 4 [57]. The related problem, Min-DS in planar graphs, is also NP-hard even for planar graphs with maximum vertex degree 3 and planar graphs that are regular of degree 4 [57]. It is well known that Min-DS in planar graphs possesses a polynomial-time approximation scheme (PTAS) based on the shifting strategy [3]: For any constant ε >0, there is a polynomial-time 1+ε-approximation algorithm. Thus, it is immediate to conclude that Min-CDS in planar graphs can be approximated within a factor 3+ε for any ε>0 in polynomial time. However, the degree of the polynomial grows with 1∕ε and hence, the approximation scheme is hardly practical.

Keywords: Planar Graphs; Maximum Vertex Degree; Polynomial Time Approximation Scheme (PTAS); Gray Neighbors; White Vertices (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-1-4614-5242-3_12

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