J Comb Optim DOI 10.1007/s10878-015-9871-0 New analysis and computational study for the planar connected dominating set problem Marjan Marzban 1 · Qian-Ping Gu 1 · Xiaohua Jia 2 © Springer Science+Business Media New York 2015 Abstract The connected dominating set (CDS) problem is a well studied NP-hard problem with many important applications. Dorn et al. (Algorithmica 58:790–810 2010) introduce a branch-decomposition based algorithm design technique for NP- hard problems in planar graphs and give an algorithm (DPBF algorithm) which solves the planar CDS problem in O (2 9.822 √ n n + n 3 ) time and O (2 8.11 √ n n + n 3 ) time, with a conventional method and fast matrix multiplication in the dynamic program- ming step of the algorithm, respectively. We show that DPBF algorithm solves the planar CDS problem in O (2 9.8 √ n n + n 3 ) time with a conventional method and in O (2 8.08 √ n n + n 3 ) time with a fast matrix multiplication. For a graph G, let bw(G) be the branchwidth of G and γ c (G) be the connected dominating number of G. We prove bw(G) ≤ 2 √ 10γ c (G) + 32. From this result, the planar CDS problem admits an O (2 23.54 √ γ c (G) γ c (G) + n 3 ) time fixed-parameter algorithm. We report computational study results on the practical performance of DPBF algorithm, which show that the size of instances can be solved by the algorithm mainly depends on the branchwidth of the instances, coinciding with the theoretical analysis. For graphs with small or A preliminary version of this paper appeared in the Proceeding of the 2010 International Conference on Combinatorial Optimization and Applications (COCOA 2010) Marzban et al. (2010). B Qian-Ping Gu qgu@cs.sfu.ca Marjan Marzban mmarzban@cs.sfu.ca Xiaohua Jia csjia@cityu.edu.hk 1 School of Computing Science, Simon Fraser University, Burnaby, BC V5A 1S6, Canada 2 Department of Computer Science, City University of Hong Kong, Kowloon, Hong Kong 123