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This paper investigates a combinatorial optimization problem motived from asecure power network design application in [D\'{a}n and Sandberg 2010]. Twoequivalent graph optimization formulations are derived. One of the formulationsis a relaxed version of the connected dominating set problem, and hence theconsidered problem is referred to as relaxed connected dominating set (RCDS)problem. The RCDS problem is shown to be NP-hard, even for planar graphs. Amixed integer linear programming formulation is presented. In addition, forplanar graphs a fixed parameter polynomial time solution methodology based onsphere-cut decomposition and dynamic programming is presented. The computationcost of the sphere-cut decomposition based approach grows linearly with probleminstance size, provided that the branchwidth of the underlying graph is fixedand small. A case study with IEEE benchmark power networks verifies that smallbranchwidth are not uncommon in practice. The case study also indicates thatthe proposed methods show promise in computation efficiency.

Relaxed Connected Dominating Set Problem with Application to Secure Power Network Design