Restricted power domination and fault-tolerant power domination on grids

  • Authors:

  • Kung-Jui Pai;Jou-Ming Chang;Yue-Li Wang

  • Affiliations:

  • Department of Industrial Engineering and Management, Mingchi University of Technology, Taipei County, Taiwan;Institute of Information Science and Management, National Taipei College of Business, Taipei, Taiwan;Department of Information Management, National Taiwan University of Science and Technology, Taipei, Taiwan

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

The power domination problem is to find a minimum placement of phase measurement units (PMUs) for observing the whole electric power system, which is closely related to the classical domination problem in graphs. For a graph G=(V,E), the power domination number of G is the minimum cardinality of a set S@?V such that PMUs placed on every vertex of S results in all of V being observed. A vertex with a PMU observes itself and all its neighbors, and if an observed vertex with degree d1 has only one unobserved neighbor, then the unobserved neighbor becomes observed. Although the power domination problem has been proved to be NP-complete even when restricted to some special classes of graphs, Dorfling and Henning in [M. Dorfling, M.A. Henning, A note on power domination in grid graphs, Discrete Applied Mathematics 154 (2006) 1023-1027] showed that it is easy to determine the power domination number of an nxm grid. Their proof provides an algorithm for giving a minimum placement of PMUs. In this paper, we consider the situation in which PMUs may only be placed within a restricted subset of V. Then, we present algorithms to solve this restricted type of power domination on grids under the conditions that consecutive rows or columns form a forbidden zone. Moreover, we also deal with the fault-tolerant measurement placement in the designed scheme and provide approximation algorithms when the number of faulty PMUs does not exceed 3.