Power domination in planar graphs with small diameter
The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graph theory.In this paper,it Was shown that the power domination number of an outerplanar graph with the diameter two or a 2-connected outerplanar graph with the diameter three is precisely one.Upper bounds on the power domination number for a general planar graph with the diameter two or three were determined as an immediate consequences of results proven by Dorfling,et al.Also,an infinite family of outerplanar graphs with the diameter four having arbitrarily large power domination numbers were given.
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