## Nordhaus–Gaddum problems for power domination

2018-12-31
##### Authors
Benson, Katherine
Ferrero, Daniela
Flagg, Mary
Furst, Veronika
Vasilevska, Violeta
Person
Hogben, Leslie
Associate Dean
##### Organizational Units
Organizational Unit
Electrical and Computer Engineering

The Department of Electrical and Computer Engineering (ECpE) contains two focuses. The focus on Electrical Engineering teaches students in the fields of control systems, electromagnetics and non-destructive evaluation, microelectronics, electric power & energy systems, and the like. The Computer Engineering focus teaches in the fields of software systems, embedded systems, networking, information security, computer architecture, etc.

History
The Department of Electrical Engineering was formed in 1909 from the division of the Department of Physics and Electrical Engineering. In 1985 its name changed to Department of Electrical Engineering and Computer Engineering. In 1995 it became the Department of Electrical and Computer Engineering.

Dates of Existence
1909-present

Historical Names

• Department of Electrical Engineering (1909-1985)
• Department of Electrical Engineering and Computer Engineering (1985-1995)

Related Units

Organizational Unit
Mathematics
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##### Abstract

A power dominating set of a graph G is a set S of vertices that can observe the entire graph under the rules that (1) the closed neighborhood of every vertex in S is observed, and (2) if a vertex and all but one of its neighbors are observed, then the remaining neighbor is observed; the second rule is applied iteratively. The power domination number of G, denoted by gamma p(G), is the minimum number of vertices in a power dominating set. A Nordhaus-Gaddum problem for power domination is to determine a tight lower or upper bound on gamma p(G) + gamma p(G) or gamma p(G).gamma p(G), where G denotes the complement of G. The upper and lower Nordhaus-Gaddum bounds over all graphs for the power domination number follow from known bounds on the domination number and examples. In this note we improve the upper sum bound for the power domination number substantially for graphs having the property that both the graph and its complement are connected. For these graphs, our bound is tight and is also significantly better than the corresponding bound for the domination number. We also improve the product upper bound for the power domination number for graphs with certain properties.