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 must be connected. For these graphs, our bound is tight and is also significantly better than the corresponding bound for domination number. We also improve the product upper bound for the power domination number for graphs with certain properties.

determined the radio number of complete m-ary trees [7]. Ruxandra Marinescu-Ghemeci found the radio number for some thorn graphs, one of which is a particular type of caterpillar graph [10]. This thesis builds off of work done on paths and trees in general to determine an improved lower bound or the actual radio number of certain types of caterpillar graphs. This thesis includes joint work with Matthew Porter and Maggy Tomova on determining the radio numbers of graphs with n vertices and diameter n − 2, a subcase of which is a particular caterpillar. This thesis also establishes the radio number of some specific caterpillar graphs as well as an improved lower bound for the radio number of more general caterpillar graphs. v TABLE OF CONTENTS LIST OF TABLES .

No abstract