Contractible Theta Complexes of Graphs

We examine properties of graphs that result in the graph having a contractible theta complex. We classify such properties for tree graphs and graphs with one loop and we introduce examples of graphs with such properties for tree graphs and graphs with one or two loops. For more general graphs, we show that having a contractible theta complex is not an elusive property, and that any skeleton of a graph with at least three loops can be made to have a contractible theta complex by strategically adding vertices to its skeleton.