  Two-sided group digraphs and graphs, introduced by Iradmusa and Praeger,
provide a generalization of Cayley digraphs and graphs in which arcs are
determined by left and right multiplying by elements of two subsets of the
group. We characterize when two-sided group digraphs and graphs are weakly and
strongly connected and count connected components, using both an explicit
elementary perspective and group actions. Our results and examples address four
open problems posed by Iradmusa and Praeger that concern connectedness and
valency. We pose five new open problems.
