  Dujmovi\'{c}, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020]
established that every planar graph is a subgraph of the strong product of a
graph with bounded treewidth and a path. Motivated by this result, this paper
systematically studies various structural properties of cartesian, direct and
strong products. In particular, we characterise when these graph products
contain a given complete multipartite subgraph, determine tight bounds for
their degeneracy, establish new lower bounds for the treewidth of cartesian and
strong products, and characterise when they have bounded treewidth and when
they have bounded pathwidth.
