Bounding Width on Graph Classes of Constant Diameter

We determine if the width of a graph class G changes from unbounded to bounded if we consider only those graphs from G whose diameter is bounded. As parameters we consider treedepth, pathwidth, treewidth and clique-width, and as graph classes we consider classes defined by forbidding some specific graph F as a minor, induced subgraph or subgraph, respectively. Our main focus is on treedepth for F-subgraphfree graphs of diameter at most d for some fixed integer d. We give classifications of boundedness of treedepth for d ∈ {4, 5, . . .} and partial classifications for d = 2 and d = 3.