Block duplicate graphs and a hierarchy of chordal graphs

A block graph is a graph whose blocks are cliques. A block duplicate (BD) graph is a graph obtained by adding true twins (i.e., adjacent vertices with the same closed neighborhood) to vertices of a block graph. If each cut-vertex belonging to three or more blocks in the block graph receives at most one true twin, the resulting graph is called a restricted block duplicate (RBD) graph. BD and RBD graphs arise in the study of RU-chordal graphs by Peled and Wu (J. Graph Theory 30 (1999) 121), who proved that the 2-connected RU-chordal graphs are equivalent to the 2-connected RBD graphs. In this note, we characterize the BD and the RBD graphs by forbidden induced subgraphs, and complete their incorporation into the hierarchy of special classes of chordal graphs.