Demonstrating the tightness of the adjacency bound in Lemma 33. (A) A history graph with 3 adjacencies. (B) A G-bounded AVG extension of the graph in (A) with 22 extra G-reducible adjacencies and 44 extra attached vertices, the maximum possible. The leaf adjacencies connecting the labeled leaves are G-irreducible, as there are no other labeled vertices in G′. All the other adjacencies are junctions or bridges. The number of elements in the red subgraph of (B) corresponds to the maximum number of adjacencies and attached vertices that can be added given the inclusion of the red subgraph in (A), and similarly for the blue subgraph. By extrapolation this demonstrates the bounds on the number of additional adjacencies and attached vertices are tight for all possible numbers of adjacencies in the original graph.