Enumeration of unlabeled graph classes A study of tree decompositions and related approaches

In this paper, we study the enumeration of certain classes of graphs that can be fully characterized by tree decompositions; these classes are particularly significant due to the algorithmic improvements derived from tree decompositions on classically NP-complete problems on these classes [12, 7, 17, 35]. Previously, Chauve et al. [6] and Iriza [26] constructed grammars from the split decomposition trees of distance hereditary graphs and 3-leaf power graphs. We extend upon these results to obtain an upper bound grammar for parity graphs. Also, Nakano et al. [25] used the vertex incremental characterization of distance hereditary graphs to obtain upper bounds. We constructively enumerate (6,2)-chordal bipartite graphs, (C5, bull, gem, co-gem)-free graphs, and parity graphs using their vertex incremental characterization and extend upon Nakano et al.’s results to analytically obtain upper bounds of O ( 7n ) and O ( 11n ) for (6,2)-chordal bipartite graphs and (C5, bull, gem, co-gem)-free graphs respectively.