Enumerative aspects of certain subclasses of perfect graphs

Abstract

We investigate the enumerative aspects of various classes of perfect graphs like cographs, split graphs, trivially perfect graphs and threshold graphs. For subclasses of permutation graphs like cographs and threshold graphs we also determine the number of permutations π of {1,2,…,n} such that the permutation graph G[π] belongs to that class. We establish an interesting bijection between permutations whose permutation graphs are cographs (P₄-free graphs) and permutations that are obtainable using an output-restricted deque (Knuth, Art of Computer Programming, Vol I, Fundamental Algorithms) and thereby enumerate such permutations. We also prove that the asymptotic number of permutations of {1,2,…,n} whose permutation graphs are split graphs is $\text{Θ}\text{(4}{}^{\mathrm{n}}\text{/}\text{n}\text{)}$. We also introduce a new class of graphs called C₅-split graphs, characterize and enumerate them. C₅-split graphs form a superclass of split graphs and are not necessarily perfect. All the classes of graphs that we enumerate have a finite family of small forbidden induced subgraphs.