Forbidden graphs for classes of split-like graphs

The class of split graphs consists of those graphs G , for which there exist partitions ( V 1 , V 2 ) of their vertex sets V ( G ) such that G [ V 1 ] is an edgeless graph and G [ V 2 ] is a complete graph. The classes of edgeless and complete graphs are members of a family L ≤ ∗ , which consists of all graph classes that are induced hereditary and closed under substitution. Graph classes considered in this paper are alike to split graphs. Namely a graph class P is an object of our interest if there exist two graph classes P 1 , P 2 ∈ L ≤ ∗ (not necessarily different) such that for each G ∈ P we can find a partition ( V 1 , V 2 ) of V ( G ) satisfying G [ V 1 ] ∈ P 1 and G [ V 2 ] ∈ P 2 . For each such class P we characterize all forbidden graphs that are strongly decomposable. The finiteness of families of forbidden graphs for P is analyzed giving a result characterizing classes with a finite number of forbidden graphs. vestigation confirms, in the class L ≤ ∗ , Zverovich’s conjecture describing all induced hereditary graph classes defined by generalized vertex 2-partitions that have finite families of forbidden graphs.