On the closure of graphs under substitution
Discrete Mathematics
r-Bounded k-complete bipartite bihypergraphs and generalized split graphs
Discrete Mathematics
Extension of hereditary classes with substitutions
Discrete Applied Mathematics
On prime inductive classes of graphs
European Journal of Combinatorics
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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. Our investigation 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.