We consider the following generalization of split graphs: A graph is said to be a (k, l)-graph if its vertex set can be partitioned into k independent sets and l cliques. (Split graphs are obtained by setting k = l = 1). Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by (k, l)-graphs in general. (For instance, being a (k, 0)-graph is equivalen...

We give characterizations of integral graphs in the family of complete split graphs and a few related families of graphs.

Wediscuss various questions around partitioning a split graph into connected parts. Our main result is a polynomial time algorithm that decides whether a given split graph is fully decomposable, that is, whether it can be partitioned into connected parts of orders α1, α2, . . . , αk for every α1, α2, . . . , αk summing up to the order of the graph. In contrast, we show that the decision problem...

A common model of robustness a graph against random failures has all vertices operational, but the edges independently operational with probability p $$ . One can ask for that communicate (all-terminal reliability) or two specific (or terminals) each other (two-terminal reliability). relatively new measure is split reliability, where fixed s and t , we consider every vertex communicates one not...