On two variants of split graphs: 2-unipolar graph and k-probe-split graph

A graph is called split if its vertex set can be partitioned into a stable set and a clique. In this article, we studied two variants of split graphs. A graph G is polar if its vertex set can be partitioned into two sets A and B such that G[A] is a complete multipartite graph and G[B] is a disjoint union of complete graphs. A 2-unipolar graph is a polar graph G such that G[A] is a clique and G[B] is the disjoint union of complete graphs with at most two vertices. We present a minimal forbidden induced subgraph characterization for 2-unipolar graphs. In addition, we show that they can be represented as an intersection of substars of special cacti. Let G be a graph class, the G-width of a graph G is the minimum positive integer k such that there exist k independent sets N1,...,Nk such that a set F of nonedges of G, whose endpoints belong to some Ni with i = 1,...,k, can be added so that the resulting graph G0 belongs to G. We say that a graph G is k-probe-G if it has G-width at most k and when G is the class of split graphs it is denominated k-probe-split. We prove that deciding, given a graph G and a positive integer k, whether G is a h-probe-split graph for some h ≤ k is NP-complete. Besides, a characterization by minimal forbidden induced subgraphs for 2-probesplit cographs is presented.