Characterization and recognition of edge intersection graphs of $3$-chromatic hypergraphs with multiplicity at most than two in the class of split graphs

Let $L^{m}k$ denote the class of edge intersection graphs of $k$-chromatic hypergraphs with multiplicity at most $m$. It is known that the problem of recognizing graphs from $L^{1}(k)$ is polynomially solvable if $k=2$ and is $NP$-complete if $k=3$. It is also known that for any $k\geq 2$ the graphs from $L^{1}k$ can be characterized by a finite list of forbidden induced subgraphs in the class of split graphs. The question of the complexity of recognizing graphs from $L^{m}k$ for fixed $k\geq 2$ and $m\geq 2$ remains open. Here it is proved that there exists a finite characterization in terms of forbidden induced subgraphs for the graphs from $L^{2}(3)$ in the class of split graphs. In particular, it follows that the problem of recognizing graphs from $G\in L^{2}(3)$ is polynomially solvable in the class of split graphs. The results are obtained on the basis of proven here characterization of the graphs from $L^{2}(3)$ in terms of vertex degrees in one of the subclasses of split graphs. In turn, this characterization is obtained using the well-known description of graphs from $L^{m}(k)$ by means of clique coverings and proven here Lemma on large clique, specifying the mutual location of cliques in the graph from $L^{m}(k)$.