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We introduce a new subclass of chordal graphs that generalizes split graphs,which we call well-partitioned chordal graphs. Split graphs are graphs thatadmit a partition of the vertex set into cliques that can be arranged in a starstructure, the leaves of which are of size one. Well-partitioned chordal graphsare a generalization of this concept in the following two ways. First, thecliques in the partition can be arranged in a tree structure, and second, eachclique is of arbitrary size. We provide a characterization of well-partitionedchordal graphs by forbidden induced subgraphs, and give a polynomial-timealgorithm that given any graph, either finds an obstruction, or outputs apartition of its vertex set that asserts that the graph is well-partitionedchordal. We demonstrate the algorithmic use of this graph class by showing thattwo variants of the problem of finding pairwise disjoint paths between k givenpairs of vertices is in FPT parameterized by k on well-partitioned chordalgraphs, while on chordal graphs, these problems are only known to be in XP.From the other end, we observe that there are problems that are polynomial-timesolvable on split graphs, but become NP-complete on well-partitioned chordalgraphs.

Well-partitioned chordal graphs: obstruction set and disjoint paths