  We introduce a new subclass of chordal graphs that generalizes split graphs,
which we call well-partitioned chordal graphs. Split graphs are graphs that
admit a partition of the vertex set into cliques that can be arranged in a star
structure, the leaves of which are of size one. Well-partitioned chordal graphs
are a generalization of this concept in the following two ways. First, the
cliques in the partition can be arranged in a tree structure, and second, each
clique is of arbitrary size. We provide a characterization of well-partitioned
chordal graphs by forbidden induced subgraphs, and give a polynomial-time
algorithm that given any graph, either finds an obstruction, or outputs a
partition of its vertex set that asserts that the graph is well-partitioned
chordal. We demonstrate the algorithmic use of this graph class by showing that
two variants of the problem of finding pairwise disjoint paths between k given
pairs of vertices is in FPT parameterized by k on well-partitioned chordal
graphs, 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-time
solvable on split graphs, but become NP-complete on well-partitioned chordal
graphs.
