Linear Algorithms for Chordal Graphs of Bounded Directed Vertex Leafage

The directed vertex leafage of a chordal graph G is the smallest integer k such that G is the intersection graph of subtrees of a rooted directed tree where each subtree has at most k leaves. In this note, we show how to find in time O(kn) an optimal colouring, a maximum independent set, a maximum clique, and an optimal clique cover of an n-vertex chordal graph G with directed vertex leafage k if a representation of G is given. In particular, this implies that for any path graph G, the four problems can be solved in time O(n) given a path representation of G.