The vertex leafage of chordal graphs

• Authors: Steven Chaplick;Juraj Stacho
• Venue: Discrete Applied Mathematics
• Year: 2014

Every chordal graph G can be represented as the intersection graph of a collection of subtrees of a host tree, a so-called tree model of G. This representation is not necessarily unique. The leafage @?(G) of a chordal graph G is the minimum number of leaves of the host tree of a tree model of G. The leafage is known to be polynomially computable. In this contribution, we introduce and study the vertex leafage. The vertex leafage v@?(G) of a chordal graph G is the smallest number k such that there exists a tree model of G in which every subtree has at most k leaves. In particular, the case v@?(G)@?2 coincides with the class of path graphs (vertex intersection graphs of paths in trees). We prove for every fixed k=3 that deciding whether the vertex leafage of a given chordal graph is at most k is NP-complete. In particular, we show that the problem is NP-complete on split graphs with vertex leafage of at most k+1. We further prove that it is NP-hard to find for a given split graph G (with vertex leafage at most three) a tree model with minimum total number leaves in all subtrees, or where maximum number of subtrees are paths. On the positive side, for chordal graphs of leafage at most @?, we show that the vertex leafage can be calculated in time n^O^(^@?^). Finally, we prove that every chordal graph G admits a tree model that realizes both the leafage and the vertex leafage of G. Notably, for every path graph G, there exists a path model with @?(G) leaves in the host tree and we describe an O(n^3) time algorithm to compute such a path model.