The vertex leafage of chordal graphs

Every chordal graph $G$ can be represented as the intersection graph of a collection of subtrees of a host tree, a so-called {\em tree model} of $G$. The leafage $\ell(G)$ of a connected chordal graph $G$ is the minimum number of leaves of the host tree of a tree model of $G$. The vertex leafage $\vl(G)$ is the smallest number $k$ such that there exists a tree model of $G$ in which every subtree has at most $k$ leaves. The leafage is a polynomially computable parameter by the result of \cite{esa}. In this contribution, we study the vertex leafage. We prove for every fixed $k\geq 3$ that deciding whether the vertex leafage of a given chordal graph is at most $k$ is NP-complete by proving a stronger result, namely that the problem is NP-complete on split graphs with vertex leafage of at most $k+1$. On the other hand, for chordal graphs of leafage at most $\ell$, we show that the vertex leafage can be calculated in time $n^{O(\ell)}$. Finally, we prove that there exists 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 $\ell(G)$ leaves in the host tree and it can be computed in $O(n^3)$ time.

💡 Research Summary

The paper introduces and studies the vertex leafage (vℓ(G)) of chordal graphs, a parameter complementary to the already known leafage ℓ(G). While leafage measures the minimum number of leaves in the host tree of a tree model of a chordal graph, vertex leafage asks for the smallest integer k such that there exists a tree model where every vertex‑subtree has at most k leaves. This notion captures how close a chordal graph is to being a path graph (the case vℓ(G) ≤ 2).

The first major contribution is a complexity classification. For any fixed integer k ≥ 3, the decision problem “Is vℓ(G) ≤ k?” is shown to be NP‑complete, even when the input graph is a split graph whose vertex leafage is at most k + 1. The reduction is from NOT‑ALL‑EQUAL‑k‑SAT. From a SAT instance I consisting of k‑element sets C₁,…,C_m over variables v₁,…,v_n, the authors construct a split graph G_I with a clique {y₁,…,y_m} and an independent set {v₁,…,v_n, z₁, z₂}. The maximal cliques of G_I are A={z₁, y₁,…,y_m}, B={z₂, y₁,…,y_m}, and Q_i={v_i}∪{y_j | v_i∈C_j}. In any clique tree of G_I each y_j belongs to exactly k+2 cliques (A, B, and the k cliques Q_i corresponding to the elements of C_j). The authors prove two facts: (a) every clique tree yields a vertex leafage at most k + 1, and (b) a clique tree with vertex leafage ≤ k exists if and only if the original SAT instance has a solution. The proof hinges on forcing the edge AB in the clique tree and making each Q_i a leaf; the presence or absence of edges A‑Q_i versus B‑Q_i encodes a selection of variables that forms a satisfying assignment. Consequently, the vertex leafage decision problem is NP‑complete for any fixed k ≥ 3.

The second contribution addresses the algorithmic side when the leafage ℓ(G) is bounded. Using the well‑known correspondence between minimal tree models and clique trees, the authors enumerate all possible clique trees whose internal nodes have degree at least three. Because a host tree with ℓ leaves can have at most O(ℓ) internal nodes, the number of such trees is n^{O(ℓ)}. For each candidate tree they compute the number of leaves in each vertex‑subtree, thereby obtaining the minimal k = vℓ(G). Hence, for any fixed ℓ, vertex leafage can be computed in time n^{O(ℓ)}; in particular, when ℓ is a constant the problem becomes polynomial‑time solvable.

The third major result is a structural theorem: for every chordal graph G there exists a tree model that simultaneously realizes the optimal leafage ℓ(G) and the optimal vertex leafage vℓ(G). Starting from any tree model (T, {T_u}), the authors contract unnecessary edges of the host tree and rearrange the subtrees so that the host tree’s leaf count is reduced to ℓ(G) while never increasing the leaf count of any individual subtree. This yields a model (T*, {T*_u}) with |L(T*)| = ℓ(G) and |L(T*_u)| ≤ vℓ(G) for all vertices u. As a corollary, for path graphs (vℓ(G) ≤ 2) there is a path model whose host tree is a simple path with exactly ℓ(G) leaves, and such a model can be constructed in O(n³) time.

Overall, the paper establishes that vertex leafage is a natural and non‑trivial graph invariant: it is computationally hard in general, but tractable when the leafage is bounded, and it can be optimized together with leafage in a single tree model. These findings deepen the understanding of tree representations of chordal graphs and open avenues for further algorithmic and structural investigations.