Split decomposition and graph-labelled trees: characterizations and fully-dynamic algorithms for totally decomposable graphs

In this paper, we revisit the split decomposition of graphs and give new combinatorial and algorithmic results for the class of totally decomposable graphs, also known as the distance hereditary graphs, and for two non-trivial subclasses, namely the cographs and the 3-leaf power graphs. Precisely, we give strutural and incremental characterizations, leading to optimal fully-dynamic recognition algorithms for vertex and edge modifications, for each of these classes. These results rely on a new framework to represent the split decomposition, namely the graph-labelled trees, which also captures the modular decomposition of graphs and thereby unify these two decompositions techniques. The point of the paper is to use bijections between these graph classes and trees whose nodes are labelled by cliques and stars. Doing so, we are also able to derive an intersection model for distance hereditary graphs, which answers an open problem.

💡 Research Summary

The paper revisits the classic split decomposition of graphs and introduces a unifying combinatorial framework called graph‑labelled trees (GLTs). In a GLT each internal node of a tree is labelled by a small graph—either a clique or a star—whose vertices (called marker‑vertices) are in one‑to‑one correspondence with the incident tree‑edges. By defining an “accessibility” relation between leaves, the authors show that the set of leaves together with this relation forms exactly the original graph, called the accessibility graph. This representation captures both split decomposition and modular decomposition, allowing the two traditionally separate techniques to be treated uniformly.

The authors focus on three important graph families that are totally decomposable with respect to split decomposition: distance‑hereditary (DH) graphs, cographs, and 3‑leaf power graphs. They prove that each of these families corresponds bijectively to a restricted class of GLTs in which the node‑labels are cliques and stars arranged according to simple distribution rules. For DH graphs this bijection yields a new intersection model: each vertex is represented by a point on a line (the centre of a star) or an interval (a clique), and two vertices are adjacent precisely when the corresponding geometric objects intersect. This answers an open question about an explicit intersection model for DH graphs.

A major contribution is the development of incremental (vertex‑insertion) characterisations for the three families. The authors define a “1‑Clique‑Star” operation: adding a new vertex x adjacent to a set S of existing vertices is possible exactly when the leaves representing S are all accessible through the same star centre in the GLT. This condition is shown to be necessary and sufficient for DH graphs (Theorem 3.4), and it reduces to known incremental characterisations for cographs (Theorem 3.7) while providing a novel one for 3‑leaf powers (Theorem 3.9).

Edge‑modification characterisations are also derived. Adding or deleting an edge {u,v} is feasible if and only if the unique path between the leaves representing u and v in the GLT has length at most four and belongs to a small finite set of patterns. This local test improves on earlier global breadth‑first‑search based criteria for DH graphs and generalises the edge‑modification test for cographs.

Using these structural insights, the paper presents optimal fully‑dynamic recognition algorithms for all three classes. For vertex insertion, the algorithm locates the minimal subtree spanned by the neighbours of the new vertex, determines the appropriate insertion point in the GLT, and performs a constant‑time local transformation of the labels. Vertex deletion is handled by a symmetric local operation because the classes are hereditary. Edge insertion and deletion are implemented by checking the path pattern and updating the relevant marker‑vertex adjacencies. The running time of each update is linear in the number of incident edges (i.e., O(deg) for vertex updates and O(1) for edge updates), which is optimal.

The authors also show that the GLT representation yields immediate graph‑isomorphism tests for these families: two graphs are isomorphic iff their GLTs are identical, leading to a simple linear‑time isomorphism algorithm (Corollary 4.3).

In summary, the paper unifies split and modular decompositions via graph‑labelled trees, provides new combinatorial characterisations (including an explicit intersection model for distance‑hereditary graphs), and leverages these results to design optimal fully‑dynamic algorithms for vertex and edge modifications in distance‑hereditary graphs, cographs, and 3‑leaf power graphs. The work advances both the theoretical understanding of these graph classes and the practical toolkit for maintaining them under dynamic changes.