Clique Separator Decomposition of Hole- and Diamond-Free Graphs and Algorithmic Consequences

Clique separator decomposition introduced by Tarjan and Whitesides is one of the most important graph decompositions. A graph is an {\em atom} if it has no clique separator. A {\em hole} is a chordless cycle with at least five vertices, and an {\em antihole} is the complement graph of a hole. A graph is {\em weakly chordal} if it is hole- and antihole-free. $K_4-e$ is also called {\em diamond}. {\em Paraglider} has five vertices four of which induce a diamond, and the fifth vertex sees exactly the two vertices of degree two in the diamond. In this paper we show that atoms of hole- and diamond-free graphs (of hole- and paraglider-free graphs, respectively) are either weakly chordal or of a very specific structure. Hole- and paraglider-free graphs are perfect graphs. The structure of their atoms leads to efficient algorithms for various problems.

💡 Research Summary

The paper investigates the structure of atoms—graphs without clique separators—within two hereditary graph classes: (i) hole‑and‑diamond‑free graphs and (ii) hole‑and‑paraglider‑free graphs (HP‑free). Using the classic Tarjan‑Whitesides clique‑separator decomposition, the authors show that every atom of an HP‑free graph falls into one of two categories. If the atom contains no induced C₆ (a 6‑cycle), it is weakly chordal, i.e., it contains neither holes nor antiholes and is therefore perfect. If it does contain a C₆, a detailed analysis of the neighborhoods of the C₆ vertices yields a very restricted configuration: the atom must be a matched co‑bipartite graph (the complement of a complete bipartite graph with a perfect matching) possibly augmented with universal vertices or a full clique. Lemma 1 proves that no vertex can be adjacent to exactly one endpoint of a matching edge of the C₆, which forces the atom into the matched co‑bipartite form.

From this structural dichotomy the authors derive two major consequences. First, they obtain a direct proof that HP‑free graphs are perfect, independent of the Strong Perfect Graph Theorem. Second, because the decomposition tree can be built in polynomial time, any problem that is polynomial on each atom can be solved in polynomial time on the whole class. The paper explicitly discusses algorithmic applications such as Maximum Weight Independent Set, Maximum Clique, Graph Coloring, and Minimum Fill‑in. For weakly chordal atoms the known O(n⁴) MWIS algorithm applies; for matched co‑bipartite atoms the authors present linear‑time procedures based on the simple bipartite matching structure. Consequently, all these problems become polynomial‑time solvable on hole‑and‑diamond‑free as well as hole‑and‑paraglider‑free graphs.

The work also situates its results within the broader literature on cycle‑restricted graph classes, noting connections to chordal bipartite, block, and (dart, gem)‑free graphs. It suggests that similar atom‑based analyses could be extended to other classes defined by forbidding small induced subgraphs, opening avenues for further structural and algorithmic research.