Colouring Diamond-free Graphs

The Colouring problem is that of deciding, given a graph $G$ and an integer $k$, whether $G$ admits a (proper) $k$-colouring. For all graphs $H$ up to five vertices, we classify the computational complexity of Colouring for $(\mbox{diamond},H)$-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for $(\mbox{diamond}, P_1+2P_2)$-free graphs. Our technique for handling this case is to reduce the graph under consideration to a $k$-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of $(H_1,H_2)$-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of $(H_1,H_2)$-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8.

💡 Research Summary

The paper investigates the computational complexity of the Graph Colouring problem when the input graphs are simultaneously free of the diamond (the graph 2P₁+P₂) and another forbidden induced subgraph H with at most five vertices. Building on the known dichotomy for single‑forbidden‑subgraph classes, the authors first observe that Colouring is polynomial‑time solvable for all linear‑forest H (including sP₁+P₂, 2P₁+P₃, P₁+P₄, P₂+P₃, P₅) and NP‑complete otherwise. Consequently, the only unresolved case among five‑vertex graphs is H = P₁+2P₂.

To settle this case, the authors turn to the graph invariant clique‑width. Since any graph class of bounded clique‑width admits polynomial‑time algorithms for all MSO₂‑definable problems—including Colouring—the key is to prove that (diamond, P₁+2P₂)-free graphs have bounded clique‑width. They introduce a novel structural notion: totally k‑decomposable graphs, which generalise the canonical decomposition of bipartite graphs to k‑partite graphs. They show that any totally k‑decomposable graph has clique‑width at most 2k.

The main technical contribution is a series of graph operations (vertex deletions, bipartite complementations, label renamings) that transform any (diamond, P₁+2P₂)-free graph into a totally k‑decomposable graph without blowing up the clique‑width. This yields a uniform bound on the clique‑width of the whole class, and therefore Colouring becomes polynomial‑time solvable for (diamond, P₁+2P₂)-free graphs (Theorem 1).

The same methodology is applied to four additional bigenic classes, proving bounded clique‑width for (diamond, P₂+P₃), (K₃, P₁+2P₂), (K₃, P₁+P₂+P₃) and (K₃, P₁+P₅). Together with earlier results, these five new bounded‑width classes reduce the number of open cases in the systematic study of (H₁, H₂)-free graphs from 13 to 8 (Theorem 2).

The paper concludes with two open problems: (1) determining the complexity of Colouring for (diamond, H)-free graphs when H has at least six vertices (the first unknown case being H = P₆), and (2) resolving the remaining eight (H₁, H₂)-free classes for which the boundedness of clique‑width is still unknown. The work thus advances both the algorithmic understanding of Colouring in restricted graph classes and the structural theory of clique‑width.