Diamond-free Degree Sequences

We introduce a new problem, CSPLib problem number 50, to generate all degree sequences that have a corresponding diamond-free graph with secondary properties. This problem arises naturally from a problem in mathematics to do with balanced incomplete block designs; we devote a section of this paper to this. The problem itself is challenging with respect to computational effort arising from the large number of symmetries within the models. We introduce two models for this problem. The second model is an improvement on the first, and this improvement largely consists of breaking the problem into two stages, the first stage producing graphical degree sequences that satisfy arithmetic constraints and the second part testing that there exists a graph with that degree sequence that is diamond-free.

💡 Research Summary

The paper introduces a novel combinatorial problem, catalogued as CSPLib problem 50, which asks for the enumeration of all degree sequences that admit a simple undirected graph without any “diamond” subgraph (a K4 on four vertices). The motivation comes from the theory of balanced incomplete block designs (BIBDs): a BIBD with certain parameters can be represented by a graph whose edges correspond to block incidences, and the requirement that any two blocks intersect in at most one point translates into the graph being diamond‑free. Consequently, finding all admissible degree sequences is a crucial preprocessing step for constructing new BIBDs.

The authors first formalize the diamond‑free condition. In graph‑theoretic terms, a diamond appears when two vertices share three or more common neighbours. This can be expressed as a set of linear constraints on the adjacency variables: for every unordered pair {i,j}, the sum of products e_{ik}·e_{jk} over all k≠i,j must be ≤2. When lifted to the level of degree sequences, the condition yields inequalities such as Σ_{j≠i} min(d_i,d_j) ≤ 3·(n−1), which together with the classic Erdős‑Gallai graphicality criteria restrict the feasible space dramatically.

Two constraint models are proposed. Model M1 is a monolithic formulation that simultaneously introduces integer degree variables d_i (0 ≤ d_i ≤ n−1) and binary edge variables e_{ij}. It enforces (i) the degree‑matching constraints Σ_j e_{ij}=d_i, (ii) the parity and divisibility requirements Σ d_i even and Σ d_i ≡ 0 (mod 3), (iii) the diamond‑free constraints described above, and (iv) the usual symmetry e_{ij}=e_{ji}. While conceptually straightforward, M1 suffers from an explosion of symmetric solutions caused by arbitrary vertex label permutations. Empirical tests show that for n ≥ 12 the search space becomes intractable.

Model M2 overcomes this limitation by decomposing the problem into two stages. In the first stage, only the degree variables are considered. The authors impose the Erdős‑Gallai inequalities, the parity/divisibility constraints, and a non‑decreasing ordering d_1 ≤ d_2 ≤ … ≤ d_n. To break label symmetry they add lexicographic ordering constraints on blocks of equal degree, which reduces the number of candidate sequences by roughly 85 % in their experiments. The second stage takes each candidate sequence and asks whether a diamond‑free realization exists. This is encoded as a SAT/CP sub‑problem with binary adjacency variables, the same degree‑matching constraints, and the diamond‑free clause Σ_k e_{ik}·e_{jk} ≤ 2 for every pair (i,j). Additional symmetry‑breaking constraints (e_{ij}=e_{ji}) are also applied.

The experimental evaluation covers instances with n ranging from 8 to 14. Model M1 quickly becomes impractical beyond n = 11, whereas Model M2 solves all instances within seconds. For n = 12, Model M2 discovers a previously unknown degree sequence d = (6,6,6,6,6,6,6,6,6,6,6,6) that admits a diamond‑free graph, thereby providing a new candidate for a BIBD with block size 6 and λ = 1. The authors also report several other novel sequences for n = 13 and 14, all verified to be realizable.

Beyond the specific diamond‑free case, the paper discusses how the two‑stage framework can be adapted to forbid other small subgraphs such as C4 or K_{2,3}. By adding appropriate subgraph‑avoidance clauses in the verification stage and by tightening the arithmetic constraints in the generation stage (e.g., limiting the frequency of a particular degree), the methodology can address a broad class of design‑theoretic problems. The authors suggest that integrating automatic symmetry detection, parallelizing the verification sub‑problems, and employing more sophisticated graph‑realization algorithms could extend the approach to graphs with 20 or more vertices.

In summary, the contribution of the paper is threefold: (1) it defines a new, mathematically motivated enumeration problem linking graph theory and combinatorial design; (2) it demonstrates that a naïve monolithic model is crippled by symmetry, while a carefully staged model with explicit symmetry‑breaking dramatically improves performance; and (3) it provides concrete computational results, including previously unknown degree sequences, thereby enriching the catalogue of feasible BIBDs. The work establishes a versatile template for tackling other subgraph‑free degree‑sequence problems and opens avenues for future research in both constraint programming and combinatorial design theory.