Instruction set for the representation of graphs

We present IsalHG, a method for representing the structure of any finite, connected hypergraph of bounded hyperedge arity as a string over a compact instruction alphabet $Σ_{\mathrm{HG}}$. The encoding is executed by a small virtual machine comprising a sparse hypergraph, a circular doubly-linked list (CDLL) of node references, and $k$ traversal pointers, where $k$ bounds the hyperedge arity. Instructions either move a pointer through the CDLL or insert a hyperedge, optionally together with new nodes, into the hypergraph. Every string over $Σ_{\mathrm{HG}}$ decodes to a valid hypergraph; the alphabet is closed. A greedy HypergraphToString (h2s) algorithm encodes any connected hypergraph into a string; a backtracking variant seeded at nodes of lexicographically maximal structural tuple produces a canonical string $w^{*}$, which we conjecture to be a complete isomorphism invariant. Canonical-string equality then decides hypergraph isomorphism natively, without the standard reduction to the Levi incidence graph followed by a graph-isomorphism engine. We verify the round-trip property $s2h(h2s(H)) \cong H$ on 150 connected random uniform hypergraphs and on named combinatorial designs, and we benchmark the canonical algorithm against the three practically available exact baselines -- nauty, Traces, and bliss operating on the 2-coloured Levi graph -- across a $(n, c)$ grid with ten seeds per cell. All four methods agree on every one of 600 isomorphism verdicts, consistent with the completeness conjecture. On wall-clock time the Levi baselines dominate every tested cell by three to five orders of magnitude (geometric-mean ratio $311\times$ to $117{,}672\times$), which we report as measured. We contribute the representation framework, a conjecture of canonical completeness, and the first native-versus-Levi benchmark for hypergraph isomorphism.

We introduce DRESS, a deterministic, parameter-free framework that iteratively refines the structural similarity of edges in a graph to produce a canonical fingerprint: a real-valued edge vector, obtained by converging a non-linear dynamical system to its unique fixed point. The fingerprint is isomorphism-invariant by construction, numerically stable (strictly bounded, precision-preserving, and mathematically well-posed), fast and embarrassingly parallel to compute: DRESS total runtime is $\mathcal{O}(I \cdot m \cdot d_{\max})$ for $I$ iterations to convergence, and convergence is guaranteed by Birkhoff contraction. We generalize the original equation to Motif-DRESS (arbitrary structural motifs) and Generalized-DRESS (abstract aggregation template), and introduce $Δ$-DRESS, which runs DRESS on each vertex-deleted subgraph to boost expressiveness. $Δ$-DRESS empirically separates all 7,983 graphs in a comprehensive Strongly Regular Graph benchmark, and on the tested CFI instances ($k = 0,1,2,3$), $k$-deletion ($Δ^k$-DRESS) empirically matches the $(k{+}2)$-WL boundary.

DRESS is a deterministic, parameter-free framework for structural graph refinement that iteratively refines the structural similarity of edges in a graph to produce a canonical fingerprint: a real-valued edge vector, obtained by converging a nonlinear dynamical system to its unique fixed point. $Δ$-DRESS is a member of the DRESS family of graph fingerprints that applies a single level of vertex deletion. We test it on a benchmark of 51,813 distinct graphs across 34 hard families, including the complete Spence collection of strongly regular graphs (43,703 SRGs, 12 families), four additional SRG families (8,015 graphs), and 18 classical hard constructions (102 family entries corresponding to 99 distinct graphs). $Δ$-DRESS produces unique fingerprints in 33 of 34 benchmark families at $k=1$, resolving all but one within-family collision among over 576 million non-isomorphic pairs. One genuine collision exists at deletion depth $k=1$, between two vertex-transitive SRGs in SRG(40,12,2,4), which is resolved by a single-step fallback to $Δ^2$-DRESS. For every family with pairwise-comparable full sorted-multiset fingerprints, the minimum observed separation margin remains at least $137 \times ε$, confirming that the reported separations are numerically robust and not artifacts of the convergence threshold. We also show that $Δ$-DRESS separates the Rook $L_2(4)$/Shrikhande pair, proving it escapes the theoretical boundary of 3-WL. The method runs in $\mathcal{O}(n \cdot I \cdot m \cdot d_{\max})$ time per graph.