Triangle Detection via Complement Vertex Covers: A Fast Certifier with an Exact Fallback

1. Introduction

Triangle detection—determining whether an undirected graph G contains three mutually adjacent vertices—is one of the most fundamental problems in graph algorithms. It arises as a primitive in network analysis [1], sparse-matrix computations [2], database join processing [3], and as the base case of exact clique solvers [4].

• Known complexity.

Let $n=\left|V\right|$ and $m=\left|E\right|$. The classical results are:

• $\mathcal{O}\left(m^{3/2}\right)$ via adjacency intersections. Chiba and Nishizeki [5] showed that listing all triangles costs $\mathcal{O}\left(a\right(G\left)·m\right)$, where $a\left(G\right)$ is the arboricity of G; since $a\left(G\right)=\mathcal{O}\left(\sqrt{m}\right)$, detection costs $\mathcal{O}\left(m^{3/2}\right)$.
• $\mathcal{O}\left(n^{\omega }\right)$ via matrix multiplication. Itai and Rodeh [6] observed that triangle detection reduces to Boolean matrix multiplication; with the current best exponent $\omega <2.371339$ [7], this gives $\mathcal{O}\left(n^{2.372}\right)$. Alon, Yuster, and Zwick [8] sharpened the combination to $\mathcal{O}\left(m^{2\omega /(\omega +1)}\right)$.
• Conditional lower bounds. Triangle detection sits at the centre of a web of fine-grained reductions [9,10]. The associated lower bounds are model-sensitive—they distinguish combinatorial from algebraic algorithms and are stated variously in terms of n or m—so they must be quoted with care. We make no claim against them in this paper.

• Our contribution.

We introduce Aegypti, a framework built on a single structural observation:

A triangle of G is precisely a three-vertex independent set of the complement $\overline{G}$, and a large independent set of $\overline{G}$ is exactly the part of V left uncovered by a small vertex cover of $\overline{G}$.

A vertex cover of $\overline{G}$ is produced in linear time by the Hvala algorithm [11], a companion work giving a 2-approximate minimum vertex cover in $\mathcal{O}(n+m)$ time. Removing it from V leaves an independent set of $\overline{G}$, i.e. a clique of G; any three such vertices are a triangle, which we certify in $\mathcal{O}\left(1\right)$. This complement route is efficient exactly when G is dense—the regime in which adjacency intersection becomes expensive—so we dispatch at the density threshold $\lceil n^{4/3}\rceil$: Chiba–Nishizeki below it, the complement cover above it. The dense branch is always sound, but it can be inconclusive (a 2-approximate cover need not leave three vertices uncovered even when G has a triangle), which yields two variants:

• Aegypti-fast (the default mode): use the dense branch as-is. A uniform $\mathcal{O}\left(n^2\right)$-time, one-sided triangle certifier: every triple it returns is a genuine triangle, but a None answer is conclusive only in the sparse regime or when the dense-branch cover condition (Theorem 4 and Hypothesis 1) holds.
• Aegypti-safe: if the dense branch is inconclusive, fall back to Chiba–Nishizeki. Unconditionally sound and complete on every graph, at a worst-case cost of $\mathcal{O}(n+m^{3/2})$.

We are explicit about what is and is not new. The safe variant’s worst case, $\mathcal{O}(n+m^{3/2})$, is the classical Chiba–Nishizeki regime; its value is correctness, not speed. The fast variant is quadratic but conditionally complete. The contribution is therefore best read as a framework rather than an unconditional fast detector:

• the complement-cover view turns a linear-time vertex cover into a certificate-producing dense preconditioner for triangle detection;
• it is sound on every graph, and every positive answer carries an explicit witness;
• it composes cleanly with an exact fallback to give an unconditionally complete detector;
• it isolates a precise structural condition (Theorem 4, Hypothesis 1) under which the fast branch alone is complete; and
• empirically it avoids the fallback on every tested dense instance, including adversarial small-clique families and an exhaustive sweep of all graphs on at most seven vertices (Section 6).

The framework is named Aegypti after Aedes aegypti, a mosquito whose compound eye detects a pattern by aggregating many simple local signals—analogous to reducing the local pattern of a triangle to a single global structure, a vertex cover of the complement.

• Paper organisation.

Section 2 fixes notation and recalls the two ingredients: the triangle/independent-set duality and the Hvala cover. Section 3 states Aegypti and its sub-routines as pseudocode that mirrors the reference implementation. Section 4 proves soundness, characterises the completeness of each variant, and establishes the running-time bounds. Section 5 reviews related work. Section 6 reports the empirical evaluation. Section 7 concludes.

2. Preliminaries

2.1. Graph Notation

All graphs $G=(V,E)$ are simple and undirected. We write $n=\left|V\right|$, $m=\left|E\right|$, $N\left(v\right)$ for the open neighbourhood of v, $d\left(v\right)=\left|N\right(v\left)\right|$, and $\Delta ={max}_{v}d\left(v\right)$. The complement$\overline{G}=(V,\overline{E})$ has the same vertex set and edge set $\overline{E}=\left({\displaystyle \genfrac{}{}{0pt}{}{V}{2}}\right)\setminus E$, so $|\overline{E}|=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)-m$.

Definition 1 

(Triangle). Atrianglein G is a set $\{u,v,w\}\subseteq V$ of three distinct vertices with $\{u,v\},\{v,w\},\{u,w\}\in E$.

Definition 2 

(Vertex cover, independent set, clique). Avertex coverof G is a set $C\subseteq V$ such that every edge has at least one endpoint in C. Anindependent setis a set $I\subseteq V$ with no edge between its members. Acliqueis a set of pairwise-adjacent vertices. We write $\alpha \left(G\right)$ for the maximum independent set size and $\omega \left(G\right)$ for the maximum clique size.

2.2. Two Structural Facts

The framework rests on two textbook dualities, stated explicitly because the dense regime is a direct application of them.

Proposition 1 

(Cover/independent-set duality). For any graph $H=(V,E_H)$, a set $C\subseteq V$ is a vertex cover of H if and only if $V\setminus C$ is an independent set of H. Consequently $\alpha \left(H\right)=n-{\mathrm{OPT}}_{\mathrm{VC}}\left(H\right)$, where ${\mathrm{OPT}}_{\mathrm{VC}}\left(H\right)$ is the minimum vertex cover size.

Proposition 2 

(Complement duality). A set $S\subseteq V$ is a clique of G if and only if S is an independent set of $\overline{G}$. In particular $\omega \left(G\right)=\alpha \left(\overline{G}\right)$, and $\{u,v,w\}$ is a triangle of G if and only if it is a three-vertex independent set of $\overline{G}$.

Combining the two: if C is any vertex cover of $\overline{G}$, then $V\setminus C$ is an independent set of $\overline{G}$ (Proposition 1), hence a clique of G (Proposition 2). If $|V\setminus C|\ge 3$, any three of its vertices form a triangle of G. This is the soundness of the dense regime; the only algorithmic content is producing a small cover C quickly.

2.3. The Hvala Vertex Cover

Hvala [11] is a linear-time ensemble heuristic for minimum vertex cover. On a graph with n vertices and m edges it returns, in $\mathcal{O}(n+m)$ time and space, a vertex cover whose size is at most twice optimal. We use it as a black box through the call $C\leftarrow Hvala-Vertex-Cover\left(H\right)$ and rely only on the following.

Lemma 1 

(Hvala guarantee). For every finite simple graph H, $Hvala-Vertex-Cover\left(H\right)$ returns a valid vertex cover C of H with $\left|C\right|\le 2{\mathrm{OPT}}_{\mathrm{VC}}\left(H\right)$, in time and space $\mathcal{O}\left(\right|V\left(H\right)|+|E\left(H\right)\left|\right)$.

Validity (that C is a genuine cover) underwrites soundness; the factor-2 size bound gives the unconditional large-clique guarantee of Theorem 4.

2.4. Chiba–Nishizeki Detection

For the sparse regime, and as the fallback of Aegypti-safe, we use the classical adjacency-intersection routine of Chiba and Nishizeki [5]. It scans each edge once and searches the smaller of the two endpoints’ adjacency sets for a common neighbour; a common neighbour completes a triangle. No vertex ordering is needed: building the adjacency sets is $\mathcal{O}(n+m)$ and the scan is ${\sum }_{\{u,v\}\in E}min(d\left(u\right),d\left(v\right))=\mathcal{O}\left(m^{3/2}\right)$, for $\mathcal{O}(n+m^{3/2})$ total (Theorem 5).

3. The Aegypti Framework

Aegypti (Algorithm 1) is a density-split detector whose pseudocode mirrors the reference implementation (find_triangle_coordinates): the same threshold $\lceil n^{4/3}\rceil$, the same two branches, the same extraction of three uncovered vertices, and the same $\mathcal{O}\left(1\right)$ certification. The Boolean parameter fallback selects the variant: with the marked line 19 it is Aegypti-safe; omitting that line yields Aegypti-fast, a one-sided certifier. The reference implementation defaults to $fallback=false$ (the fast variant); the safe variant is opt-in. Algorithm 2 states the Chiba–Nishizeki sub-routine (find_triangle_chiba_nishizeki) used in the sparse branch and as the fallback.

Algorithm 1 Aegypti(G,Fallback): a sound triangle certifier (with optional exact fallback).

Require:  Simple undirected graph $G=(V,E)$, no self-loops; flag fallback Ensure:  A certified triangle $\{u,v,w\}$, or None 1: if $\left|V\right|<3$or$\left|E\right|=0$then 2:     return None 3: end if 4: $n\leftarrow \left|V\right|$;   $m\leftarrow \left|E\right|$;   $\mathrm{bound}\leftarrow \lceil n^{4/3}\rceil$ 5: if $m\le \mathrm{bound}$then                                                                                                       ▹Sparse regime: exact 6:     return Chiba-Nishizeki$\left(G\right)$ 7: end if                                                                                                                                                  ▹Dense regime 8: $H\leftarrow \overline{G}$;   $C\leftarrow Hvala-Vertex-Cover\left(H\right)$                                                  ▹ linear-time 2-approx cover of $\overline{G}$ 9: $I\leftarrow V\setminus C$                                                                                              ▹ independent set of $\overline{G}$ = clique of G 10: if $\left|I\right|\ge 3$then 11:     pick three distinct vertices $u,v,w\in I$ 12:     if $\{u,v\},\{v,w\},\{u,w\}\in E$ then                                                                                     ▹$\mathcal{O}\left(1\right)$ certification 13:         return $\{u,v,w\}$ 14:     end if 15: end if 16: iffallbackthen                                                                                                              ▹Aegypti-safe only 17:     return Chiba-Nishizeki$\left(G\right)$ 18: end if 19: return None▹Aegypti-fast: inconclusive, not a proof of triangle-freeness

Algorithm 2 Chiba-Nishizeki$\left(G\right)$: exact $\mathcal{O}(n+m^{3/2})$ adjacency-intersection detection.

Require:  Simple undirected graph $G=(V,E)$, no self-loops Ensure:  A triangle $\{u,v,w\}$ if one exists, else None 1: if $\left|V\right|<3$or $\left|E\right|=0$ then 2:     return None 3: end if 4: $\mathrm{adj}\left[v\right]\leftarrow \mathrm{set}\left(N\right(v\left)\right)$ for every v                                                        ▹ static adjacency sets 5: for each edge $(u,v)\in E$ do 6:     let (small, large) be the smaller and larger of $\mathrm{adj}\left[u\right],\mathrm{adj}\left[v\right]$ 7:     for each $w\in \mathrm{small}$ do 8:         if $w\ne u$ and $w\ne v$ and $w\in \mathrm{large}$ then 9:            return $\{u,v,w\}$ 10:         end if 11:     end for 12: end for 13: return None

The matrix-multiplication routine (is_triangle_free_brute_force), which forms $A^3$ and tests its diagonal for the adjacency matrix A, is retained in the package only as a reference baseline and is not part of Aegypti.

4. Correctness and Complexity Analysis

4.1. Soundness

Theorem 1 

(Soundness). For either value offallback, ifAegypti$(G,fallback)$ returns a set $\{u,v,w\}$, then $\{u,v,w\}$ is a triangle of G.

Proof. 

A non-None answer is returned at one of three sites. The sparse branch and the fallback both return the output of Chiba-Nishizeki$\left(G\right)$, which returns $\{u,v,w\}$ only after checking $w\in \mathrm{adj}\left[u\right]$ and $w\in \mathrm{adj}\left[v\right]$ for an edge $(u,v)\in E$, so the three pairs are edges. The dense branch returns $\{u,v,w\}$ only after the explicit $\mathcal{O}\left(1\right)$ test $\{u,v\},\{v,w\},\{u,w\}\in E$. In all cases the returned set is a triangle. (The dense branch is sound by construction even without the test: by Lemma 1 the set C is a valid cover of $\overline{G}$, so $V\setminus C$ is an independent set of $\overline{G}$, hence a clique of G by Proposition 2; the test is a constant-time safeguard against a cover-routine fault.)    □

4.2. Completeness

Completeness holds outright in the sparse regime. In the dense regime it depends on the variant.

Theorem 2 

(Sparse-regime completeness). If $m\le \lceil n^{4/3}\rceil$ and G contains a triangle, thenAegypti$(G,fallback)$ returns one, for either value offallback.

Proof. 

Here the algorithm returns the output of the complete detector Chiba-Nishizeki$\left(G\right)$: its edge scan searches, for every edge, the smaller endpoint neighbourhood for a common neighbour, so a triangle $\{a,b,c\}$ is found when the first of its three edges is scanned [5].    □

Theorem 3 (Unconditional completeness ofAegypti-safe)With $fallback=true$,Aegypti$\left(G\right)$ returns a triangle if and only if G contains one.

Proof. 

“Only if” is Theorem 1. For “if”, suppose G has a triangle. If $m\le \lceil n^{4/3}\rceil$, apply Theorem 2. Otherwise the dense branch either returns a certified triangle, or reaches line 19 and returns $Chiba-Nishizeki\left(G\right)$, an exact detector that finds the triangle.    □

The fast variant is, by contrast, a one-sided certifier: it never returns a false triangle, but a None answer is conclusive only under the condition below. We state this prominently to forestall misreading.

Theorem 4 (Dense-regime completeness ofAegypti-fast)Let $m>\lceil n^{4/3}\rceil$ and $fallback=false$. The dense branch returns a triangle if and only if the cover C returned byHvalaon $\overline{G}$ satisfies $\left|C\right|\le n-3$. Since $\left|C\right|\le 2{\mathrm{OPT}}_{\mathrm{VC}}\left(\overline{G}\right)=2\left(n-\omega \left(G\right)\right)$ by Lemma 1 and Proposition 2, this is guaranteed whenever

$$\omega \left(G\right)\ge ⌈\frac{n+3}{2}⌉.$$

(1)

Proof. 

The branch returns a triangle exactly when $\left|I\right|=n-\left|C\right|\ge 3$, i.e. $\left|C\right|\le n-3$; soundness is Theorem 1. For the sufficient condition, $\left|C\right|\le 2{\mathrm{OPT}}_{\mathrm{VC}}\left(\overline{G}\right)$ and ${\mathrm{OPT}}_{\mathrm{VC}}\left(\overline{G}\right)=n-\alpha \left(\overline{G}\right)=n-\omega \left(G\right)$, so $n-\left|C\right|\ge 2\omega \left(G\right)-n\ge 3$ exactly when (1) holds.    □

Remark 1 (The fast variant is a certifier; None is not a triangle-free proof). Condition (1) is what the factor-2 bound alone guarantees: a clique on more than half the vertices. Outside it—a dense graph with a triangle but only small cliques—the fast variant may returnNoneon a triangle-containing input. Its failure to return a triangle is therefore not evidence of triangle-freeness unless Hypothesis 1 below is known to hold for the Hvala implementation. Use Aegypti-safe when a decision (not just a certificate) is required.

Hypothesis 1 

(Hvala independent-set completeness). For every finite simple graph H with $\alpha \left(H\right)\ge 3$, the cover returned by $Hvala-Vertex-Cover\left(H\right)$ leaves at least three vertices uncovered.

Remark 2 

(Hypothesis 1 is the central open problem, not a mild detail). Applied to $H=\overline{G}$ (so $\alpha \left(\overline{G}\right)=\omega \left(G\right)$), Hypothesis 1 says exactly: whenever G has a triangle,Hvalaexposes at least three uncovered vertices in $\overline{G}$. This is essentially the dense-branch completeness problem itself, so we present it as the framework’s principal open question rather than a supporting assumption. It isnota consequence of the 2-approximation: Hvalareturns amaximalindependent set of $\overline{G}$, and a maximal independent set can be strictly smaller than a maximum one—the independent domination number can be 2 while $\alpha =3$ (e.g. the five-vertex graph with edges $\{ax,bx,cy\}$). A worst-case 2-approximate cover can leave only $2\omega \left(G\right)-n<3$ vertices uncovered. Aegypti-safe removes the dependency entirely; Section 6 reports the empirical behaviour, including on families built to attack Hypothesis 1.

4.3. Running Time and Space

Theorem 5 

(Running time). On every simple undirected graph $G=(V,E)$:Aegypti-fastruns in $\mathcal{O}\left(n^2\right)$ time and space;Aegypti-saferuns in $\mathcal{O}(n+m^{3/2})$ time and $\mathcal{O}\left(n^2\right)$ space. The Chiba–Nishizeki sub-routine runs in $\mathcal{O}(n+m^{3/2})$ time.

Proof. 

Chiba–Nishizeki. Building the adjacency sets is $\mathcal{O}(n+m)$. The scan does, per edge $\{u,v\}$, work $min\left(d\right(u),d(v\left)\right)$. Splitting vertices at degree $\sqrt{m}$: an edge with both endpoints of degree $\le \sqrt{m}$ contributes $\le \sqrt{m}$, and there are at most m such edges, for $\mathcal{O}\left(m^{3/2}\right)$; an edge with an endpoint of degree $>\sqrt{m}$ is charged to that endpoint, and since ${\sum }_{v}d\left(v\right)=2m$ there are at most $2\sqrt{m}$ such vertices, each contributing ${\sum }_{u}min\left(·\right)\le {\sum }_{u}d\left(u\right)=2m$ over its incident edges, again $\mathcal{O}\left(m^{3/2}\right)$. Hence ${\sum }_{\{u,v\}\in E}min(d\left(u\right),d\left(v\right))=\mathcal{O}\left(m^{3/2}\right)$ [5]; adding the $\mathcal{O}(n+m)$ initialisation of the adjacency sets (over all n vertices, including isolated ones), the routine runs in $\mathcal{O}(n+m^{3/2})$ time and $\mathcal{O}(n+m)$ space.

Sparse branch ($m\le \lceil n^{4/3}\rceil$, both variants). The above is $\mathcal{O}\left({\left(n^{4/3}\right)}^{3/2}\right)=\mathcal{O}\left(n^2\right)$ time and $\mathcal{O}\left(n^2\right)$ space.

Dense branch ($m>\lceil n^{4/3}\rceil$). Forming $\overline{G}$ touches each of the $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)$ pairs once, $\mathcal{O}\left(n^2\right)$ time and a graph of size $\mathcal{O}\left(n^2\right)$. Hvala on $\overline{G}$ costs $\mathcal{O}\left(\right|V|+|\overline{E}\left|\right)=\mathcal{O}\left(n^2\right)$ (Lemma 1); cover removal, selection, and the $\mathcal{O}\left(1\right)$ certification are $\mathcal{O}\left(n\right)$. So the dense branch up to line 19 is $\mathcal{O}\left(n^2\right)$.

For Aegypti-fast the dense branch ends there: total $\mathcal{O}\left(n^2\right)$ across both regimes. For Aegypti-safe, an inconclusive dense branch additionally runs Chiba–Nishizeki, adding $\mathcal{O}(n+m^{3/2})$, so the worst-case time is $\mathcal{O}(n+m^{3/2})$ and the space $\mathcal{O}\left(n^2\right)$.    □

Remark 3 

(Why the threshold is $n^{4/3}$). The crossover is the m at which the sparse cost $m^{3/2}$ reaches the dense cost $n^2$: $m^{3/2}=n^2\iff m=n^{4/3}$. Splitting at $\lceil n^{4/3}\rceil$ caps the fast variant at $\mathcal{O}\left(n^2\right)$ uniformly.

4.4. Position Relative to the Barriers

The fast branch has a quadratic worst-case running time, but because its dense-regime completeness is conditional (Remark 1 and Remark 2), this should not be read as an unconditional improvement over complete algebraic triangle detection, whose best exponent stands at $\omega <2.371339$ [7]. Comparing a one-sided certifier to a complete algorithm would be unfair, and we do not. The unconditionally complete variant, Aegypti-safe, runs in $\mathcal{O}\left(m^{3/2}\right)$—the classical combinatorial regime—so the framework refutes no fine-grained lower-bound conjecture [9,10]; those bounds are model-sensitive and connect triangle, matrix, and path problems by subcubic reductions. What Aegypti contributes is structural: a sound, certificate-producing, linear-time dense preconditioner that, on every instance we tested, lets the quadratic branch decide the problem without invoking the fallback.

5. Related Work

Combinatorial triangle detection.

Itai and Rodeh [6] gave the first linear-space $\mathcal{O}\left(m^{3/2}\right)$ algorithm, still the best combinatorial bound for sparse graphs. Chiba and Nishizeki [5] gave the output-sensitive arboricity analysis underlying the sparse branch and the fallback. Latapy [12] and Ortmann and Brandes [13] are detailed engineering studies of these adjacency-intersection methods.

Algebraic algorithms.

Alon, Yuster, and Zwick [8] obtained $\mathcal{O}\left(m^{2\omega /(\omega +1)}\right)$ by combining matrix multiplication with combinatorial techniques; for dense graphs $\mathcal{O}\left(n^{\omega }\right)$ dominates, with the exponent still being lowered [7,14]. Aegypti stays combinatorial.

Fine-grained barriers.

Vassilevska Williams and Williams [9] established the subcubic equivalence of triangle, matrix, and path problems; Abboud and Vassilevska Williams [10] derived conditional lower bounds for many problems with triangle detection at their core. These bounds are model-sensitive, and our work is consistent with them.

Vertex cover and independent set.

The vertex-cover/independent-set duality (Proposition 1) and the complement duality (Proposition 2) are classical. What is new is using a fast approximate cover of the complement as a sound triangle certificate producer, composed with a Chiba–Nishizeki fallback to recover unconditional completeness. The cover is supplied by Hvala [11]; we depend on it only through Lemma 1.

6. Experimental Evaluation

We validate the implementation with the reproducible car/ experiment in the repository. It runs four routines, scoring each against an independent exact oracle (triangle existence by direct neighbourhood intersection): Aegypti-safe (fallback=True), Aegypti-fast (fallback=False), find_triangle_chiba_nishizeki, and the is_triangle_free_brute_force baseline. For every dense-regime instance the harness additionally records the cover diagnostics that decide Aegypti-fast completeness: $\left|C\right|$, the uncovered count $|V\setminus C|$, $\omega \left(G\right)$ (exact, for $n\le 24$), ${\mathrm{OPT}}_{\mathrm{VC}}\left(\overline{G}\right)=n-\omega \left(G\right)$, the ratio $\left|C\right|/{\mathrm{OPT}}_{\mathrm{VC}}\left(\overline{G}\right)$, and the miss / fallback flags.

Benchmark.

The suite combines three groups. (i) Random and structured families: sparse Erdos–Rényi ($n\in [12,60]$, $p\in \{0.02,0.04,0.06,0.08\}$), dense Erdos–Rényi ($n\in [10,40]$, $p\in \{0.4,0.55,0.7,0.85\}$), triangle-free bipartite, planted-triangle sparse graphs, planted-clique dense graphs, and structured graphs (complete, even cycles, wheels, complete bipartite, random regular). (ii) Adversarial dense, small clique number: complete tripartite $K_{a,b,c}$ (dense, $\omega =3$, $K_4$-free), complete 4-partite $K_{a,b,c,d}$ ($\omega =4$, $K_5$-free), and balanced complete bipartite plus one intra-part edge (dense, $\omega =3$)—families where “has a triangle” is far from “has a large clique”, precisely the regime that stresses Hypothesis 1. (iii) Exhaustive: all graphs on at most seven vertices (the NetworkX Graph Atlas), giving complete coverage of small cases. A fixed seed makes the run deterministic.

Reproducibility.

The baseline forms $A^3$ by sparse int8 matrix products (SciPy CSR) and tests the diagonal; it is a practical linear-algebra check, not an implementation of an $\mathcal{O}\left(n^{\omega }\right)$ algorithm. The dense/sparse split uses $\lceil n^{4/3}\rceil$ exactly as in the dispatch. Each run records the platform string and the NetworkX, NumPy, and SciPy versions in car_experiment.json; the cover is the hvala package, the detectors the aegypti package (both at the repository version). The run reported below used Python 3.12.3, NetworkX 3.6.1, and NumPy 2.4.6 on a Windows 11 workstation. Per-instance and aggregate outputs (car_by_instance.csv, car_summary.csv, car_experiment.json) are emitted next to the script.

Correctness.

Across the full benchmark of $12,330$ instances ($8,860$ triangle-containing), all four routines returned the oracle-correct verdict on every instance; the three witness-returning routines (both Aegypti variants and Chiba–Nishizeki) returned a valid triangle on every positive instance, with 0 disagreements and 0 invalid witnesses. Decisively for the one-sided fast variant, Aegypti-fast recorded 0 misses among the $4,799$ triangle-containing dense instances: its dense branch left at least three vertices uncovered on every one. The Aegypti-safe fallback was entered on only 840 dense instances—all of them triangle-free, where returning None is correct—and never on a triangle-containing graph. Table 1 reports the mean running times. On these deliberately small instances the asymptotic gap between the $\mathcal{O}\left(n^2\right)$ and matrix-multiplication methods is not visible; the value of the dispatch is the uniform $\mathcal{O}\left(n^2\right)$ ceiling for the fast variant (Theorem 5), not a constant-factor win.

Stressing the dense branch.

Table 2 reports the dense-branch diagnostics on the families where “triangle” and “large clique” come apart. On the adversarial small-clique families—complete tripartite ($\omega =3$), complete 4-partite ($\omega =4$), and balanced bipartite plus one edge ($\omega =3$)—and on the exhaustive sweep of all graphs with $n\le 7$, Hvala returned an optimal cover of the complement (ratio $\left|C\right|/{\mathrm{OPT}}_{\mathrm{VC}}\left(\overline{G}\right)=1.0$) on every instance, leaving exactly $\omega \left(G\right)\ge 3$ vertices uncovered, and the fast branch certified a triangle on all of them. The largest cover ratio anywhere in the benchmark was $1.2$ (dense Erdos–Rényi), and even there the minimum uncovered count over triangle-containing dense instances was 3—never below the threshold of Theorem 4. The exhaustive $n\le 7$ sweep ($1,244$ graphs, 91 of them in the dense regime) passed with no miss, a complete small-case certificate: a single miss in any of these families would have been the sharpest signal against Hypothesis 1. This remains finite evidence—it does not prove Hypothesis 1—and Aegypti-safe is what guarantees completeness in its absence.

7. Conclusion

We presented Aegypti, a triangle-detection framework that dispatches on density at $\lceil n^{4/3}\rceil$: exact Chiba–Nishizeki on sparse inputs, and on dense inputs a single linear-time Hvala vertex cover of the complement whose uncovered vertices yield a certified triangle. The framework is sound on every graph. Aegypti-fast is a uniform $\mathcal{O}\left(n^2\right)$, one-sided certifier—complete in the sparse regime and, in the dense regime, under the explicit cover condition of Theorem 4—while Aegypti-safe adds an exact fallback and is unconditionally complete (Theorem 3) at a worst-case cost of $\mathcal{O}(n+m^{3/2})$. The honest reading is a sound quadratic certifier with an unconditional $\mathcal{O}(n+m^{3/2})$ fallback detector, not an unconditional quadratic detector. A reference implementation is available in the aegypti Python package [15], which depends on hvala [16].

Open problems.

• Hypothesis 1 (central). Does the Hvala ensemble leave at least three vertices uncovered on every graph with $\alpha \ge 3$? A proof would make Aegypti-fast an unconditionally complete $\mathcal{O}\left(n^2\right)$ detector; a counterexample would pinpoint exactly where the fallback is indispensable. This is the framework’s principal open question.
• Avoiding explicit complementation. The dense branch spends $\mathcal{O}\left(n^2\right)$ materialising $\overline{G}$. Can Hvala run implicitly on the complement to lower the constant?
• Counting. Does the cover/complement view extend from detecting one triangle to counting or listing triangles within the same budget?