GraphHull: Convex Hulls in Graphs & Models

• GraphHull is defined as the convex hull structure in graph theory that models closures, hull sets, and geodesic convexity.
• It underpins scalable approximations in large-network mining by employing outerplanar subgraph heuristics to estimate geodesic convex hulls efficiently.
• GraphHull also refers to a two-level generative model in graph representation learning that organizes global archetypes and community-specific prototypes as nested convex hulls.

GraphHull is a convexity-centered term used in several distinct research settings. In graph theory, it denotes the computation or analysis of convex hulls, hull sets, hull numbers, and related isometric hulls under geodetic or metric convexity. In large-network mining, it denotes a heuristic for approximating geodesic convex hulls via outerplanar spanning subgraphs. In graph representation learning, it names the two-level archetypal graph generative model of Nakis et al., in which global archetypes and community-specific prototypes are organized as nested convex hulls. In optimization notation, $\GraphHull$ also denotes the convex hull of the graph of a monomial over a box domain (Knauer et al., 2017, Seiffarth et al., 2022, Nakis et al., 24 Feb 2026, Lee et al., 2 May 2026).

1. Terminological scope and core mathematical objects

The term appears in the literature with different formal meanings, but all of them are organized around closure, convexity, or convex-hull structure.

Usage	Formal object	Source
Graph convexity	convex hull, hull set, hull number, isometric hull	(Knauer et al., 2017, Coelho et al., 2018, Albenque et al., 2013)
Large-network heuristic	approximate geodesic convex hull by sampled outerplanar subgraphs	(Seiffarth et al., 2022)
Graph ML model	two-level convex-hull generative model with archetypes and prototypes	(Nakis et al., 24 Feb 2026)
Optimization notation	$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$ over a box domain	(Lee et al., 2 May 2026)

In geodetic convexity, for a finite simple graph $G$ and vertices $u,v$, the interval $I[u,v]$ consists of $u,v$ and all vertices on any shortest $u$–$v$ path. For $S\subseteq V(G)$, one defines

$I[S]=\bigcup\{I[u,v]:u,v\in S\}.$

A set $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$0 is convex if $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$1, and its convex hull $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$2 is the smallest convex set containing $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$3. Equivalently, one iterates $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$4, $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$5 until convergence; the limit is $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$6. A hull set is a set $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$7 with $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$8, and the hull number $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$9 is the minimum cardinality of a hull set (Coelho et al., 2018).

In metric-hull language, the same closure notion is written as

$G$0

The convex-hull–number is denoted $G$1 in one formulation, while the isometric hull $G$2 is defined as a minimal-vertex isometric subgraph containing $G$3; its associated invariant is the isometric-hull–number $G$4 (Knauer et al., 2017). A related partial-cube literature uses the notation $G$5 for the minimum size of a hull set and characterizes hull sets through convex cut-partitions (Albenque et al., 2013).

A useful consequence is that “GraphHull” is not a single universally standardized object. The literature instead separates a graph-theoretic closure problem, a scalable approximation heuristic, and a latent-variable network model. The common thread is the use of convex-hull structure to control combinatorial or geometric explanation.

2. Closure systems, hull-number computation, and partial-cube structure

The closure-theoretic formulation begins with a finite set $G$6 and a closure function $G$7 satisfying extensivity, monotonicity, and idempotence. An equivalent description is

$G$8

A weaker pseudo-closure $G$9 satisfies

$u,v$0

which still implies idempotence. Graph convex hull is an instance of such a closure operator (Knauer et al., 2017).

For the Minimum Generator Set problem on a pseudo-closure $u,v$1, the paper on metric hulls gives Algorithm MinGen. If computing $u,v$2 from $u,v$3 takes $u,v$4 time, the runtime is $u,v$5 in general, and $u,v$6 when $u,v$7 is size-increasing. Specializing to graphs, if all convex sets are given, then $u,v$8 is computable in $u,v$9 time (Knauer et al., 2017).

For partial cubes, the hull-number problem admits a particularly clean reformulation. A partial cube is a graph that admits an isometric embedding into a hypercube $I[u,v]$0. With a convex cut-partition $I[u,v]$1, a set $I[u,v]$2 is a hull set if and only if it meets both sides of every cut. This turns GraphHull on partial cubes into a minimum hitting-set problem over the cut-partition (Albenque et al., 2013).

That perspective yields both positive and negative algorithmic results. Determining whether $I[u,v]$3 for a partial cube $I[u,v]$4 is NP-complete, via a reduction from SAT-AM3. At the same time, for planar partial cube quadrangulations, the problem becomes polynomial-time: the graph is represented as the region graph of a non-separating arrangement of Jordan curves, the associated intersection graph is chordal, and the hull number reduces to a minimum clique-cover problem on that chordal graph (Albenque et al., 2013).

The same work also gives a lattice-theoretic characterization. Fixing $I[u,v]$5, the poset $I[u,v]$6 of convex subgraphs containing $I[u,v]$7 is atomistic, and partial-cube-ness is characterized by $I[u,v]$8 being upper locally distributive with a Hasse diagram containing $I[u,v]$9 isometrically (Albenque et al., 2013). This places GraphHull in a broader program connecting graph convexity, lattice theory, poset dimension, and discrete geometry.

3. Structural bounds and complexity barriers

A substantial body of GraphHull research studies how hull number depends on graph class. For complementary prisms, defined from the disjoint union of $u,v$0 and $u,v$1 plus a perfect matching $u,v$2, one central lemma states that every simplicial vertex of $u,v$3 or of $u,v$4 must be in any hull set of $u,v$5. Lower bounds then follow by counting forced simplicial vertices, and upper bounds are established by explicit hull-set constructions (Coelho et al., 2018).

For trees, if $u,v$6 is a tree on $u,v$7 vertices and $u,v$8 is the star $u,v$9 with $u$0, then

$u$1

For disconnected $u$2 with $u$3 components, of which at least two are nontrivial, the hull number of the complementary prism is $u$4. If $u$5 has exactly one nontrivial component $u$6 and $u$7 isolated vertices, then $u$8, while the upper bounds depend on $u$9 and $v$0. For connected cographs whose complement has $v$1 nontrivial components and $v$2 isolated vertices, the cases $v$3, $v$4, and $v$5 are determined separately (Coelho et al., 2018).

One notable consequence is a contrast with $v$6-convexity. In $v$7-convexity, Duarte et al. proved that if both $v$8 and $v$9 are connected, then $S\subseteq V(G)$0. In geodetic convexity, by contrast, the hull number on complementary prisms cannot be limited: for each $S\subseteq V(G)$1, there exists connected $S\subseteq V(G)$2 with connected $S\subseteq V(G)$3 and $S\subseteq V(G)$4 (Coelho et al., 2018). This removes a common misconception that connectedness of both layers alone should force small hull sets.

Complexity barriers are also strong outside special classes. The LOGMGS problem is LOGSNP-complete for atomistic closures when $S\subseteq V(G)$5. Computing an isometric hull of a prescribed set $S\subseteq V(G)$6 is NP-complete even for $S\subseteq V(G)$7, and computing the isometric-hull–number $S\subseteq V(G)$8 is $S\subseteq V(G)$9-complete (Knauer et al., 2017). These hardness results indicate that exact GraphHull computation is structurally difficult even when the defining closure concept is mathematically elementary.

4. Approximate geodesic GraphHull for large networks

A distinct line of work treats GraphHull as a scalable heuristic for approximating geodesic convex hulls in large graphs. The target object is the geodesic convex hull $I[S]=\bigcup\{I[u,v]:u,v\in S\}.$0 of a node set $I[S]=\bigcup\{I[u,v]:u,v\in S\}.$1, where

$I[S]=\bigcup\{I[u,v]:u,v\in S\}.$2

and a set is closed if $I[S]=\bigcup\{I[u,v]:u,v\in S\}.$3 for all $I[S]=\bigcup\{I[u,v]:u,v\in S\}.$4. The exact closure may be computed by iterative BFS, with worst-case time $I[S]=\bigcup\{I[u,v]:u,v\in S\}.$5 (Seiffarth et al., 2022).

The heuristic replaces the original graph by a family of sampled almost-maximal outerplanar spanning subgraphs. For each sampled subgraph $I[S]=\bigcup\{I[u,v]:u,v\in S\}.$6, one computes $I[S]=\bigcup\{I[u,v]:u,v\in S\}.$7 and then declares a vertex to belong to the approximate hull if it appears in at least $I[S]=\bigcup\{I[u,v]:u,v\in S\}.$8 of the $I[S]=\bigcup\{I[u,v]:u,v\in S\}.$9 sampled closures: $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$00 The outerplanar sampling stage runs in $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$01 time, and the closure routine on each $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$02 exploits the block-bridge tree together with blockwise generators and BFS on biconnected blocks (Seiffarth et al., 2022).

The total time complexity is

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$03

where $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$04 is the maximum number of interior faces per block. Since $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$05 in practice, this is described as roughly $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$06, linear in the number of edges. The standard experimental choice is $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$07 and $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$08 (Seiffarth et al., 2022).

The empirical motivation is core-periphery decomposition based on convexity. On real-world networks from SNAP, with sizes up to $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$09 million nodes and $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$10 million edges, the exact core computation did not finish within $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$11 days for the largest graphs, whereas the heuristic completed approximate core-periphery decomposition in $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$12 hours for the $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$13 million-edge graph. Jaccard similarity between exact and approximate cores ranged from $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$14 to $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$15, and with $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$16, $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$17 achieved $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$18 in $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$19 cases (Seiffarth et al., 2022).

This version of GraphHull is therefore algorithmic rather than latent-geometric. Its purpose is not to parameterize graphs, but to accelerate closure computation while preserving enough shortest-path structure to approximate geodesic convexity accurately.

5. GraphHull as a two-level archetypal graph generative model

In the 2026 representation-learning usage, GraphHull is a generative model for undirected graphs with adjacency $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$20, $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$21, and a $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$22-dimensional latent embedding constrained to a $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$23-polytope with $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$24 “pure” community archetypes. The global archetypes are collected in

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$25

with global convex hull

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$26

where $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$27. To ensure numerical stability and diversity of directions, $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$28 is parameterized by a boxed SVD,

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$29

with $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$30, $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$31 having orthonormal columns and $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$32 (Nakis et al., 24 Feb 2026).

Each community $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$33 is refined by a local convex hull

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$34

whose $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$35 vertices are local “prototypes”. The matrix $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$36 contains $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$37 community-specific convex-weight rows $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$38 and one anchor row $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$39, so the last row of $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$40 is exactly $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$41. The anchor-dominance constraint

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$42

ensures that $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$43 for $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$44 (Nakis et al., 24 Feb 2026).

Each node is assigned to exactly one community $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$45, encoded by a one-hot vector $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$46, and receives barycentric weights $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$47 over the local prototypes of its community. Its latent embedding is

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$48

The edge model is logistic: $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$49 where $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$50 is a degree bias and $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$51 Half-Normal$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$52 is a global scale. The complete-data log-likelihood is

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$53

(Nakis et al., 24 Feb 2026).

To encourage diversity and stability, the model places determinantal point process priors on both the global archetypes $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$54 and each local hull $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$55. With row-normalized $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$56 and Gram matrix $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$57, the $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$58-ensemble DPP prior is

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$59

Additional priors are Dirichlet on $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$60, truncated or anchor-dominant Dirichlet on each $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$61, Gaussian on $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$62, and half-normal on $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$63 (Nakis et al., 24 Feb 2026).

MAP estimation combines the likelihood and priors into a joint log-posterior $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$64. To avoid the naïve $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$65 edge cost, the method evaluates positive edges exactly in $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$66 time and unbiasedly subsamples non-edges. Each iteration therefore costs

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$67

which is linear in the number of edges when $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$68. Optimization uses Adam with projection of simplex-constrained variables back onto their domains (Nakis et al., 24 Feb 2026).

6. Interpretability, empirical behavior, and conceptual relations

The interpretability claim of the GraphHull generative model is tied directly to its geometry. Global archetypes $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$69 are “pure” community prototypes, local hulls $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$70 capture intra-community variation, and every node has the unique decomposition

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$71

The edge log-odds

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$72

therefore decompose into interpretable geometric interactions and degree biases. Reported visual diagnostics include adjacency reordering by community and prototype $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$73, PCA projections of $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$74, and circular “prototype-space” plots (Nakis et al., 24 Feb 2026).

The empirical study uses citation networks Cora and CiteSeer, collaboration networks DBLP, AstroPh, GrQc, and HepTh, and social networks LastFM and Pol. Baselines include DeepWalk, node2vec, Role2Vec, NetMF, GraRep, RandNE, MNMF, SymmNMF, NNSED, mixed-membership methods, and Dmon. In link prediction, the protocol removes $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$75 of edges while keeping the graph connected, trains on the remainder, and tests on held-out edges against an equal-size set of non-edges; the reported metrics are AUC-ROC and AUC-PR. Table 2 reports that GraphHull matches or exceeds baselines for embedding sizes $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$76. For community detection, evaluated by NMI and ARI against ground truth, Table 3 reports best or near-best performance on CiteSeer, LastFM, and Pol, and competitive performance on Cora (Nakis et al., 24 Feb 2026).

The parameter $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$77 governs a tension between identifiability and overlap. For $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$78, hulls remain strictly disjoint; as $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$79, the model permits overlapping “mixed” communities, often improving predictive performance at some loss of strict identifiability. DPP priors on $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$80 and $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$81 are used to prevent collapse and to remove spurious low-rank degeneracies by increasing repulsion strength (Nakis et al., 24 Feb 2026). This suggests a continuum between identifiable archetypal communities and more permissive mixed-membership geometry.

The graph-theoretic and graph-ML senses of GraphHull are related by vocabulary rather than by objective. In hull-number theory, convex hull is a closure operator on vertices, and the central questions concern hull sets, generators, shortest-path structure, and computational complexity. In the generative model, convex hull is a latent Euclidean constraint used to encode community purity, prototype refinement, and self-explanation. The shared language is genuine, but the formal targets are different.

A final notational extension occurs in global optimization, where $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$82 denotes the convex hull of the graph of the monomial $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$83 over a box domain $\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$84. In that setting,

$\GraphHull=\conv\{(x,y): y=\prod_i x_i\}$85

and the object is a full-dimensional polytope with an explicit linear-inequality description and a closed-form volume formula when at most one lower bound is positive (Lee et al., 2 May 2026). This usage is not graph-theoretic, but it reinforces the broader pattern that “GraphHull” often labels the convex hull of a graph, either of a function or of latent positions, rather than a single canonical algorithm or invariant.