--- title: 'GraphHull: Convex Hulls in Graphs & Models' url: https://www.emergentmind.com/topics/graphhull type: topic --- # GraphHull: Convex Hulls in Graphs & Models 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 [1710.02958] [2206.07350] [2602.21342] [2605.01493]. ## 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 | [1710.02958], [1807.08295], [1309.5724] | | Large-network heuristic | approximate geodesic convex hull by sampled outerplanar subgraphs | [2206.07350] | | Graph ML model | two-level convex-hull generative model with archetypes and prototypes | [2602.21342] | | Optimization notation | \(\GraphHull=\conv\{(x,y): y=\prod_i x_i\}\) over a box domain | [2605.01493] | 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 \(S\) is convex if \(I[S]=S\), and its convex hull \(H(S)\) is the smallest convex set containing \(S\). Equivalently, one iterates \(I^0[S]=S\), \(I^{p+1}[S]=I[I^p[S]]\) until convergence; the limit is \(H(S)\). A hull set is a set \(S\) with \(H(S)=V(G)\), and the hull number \(h(G)\) is the minimum cardinality of a hull set [1807.08295]. In metric-hull language, the same closure notion is written as \[ \mathrm{conv}(S)=\bigcap\{C\subseteq V:\;S\subseteq C,\;C\text{ is convex}\}. \] The convex-hull–number is denoted \(\chi_c(G)\) in one formulation, while the isometric hull \(\mathrm{iso}(S)\) is defined as a minimal-vertex isometric subgraph containing \(S\); its associated invariant is the isometric-hull–number \(\chi_i(G)\) [1710.02958]. A related partial-cube literature uses the notation \(h(G)\) for the minimum size of a hull set and characterizes hull sets through convex cut-partitions [1309.5724]. 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 \(A\) and a closure function \(\phi:2^A\to2^A\) satisfying extensivity, monotonicity, and idempotence. An equivalent description is \[ \phi(X)=\bigcap\{C\subseteq A:X\subseteq C,\;C=\phi(C)\}. \] A weaker pseudo-closure \(f\) satisfies \[ f(X\cup Y)=f\bigl(f(X)\cup f(Y)\bigr)\quad\forall X,Y\subseteq A, \] which still implies idempotence. Graph convex hull is an instance of such a closure operator [1710.02958]. For the Minimum Generator Set problem on a pseudo-closure \(f\), the paper on metric hulls gives Algorithm MinGen. If computing \(f(R)\) from \(f(R\setminus\{z\})\) takes \(O(c_f)\) time, the runtime is \(O\bigl(c_f\,|A|^2\,|\mathrm{Im}(f)|^2\bigr)\) in general, and \(O(c_f\,|A|\,|\mathrm{Im}(f)|)\) when \(f\) is size-increasing. Specializing to graphs, if all convex sets are given, then \(\chi_c(G)\) is computable in \(O(|V|\cdot|\mathrm{Im}(f)|)\) time [1710.02958]. 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 \(Q^d\). With a convex cut-partition \(\mathcal C\), a set \(H\subseteq V(G)\) 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 [1309.5724]. That perspective yields both positive and negative algorithmic results. Determining whether \(h(G)\le k\) for a partial cube \(G\) 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 [1309.5724]. The same work also gives a lattice-theoretic characterization. Fixing \(v\in V(G)\), the poset \(\mathcal L^v_G\) of convex subgraphs containing \(v\) is atomistic, and partial-cube-ness is characterized by \(\mathcal L^v_G\) being upper locally distributive with a Hasse diagram containing \(G\) isometrically [1309.5724]. 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 \(G\) and \(\overline G\) plus a perfect matching \(v_i\overline v_i\), one central lemma states that every simplicial vertex of \(G\) or of \(\overline G\) must be in any hull set of \(G\overline G\). Lower bounds then follow by counting forced simplicial vertices, and upper bounds are established by explicit hull-set constructions [1807.08295]. For trees, if \(T\) is a tree on \(n+1\) vertices and \(S_n\) is the star \(K_{1,n}\) with \(n\ge 3\), then \[ h(T\overline T)= \begin{cases} n+1,& \text{if }T=S_n,\\ 2,& \text{otherwise.} \end{cases} \] For disconnected \(G\) with \(k\ge 2\) components, of which at least two are nontrivial, the hull number of the complementary prism is \(k+1\). If \(G\) has exactly one nontrivial component \(G_1\) and \(t>0\) isolated vertices, then \(h(G\overline G)\ge t+2\), while the upper bounds depend on \(\mathrm{diam}(G_1)\) and \(\mathrm{diam}(\overline G_1)\). For connected cographs whose complement has \(k\) nontrivial components and \(t\) isolated vertices, the cases \(k=0\), \(k=1\), and \(k\ge 2\) are determined separately [1807.08295]. One notable consequence is a contrast with \(P_3\)-convexity. In \(P_3\)-convexity, Duarte et al. proved that if both \(G\) and \(\overline G\) are connected, then \(h_{P_3}(G\overline G)\le 5\). In geodetic convexity, by contrast, the hull number on complementary prisms cannot be limited: for each \(n\ge 2\), there exists connected \(G\) with connected \(\overline G\) and \(h(G\overline G)=n\) [1807.08295]. 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 \(k\le \log|A|\). Computing an isometric hull of a prescribed set \(S\) is NP-complete even for \(|S|=3\), and computing the isometric-hull–number \(\chi_i(G)\) is \(\Sigma^P_2\)-complete [1710.02958]. 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 \(C_G(S)\) of a node set \(S\), where \[ I(u,v)=\bigcup\{\text{vertices on shortest paths between }u\text{ and }v\}, \] and a set is closed if \(I(u,v)\subseteq X\) for all \(u,v\in X\). The exact closure may be computed by iterative BFS, with worst-case time \(O(n\cdot m)\) [2206.07350]. The heuristic replaces the original graph by a family of sampled almost-maximal outerplanar spanning subgraphs. For each sampled subgraph \(H_i\), one computes \(C_{H_i}(S)\) and then declares a vertex to belong to the approximate hull if it appears in at least \(t\) of the \(k\) sampled closures: \[ \widetilde C_G(S)=\Bigl\{v\in V\;\Big|\;\bigl|\{i:v\in C_{H_i}(S)\}\bigr|\ge t\Bigr\}. \] The outerplanar sampling stage runs in \(O(m)\) time, and the closure routine on each \(H_i\) exploits the block-bridge tree together with blockwise generators and BFS on biconnected blocks [2206.07350]. The total time complexity is \[ O(k\cdot m+k\cdot n f)=O\bigl(k\cdot(m+n f)\bigr), \] where \(f\) is the maximum number of interior faces per block. Since \(f\ll n\) in practice, this is described as roughly \(O(km)\), linear in the number of edges. The standard experimental choice is \(k=100\) and \(t=1\%\cdot k=1\) [2206.07350]. The empirical motivation is core-periphery decomposition based on convexity. On real-world networks from SNAP, with sizes up to \(4.8\) million nodes and \(117\) million edges, the exact core computation did not finish within \(50\) days for the largest graphs, whereas the heuristic completed approximate core-periphery decomposition in \(\le 5\) hours for the \(117\) million-edge graph. Jaccard similarity between exact and approximate cores ranged from \(0.82\) to \(0.99\), and with \(k=100\), \(t=1\%\) achieved \(\ge 0.9\) in \(9/15\) cases [2206.07350]. 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 \(Y\in\{0,1\}^{N\times N}\), \(Y_{ii}=0\), and a \(D\)-dimensional latent embedding constrained to a \((D-1)\)-polytope with \(K\) “pure” community archetypes. The global archetypes are collected in \[ A=[a_1^\top;\dots;a_K^\top]\in\mathbb R^{K\times D}, \] with global convex hull \[ \mathrm{conv}(A):=\Bigl\{\sum_{k=1}^K \alpha_k a_k:\alpha\in\Delta_{K-1}\Bigr\}, \] where \(\Delta_{K-1}=\{\alpha:\alpha_k\ge 0,\ \sum \alpha_k=1\}\). To ensure numerical stability and diversity of directions, \(A\) is parameterized by a boxed SVD, \[ A=U\,\mathrm{diag}(\sigma)\,V^\top, \] with \(U\in\mathbb R^{K\times K}\), \(V\in\mathbb R^{D\times K}\) having orthonormal columns and \(\sigma\in[\sigma_{\min},\sigma_{\max}]^K\) [2602.21342]. Each community \(k\) is refined by a local convex hull \[ B_k=\tilde W_k A\in\mathbb R^{K\times D}, \] whose \(K\) vertices are local “prototypes”. The matrix \(\tilde W_k\) contains \(K-1\) community-specific convex-weight rows \(w_{k,1},\dots,w_{k,K-1}\in\Delta_{K-1}\) and one anchor row \(e_k^\top\), so the last row of \(B_k\) is exactly \(a_k^\top\). The anchor-dominance constraint \[ (w_{k,r})_k\ge 1-\epsilon,\qquad 0<\epsilon<\tfrac12, \] ensures that \(\mathrm{conv}(B_k)\cap \mathrm{conv}(B_\ell)=\varnothing\) for \(k\ne \ell\) [2602.21342]. Each node is assigned to exactly one community \(c_i\in\{1,\dots,K\}\), encoded by a one-hot vector \(m_i\), and receives barycentric weights \(\omega_i\in\Delta_{K-1}\) over the local prototypes of its community. Its latent embedding is \[ z_i=\omega_i^\top B_{c_i}\in\mathbb R^D. \] The edge model is logistic: \[ \eta_{ij}=s\langle z_i,z_j\rangle + g_i + g_j,\qquad Y_{ij}\mid \{z,g,s\}\sim \mathrm{Bernoulli}(\sigma(\eta_{ij})), \] where \(g_i\sim\mathcal N(0,\tau_g^2)\) is a degree bias and \(s\sim\) Half-Normal\((0,\tau_s^2)\) is a global scale. The complete-data log-likelihood is \[ \ell(Y\mid Z,g,s)=\sum_{i