diff --git a/src/finite_function/arrow.rs b/src/finite_function/arrow.rs
index cb3467d..359ec67 100644
--- a/src/finite_function/arrow.rs
+++ b/src/finite_function/arrow.rs
@@ -216,6 +216,21 @@ impl<K: ArrayKind> FiniteFunction<K> {
     }
 }
 
+impl<K: ArrayKind> FiniteFunction<K>
+where
+    K::Type<K::I>: NaturalArray<K>,
+{
+    /// Check if this finite function is injective.
+    pub fn is_injective(&self) -> bool {
+        if self.source().is_zero() {
+            return true;
+        }
+
+        let counts = self.table.bincount(self.target.clone());
+        counts.max().map_or(true, |m| m <= K::I::one())
+    }
+}
+
 /// Compute the universal map for a coequalizer `q : B → Q` and arrow `f : B → T`, generalised to
 /// the case where `T` is an arbitrary set (i.e., `f` is an array of `T`)
 pub fn coequalizer_universal<K: ArrayKind, T>(
diff --git a/src/strict/graph.rs b/src/strict/graph.rs
index 3af6ae3..2cf4638 100644
--- a/src/strict/graph.rs
+++ b/src/strict/graph.rs
@@ -68,7 +68,21 @@ pub(crate) fn node_adjacency<K: ArrayKind, O, A>(
 where
     K::Type<K::I>: NaturalArray<K>,
 {
-    converse(&h.s).flatmap(&h.t)
+    node_adjacency_from_incidence(&h.s, &h.t)
+}
+
+/// Return the node-level adjacency map from incidence data.
+///
+/// If `W` is the finite set of nodes, then the result is an indexed coproduct
+/// `adjacency : W → W*`, where `adjacency(w)` is all nodes reachable in a single step from `w`.
+pub(crate) fn node_adjacency_from_incidence<K: ArrayKind>(
+    s: &IndexedCoproduct<K, FiniteFunction<K>>,
+    t: &IndexedCoproduct<K, FiniteFunction<K>>,
+) -> IndexedCoproduct<K, FiniteFunction<K>>
+where
+    K::Type<K::I>: NaturalArray<K>,
+{
+    converse(s).flatmap(t)
 }
 
 /// A kahn-ish algorithm for topological sorting of an adjacency relation, encoded as an
diff --git a/src/strict/hypergraph/arrow.rs b/src/strict/hypergraph/arrow.rs
index 9f8a676..2fcffbd 100644
--- a/src/strict/hypergraph/arrow.rs
+++ b/src/strict/hypergraph/arrow.rs
@@ -1,13 +1,51 @@
 use super::object::Hypergraph;
-use crate::array::{Array, ArrayKind};
+use crate::array::{Array, ArrayKind, NaturalArray};
 use crate::finite_function::FiniteFunction;
+use crate::indexed_coproduct::IndexedCoproduct;
+use crate::strict::graph::{
+    node_adjacency, node_adjacency_from_incidence, sparse_relative_indegree,
+};
 
 use core::fmt::Debug;
+use num_traits::{One, Zero};
+
+fn successors<K: ArrayKind>(
+    adjacency: &IndexedCoproduct<K, FiniteFunction<K>>,
+    frontier: &K::Index,
+) -> K::Index
+where
+    K::Type<K::I>: NaturalArray<K>,
+{
+    if frontier.is_empty() {
+        return K::Index::empty();
+    }
+
+    let f = FiniteFunction::new(frontier.clone(), adjacency.len()).unwrap();
+    let (g, _) = sparse_relative_indegree(adjacency, &f);
+    g.table
+}
+
+fn filter_unvisited<K: ArrayKind>(visited: &K::Index, candidates: &K::Index) -> K::Index
+where
+    K::Type<K::I>: NaturalArray<K>,
+{
+    if candidates.is_empty() {
+        return K::Index::empty();
+    }
+
+    let visited_on_candidates = visited.gather(candidates.get_range(..));
+    let unvisited_ix = visited_on_candidates.zero();
+    candidates.gather(unvisited_ix.get_range(..))
+}
 
 #[derive(Debug)]
 pub enum InvalidHypergraphArrow {
+    TypeMismatchW,
+    TypeMismatchX,
     NotNaturalW,
     NotNaturalX,
+    NotNaturalS,
+    NotNaturalT,
 }
 
 pub struct HypergraphArrow<K: ArrayKind, O, A> {
@@ -35,7 +73,10 @@ where
         target: Hypergraph<K, O, A>,
         w: FiniteFunction<K>,
         x: FiniteFunction<K>,
-    ) -> Result<Self, InvalidHypergraphArrow> {
+    ) -> Result<Self, InvalidHypergraphArrow>
+    where
+        K::Type<K::I>: NaturalArray<K>,
+    {
         HypergraphArrow {
             source,
             target,
@@ -46,7 +87,10 @@ where
     }
 
     /// Check validity of a HypergraphArrow.
-    pub fn validate(self) -> Result<Self, InvalidHypergraphArrow> {
+    pub fn validate(self) -> Result<Self, InvalidHypergraphArrow>
+    where
+        K::Type<K::I>: NaturalArray<K>,
+    {
         // for self : g → h
         let g = &self.source;
         let h = &self.target;
@@ -55,22 +99,136 @@ where
         // wire labels, operation labels, and operation types should be preserved under the natural
         // transformations w and x.
 
-        // Check naturality of w
-        if g.w != (&self.w >> &h.w).unwrap() {
+        // Check naturality on node labels:
+        // g.w = w ; h.w
+        let composed_w = (&self.w >> &h.w).ok_or(InvalidHypergraphArrow::TypeMismatchW)?;
+        if g.w != composed_w {
             return Err(InvalidHypergraphArrow::NotNaturalW);
         }
 
-        // Check naturality of x
-        if g.x != (&self.x >> &h.x).unwrap() {
+        // Check naturality on operation labels:
+        // g.x = x ; h.x
+        let composed_x = (&self.x >> &h.x).ok_or(InvalidHypergraphArrow::TypeMismatchX)?;
+        if g.x != composed_x {
             return Err(InvalidHypergraphArrow::NotNaturalX);
         }
 
+        // Check naturality of incidence (sources):
+        // g.s; w = h.s reindexed along x.
+        let s_lhs =
+            g.s.map_values(&self.w)
+                .ok_or(InvalidHypergraphArrow::NotNaturalS)?;
+        let s_rhs =
+            h.s.map_indexes(&self.x)
+                .ok_or(InvalidHypergraphArrow::NotNaturalS)?;
+        if s_lhs != s_rhs {
+            return Err(InvalidHypergraphArrow::NotNaturalS);
+        }
+
+        // Check naturality of incidence (targets).
+        // g.t; w = h.t reindexed along x.
+        let t_lhs =
+            g.t.map_values(&self.w)
+                .ok_or(InvalidHypergraphArrow::NotNaturalT)?;
+        let t_rhs =
+            h.t.map_indexes(&self.x)
+                .ok_or(InvalidHypergraphArrow::NotNaturalT)?;
+        if t_lhs != t_rhs {
+            return Err(InvalidHypergraphArrow::NotNaturalT);
+        }
+
         Ok(self)
+    }
+
+    /// True when this arrow is injective on both nodes and edges.
+    pub fn is_monomorphism(&self) -> bool
+    where
+        K::Type<K::I>: NaturalArray<K>,
+    {
+        self.w.is_injective() && self.x.is_injective()
+    }
+
+    /// Check convexity of a subgraph `H → G`.
+    ///
+    pub fn is_convex_subgraph(&self) -> bool
+    where
+        K::Type<K::I>: NaturalArray<K>,
+    {
+        if !self.is_monomorphism() {
+            return false;
+        }
+
+        let g = &self.target;
+        let n_nodes = g.w.len();
+        let n_edges = g.x.len();
+
+        // Build the complement set of edges (those not in the image of x).
+        let mut edge_mask = K::Index::fill(K::I::zero(), n_edges.clone());
+        edge_mask.scatter_assign_constant(&self.x.table, K::I::one());
+        let outside_edge_ix = edge_mask.zero();
+        let outside_edges = FiniteFunction::new(outside_edge_ix, n_edges).unwrap();
+
+        // Adjacency restricted to edges in the subobject (inside edges).
+        let s_in = g.s.map_indexes(&self.x).unwrap();
+        let t_in = g.t.map_indexes(&self.x).unwrap();
+        let adj_in = node_adjacency_from_incidence(&s_in, &t_in);
+
+        // Adjacency restricted to edges outside the subobject.
+        let s_out = g.s.map_indexes(&outside_edges).unwrap();
+        let t_out = g.t.map_indexes(&outside_edges).unwrap();
+        let adj_out = node_adjacency_from_incidence(&s_out, &t_out);
+
+        // Full adjacency (used after we've already left the subobject).
+        let adj_all = node_adjacency(g);
+
+        // Two-layer reachability:
+        // - layer 0: paths that have used only inside edges
+        // - layer 1: paths that have used at least one outside edge
+        //
+        // Convexity fails iff some selected node is reachable in layer 1.
+        let mut visited0 = K::Index::fill(K::I::zero(), n_nodes.clone());
+        let mut visited1 = K::Index::fill(K::I::zero(), n_nodes.clone());
+        let mut frontier0 = self.w.table.clone();
+        let mut frontier1 = K::Index::empty();
+
+        // Seed search from the selected nodes.
+        visited0.scatter_assign_constant(&frontier0, K::I::one());
+
+        while !frontier0.is_empty() || !frontier1.is_empty() {
+            // From layer 0, inside edges stay in layer 0.
+            let next0: K::Index = successors::<K>(&adj_in, &frontier0);
+            // From layer 0, outside edges move to layer 1.
+            let next1_from0 = successors::<K>(&adj_out, &frontier0);
+            // From layer 1, any edge keeps you in layer 1.
+            let next1_from1 = successors::<K>(&adj_all, &frontier1);
+
+            // Avoid revisiting nodes we've already seen in the same layer.
+            let next0: K::Index = filter_unvisited::<K>(&visited0, &next0);
+
+            let next1: K::Index = {
+                let merged = next1_from0.concatenate(&next1_from1);
+                if merged.is_empty() {
+                    K::Index::empty()
+                } else {
+                    let (unique, _) = merged.sparse_bincount();
+                    filter_unvisited::<K>(&visited1, &unique)
+                }
+            };
+
+            if next0.is_empty() && next1.is_empty() {
+                break;
+            }
+
+            // Mark and advance frontiers.
+            visited0.scatter_assign_constant(&next0, K::I::one());
+            visited1.scatter_assign_constant(&next1, K::I::one());
+            frontier0 = next0;
+            frontier1 = next1;
+        }
 
-        // TODO: add this check.
-        // Types of operations are also preserved under w and x.
-        //assert_eq!(g.s.values >> g.w, h.s.indexed_values(f.x) >> h.w);
-        //assert_eq!(g.t.values >> g.w, h.t.indexed_values(f.x) >> h.w);
+        // If any selected node is reachable in layer 1, it's not convex.
+        let reached_selected = visited1.gather(self.w.table.get_range(..));
+        !reached_selected.max().map_or(false, |m| m >= K::I::one())
     }
 }
 
diff --git a/tests/graph/mod.rs b/tests/graph/mod.rs
deleted file mode 100644
index 8b13789..0000000
--- a/tests/graph/mod.rs
+++ /dev/null
@@ -1 +0,0 @@
-
diff --git a/tests/hypergraph/mod.rs b/tests/hypergraph/mod.rs
index 31e4727..6fb56c2 100644
--- a/tests/hypergraph/mod.rs
+++ b/tests/hypergraph/mod.rs
@@ -1,3 +1,5 @@
 pub mod equality;
 pub mod strategy;
+pub mod test_arrow;
 pub mod test_hypergraph;
+pub mod test_monogamous;
diff --git a/tests/hypergraph/strategy.rs b/tests/hypergraph/strategy.rs
index 8022f2a..a198b24 100644
--- a/tests/hypergraph/strategy.rs
+++ b/tests/hypergraph/strategy.rs
@@ -174,15 +174,12 @@ pub fn arb_inclusion<
     labels: Labels<O, A>,
     g: Hypergraph<VecKind, O, A>,
 ) -> BoxedStrategy<HypergraphArrow<VecKind, O, A>> {
-    // We have an arbitrary hypergraph g, and some arbitrary label arrays.
-    let h_labels = Labels {
-        w: g.w.clone() + labels.w,
-        x: g.x.clone() + labels.x,
-    };
-    arb_hypergraph(h_labels)
-        .prop_flat_map(move |h| {
-            let w = FiniteFunction::inj0(g.w.len(), h.w.len() - g.w.len());
-            let x = FiniteFunction::inj0(g.x.len(), h.x.len() - g.x.len());
+    // Build target as a coproduct g + k so inj0 is incidence-natural by construction.
+    arb_hypergraph(labels)
+        .prop_flat_map(move |k| {
+            let h = g.coproduct(&k);
+            let w = FiniteFunction::inj0(g.w.len(), k.w.len());
+            let x = FiniteFunction::inj0(g.x.len(), k.x.len());
 
             Just(HypergraphArrow::new(g.clone(), h, w, x).expect("valid HypergraphArrow"))
         })
diff --git a/tests/hypergraph/test_arrow.rs b/tests/hypergraph/test_arrow.rs
new file mode 100644
index 0000000..d19557b
--- /dev/null
+++ b/tests/hypergraph/test_arrow.rs
@@ -0,0 +1,92 @@
+use open_hypergraphs::array::vec::*;
+use open_hypergraphs::finite_function::FiniteFunction;
+use open_hypergraphs::lax::{Hyperedge, Hypergraph as LaxHypergraph, NodeId};
+use open_hypergraphs::strict::hypergraph::arrow::{HypergraphArrow, InvalidHypergraphArrow};
+use open_hypergraphs::strict::hypergraph::Hypergraph;
+
+type Obj = usize;
+type Arr = usize;
+
+fn build_hypergraph(
+    node_labels: Vec<usize>,
+    edges: &[(usize, Vec<usize>, Vec<usize>)],
+) -> Hypergraph<VecKind, Obj, Arr> {
+    let mut h = LaxHypergraph::empty();
+    h.nodes = node_labels;
+    for (label, sources, targets) in edges {
+        let edge = Hyperedge {
+            sources: sources.iter().map(|&i| NodeId(i)).collect(),
+            targets: targets.iter().map(|&i| NodeId(i)).collect(),
+        };
+        h.new_edge(*label, edge);
+    }
+    h.to_hypergraph()
+}
+
+fn ff(table: Vec<usize>, target: usize) -> FiniteFunction<VecKind> {
+    FiniteFunction::new(VecArray(table), target).unwrap()
+}
+
+#[test]
+fn test_validate_rejects_non_natural_w() {
+    let source = build_hypergraph(vec![0], &[]);
+    let target = build_hypergraph(vec![1], &[]);
+    let w = ff(vec![0], 1);
+    let x = ff(vec![], 0);
+    let err = HypergraphArrow::new(source, target, w, x).unwrap_err();
+    assert!(matches!(err, InvalidHypergraphArrow::NotNaturalW));
+}
+
+#[test]
+fn test_validate_rejects_non_natural_x() {
+    let source = build_hypergraph(vec![0, 0], &[(0, vec![0], vec![1])]);
+    let target = build_hypergraph(vec![0, 0], &[(1, vec![0], vec![1])]);
+    let w = ff(vec![0, 1], 2);
+    let x = ff(vec![0], 1);
+    let err = HypergraphArrow::new(source, target, w, x).unwrap_err();
+    assert!(matches!(err, InvalidHypergraphArrow::NotNaturalX));
+}
+
+#[test]
+fn test_validate_rejects_undefined_w_composition() {
+    let source = build_hypergraph(vec![0], &[]);
+    let target = build_hypergraph(vec![0], &[]);
+    // valid finite function, but codomain size (2) does not match h.w.len() (1),
+    // so w ; h.w is undefined and validate should return TypeMismatchW.
+    let w = ff(vec![0], 2);
+    let x = ff(vec![], 0);
+    let err = HypergraphArrow::new(source, target, w, x).unwrap_err();
+    assert!(matches!(err, InvalidHypergraphArrow::TypeMismatchW));
+}
+
+#[test]
+fn test_validate_accepts_incidence_natural_arrow() {
+    let source = build_hypergraph(vec![0, 0], &[(0, vec![0], vec![1])]);
+    let target = build_hypergraph(vec![0, 0], &[(0, vec![1], vec![1])]);
+
+    let w = ff(vec![1, 1], 2);
+    let x = ff(vec![0], 1);
+    assert!(HypergraphArrow::new(source, target, w, x).is_ok());
+}
+
+#[test]
+fn test_validate_rejects_non_natural_s() {
+    let source = build_hypergraph(vec![0, 0], &[(0, vec![0], vec![1])]);
+    let target = build_hypergraph(vec![0, 0], &[(0, vec![0], vec![1])]);
+
+    let w = ff(vec![1, 1], 2);
+    let x = ff(vec![0], 1);
+    let err = HypergraphArrow::new(source, target, w, x).unwrap_err();
+    assert!(matches!(err, InvalidHypergraphArrow::NotNaturalS));
+}
+
+#[test]
+fn test_validate_rejects_non_natural_t() {
+    let source = build_hypergraph(vec![0, 0], &[(0, vec![0], vec![1])]);
+    let target = build_hypergraph(vec![0, 0], &[(0, vec![0], vec![1])]);
+
+    let w = ff(vec![0, 0], 2);
+    let x = ff(vec![0], 1);
+    let err = HypergraphArrow::new(source, target, w, x).unwrap_err();
+    assert!(matches!(err, InvalidHypergraphArrow::NotNaturalT));
+}
diff --git a/tests/hypergraph/test_hypergraph.rs b/tests/hypergraph/test_hypergraph.rs
index 1ba1b41..83e3bab 100644
--- a/tests/hypergraph/test_hypergraph.rs
+++ b/tests/hypergraph/test_hypergraph.rs
@@ -125,7 +125,6 @@ fn test_is_acyclic_false_self_loop() {
     let h = build_hypergraph(1, &[(vec![0], vec![0])]);
     assert!(!h.is_acyclic());
 }
-
 #[test]
 fn test_in_out_degree_counts_multiplicity() {
     let sources = IndexedCoproduct::from_semifinite(
diff --git a/tests/hypergraph/test_monogamous.rs b/tests/hypergraph/test_monogamous.rs
new file mode 100644
index 0000000..270803f
--- /dev/null
+++ b/tests/hypergraph/test_monogamous.rs
@@ -0,0 +1,78 @@
+use open_hypergraphs::array::vec::*;
+use open_hypergraphs::finite_function::FiniteFunction;
+use open_hypergraphs::lax::{Hyperedge, Hypergraph as LaxHypergraph, NodeId};
+use open_hypergraphs::strict::hypergraph::arrow::HypergraphArrow;
+use open_hypergraphs::strict::hypergraph::Hypergraph;
+
+type Obj = usize;
+type Arr = usize;
+
+fn build_hypergraph(
+    node_count: usize,
+    edges: &[(Vec<usize>, Vec<usize>)],
+) -> Hypergraph<VecKind, Obj, Arr> {
+    let mut h = LaxHypergraph::empty();
+    h.nodes = vec![0; node_count];
+    for (sources, targets) in edges {
+        let edge = Hyperedge {
+            sources: sources.iter().map(|&i| NodeId(i)).collect(),
+            targets: targets.iter().map(|&i| NodeId(i)).collect(),
+        };
+        h.new_edge(0, edge);
+    }
+    h.to_hypergraph()
+}
+
+fn ff(table: Vec<usize>, target: usize) -> FiniteFunction<VecKind> {
+    FiniteFunction::new(VecArray(table), target).unwrap()
+}
+
+#[test]
+fn test_is_monomorphism_false_for_non_injective_w() {
+    let h = build_hypergraph(2, &[]);
+    let g = build_hypergraph(1, &[]);
+    let w = ff(vec![0, 0], 1);
+    let x = ff(vec![], 0);
+    let arrow = HypergraphArrow::new(h, g, w, x).unwrap();
+    assert!(!arrow.is_monomorphism());
+}
+
+#[test]
+fn test_convex_subgraph_false_with_shortcut() {
+    let g = build_hypergraph(
+        3,
+        &[(vec![0], vec![1]), (vec![1], vec![2]), (vec![0], vec![2])],
+    );
+    let h = build_hypergraph(3, &[(vec![0], vec![1]), (vec![1], vec![2])]);
+
+    let w = ff(vec![0, 1, 2], 3);
+    let x = ff(vec![0, 1], 3);
+    let arrow = HypergraphArrow::new(h, g, w, x).unwrap();
+    assert!(!arrow.is_convex_subgraph());
+}
+
+#[test]
+fn test_convex_subgraph_true_identity() {
+    let g = build_hypergraph(
+        3,
+        &[(vec![0], vec![1]), (vec![1], vec![2]), (vec![0], vec![2])],
+    );
+    let w = ff(vec![0, 1, 2], 3);
+    let x = ff(vec![0, 1, 2], 3);
+    let arrow = HypergraphArrow::new(g.clone(), g, w, x).unwrap();
+    assert!(arrow.is_convex_subgraph());
+}
+
+#[test]
+fn test_convex_subgraph_true_small() {
+    let g = build_hypergraph(
+        3,
+        &[(vec![0], vec![1]), (vec![1], vec![2]), (vec![0], vec![2])],
+    );
+    let h = build_hypergraph(2, &[(vec![0], vec![1])]);
+
+    let w = ff(vec![0, 1], 3);
+    let x = ff(vec![0], 3);
+    let arrow = HypergraphArrow::new(h, g, w, x).unwrap();
+    assert!(arrow.is_convex_subgraph());
+}
diff --git a/tests/lib.rs b/tests/lib.rs
index 555f164..a38ebda 100644
--- a/tests/lib.rs
+++ b/tests/lib.rs
@@ -13,6 +13,5 @@ pub mod hypergraph;
 pub mod open_hypergraph;
 
 pub mod eval;
-pub mod graph;
 pub mod lax;
 pub mod layer;