Closed Neighborhood Complex

• Closed neighborhood complex is an abstract simplicial complex defined by finite subsets of vertices that lie within the closed neighborhood (vertex and its adjacent vertices) of some vertex.
• It generalizes Lovász’s neighborhood complex by incorporating reflexivity, thereby revealing key topological properties such as contractibility and homotopy equivalences in standard graph families.
• The complex connects graph theory with applied topology, linking concepts like Čech complexes in data analysis and path homology, which broadens its utility in combinatorial and topological studies.

The closed neighborhood complex $\mathcal{N}[G]$ of a simple graph $G$ is the abstract simplicial complex with vertex set $V(G)$ whose simplices are all finite subsets of $V(G)$ contained in the closed neighborhood of some vertex. Formally, $$\mathcal{N}[G] = \{\sigma \subset V(G) : \#\sigma < \infty$$ and $\exists v \in V(G)$ with $\sigma \subset N_G[v]\}$, where $N_G[v]$ is the set consisting of $v$ together with all vertices adjacent to $v$. This construction generalizes Lovász’s neighborhood complex by incorporating reflexivity, and it provides a topological object closely linked with several classical and contemporary combinatorial, topological, and homological notions (Matsushita, 16 Nov 2025).

1. Formal Definitions and Basic Properties

Let $G$ denote an undirected simple (possibly infinite) graph with vertex set $V(G)$. For $v \in V(G)$, define the open and closed neighborhoods as: $$N_G(v) = \{\,w \mid (v,w) \in E(G)\}, \quad N_G[v] = N_G(v) \cup \{v\}.$$ The closed neighborhood complex $\mathcal{N}[G]$ is the abstract simplicial complex with ground set $V(G)$ such that

$$\mathcal{N}[G] = \left\{\,\sigma \subset V(G)\,\middle|\, \#\sigma < \infty\ \text{and}\ \exists v \in V(G): \sigma \subset N_G[v]\right\}.$$

Equivalently, if $G^\circ$ denotes the reflexive closure of $G$ (i.e., add a loop at each vertex), then $\mathcal{N}[G] = N(G^\circ)$, the usual neighborhood complex of $G^\circ$.

2. Illustrative Examples

Closed neighborhood complexes exhibit characteristic topology in basic graph families:

• Path $P_n$:
  • For $P_2$, $N_{P_2}[1]=N_{P_2}[2]=\{1,2\}$, yielding $\mathcal{N}[P_2]$ as a full 1-simplex (contractible).
  • For $P_3$, $N[1] = \{1,2\}, N[2]=\{1,2,3\}, N[3]=\{2,3\}$; $\mathcal{N}[P_3]$ consists of two maximal edges, contractible.
  • In general, $\mathcal{N}[P_n]$ is always contractible: either a cone on a central vertex (odd $n$) or a suspension of a smaller complex (even $n$).
• Cycle $C_n$:
  • $N_{C_n}[v]=\{v-1, v, v+1\}$ (indices mod $n$).
  • $\mathcal{N}[C_n]$ is a cycle of $n$ overlapping triangles, with

  $$\mathcal{N}[C_n] \simeq S^1 \quad (n \geq 4), \quad \pi_1(\mathcal{N}[C_n]) \cong \mathbb{Z}.$$

• Complete graph $K_n$:

  • Each closed neighborhood is $V(K_n)$, so $\mathcal{N}[K_n]$ is the full $(n-1)$-simplex (contractible).

These cases exemplify that closed neighborhood complexes can be contractible, homotopy equivalent to spheres, or possess other accessible topological structures depending on $G$ (Matsushita, 16 Nov 2025).

3. Relations to Double Covers and Independence Complexes

The closed neighborhood complex is closely linked via homotopy to the independence complex of the canonical double covering:

Let $K_2$ be the complete graph on two vertices and form the categorical product $K_2 \times G$ (the bipartite double cover). Consider also the complement graph $\overline{G}$. The fundamental result is: $$\Sigma\, \mathcal{N}[\overline{G}] \simeq I(K_2 \times G),$$ where $I(H)$ is the independence complex of $H$ and $\Sigma$ denotes suspension. In the special case when $G$ is bipartite, $K_2 \times G \cong G \sqcup G$, yielding $$\Sigma\, \mathcal{N}[\overline{G}] \simeq I(G)*I(G)$$, where $*$ denotes the simplicial join. If $\overline{G}$ is bipartite (i.e., $V(G)=C_1 \sqcup C_2$ is a disjoint union of two cliques), then

$\Sigma\, \mathcal{N}[G] \simeq X(G) * X(G),$

with $X(G)$ the clique complex of $G$.

The core argument utilizes a construction associating to every $y \in V(G)$ a set $\varphi(y) = N_{\overline{G}}[y]$ and considers the independence complex of a bipartite graph $H$ built over these sets, which is isomorphic to $K_2 \times G$. Homotopy equivalence follows from a nerve lemma argument (Matsushita, 16 Nov 2025).

4. Connection to Neighborhood Hypergraphs and Duality

For a finite graph $G$, define the (open) neighborhood hypergraph $\mathcal{N}_G$ with vertex set $V(G)$ and edges $E(\mathcal{N}_G)=\{N_G(v) : v \in V(G)\}$. Its independence complex $I(\mathcal{N}_G)$ consists of all finite subsets of $V(G)$ not including any hyperedge in full.

Notably,

$\mathcal{N}[\overline{G}] = I(\mathcal{N}_G)^\vee$

where $I(\mathcal{N}_G)^\vee$ is the combinatorial Alexander dual. The facets of $\mathcal{N}[\overline{G}]$ correspond precisely to the minimal hyperedges $N_G(v)$. This duality provides a combinatorial link between neighborhood systems and dominating set structures in $G$ (Matsushita, 16 Nov 2025).

5. Fundamental Group and Path Homology

A central result demonstrates that the fundamental group of the closed neighborhood complex coincides with the Grigor’yan–Lin–Muranov–Yau (GLMY) route-homotopy fundamental group associated with path homology.

Generalizing, for $k \geq 1$ define: $$N_G^1[v] = N_G[v], \quad N_G^{k+1}[v] = \bigcup_{w \in N_G^k[v]} N_G[w].$$ The closed $k$-neighborhood complex $N^k[G]$ comprises finite $\sigma \subset V(G)$ contained in some $N_G^k[v]$. The corresponding closed $2k$-fundamental group $\pi_1^{2k}[G,v]$ is defined via combinatorial path-homotopy. When $k=1$, this group matches the GLMY fundamental group $\pi_1^{\text{GLMY}}(G,v)$. For every $k \geq 1$,

$\pi_1(N^k[G], v) \cong \pi_1^{2k}[G, v].$

The proof identifies edge-paths in $N^k[G]$ with routes in $G$ of length a multiple of $2k$, showing a tight correspondence between topological and path-homotopical invariants in graphs. A plausible implication is that this isomorphism extends the interpretive power of simplicial models for graph homotopy (Matsushita, 16 Nov 2025).

6. Additional Properties and Applications

Closed neighborhood complexes interface with diverse areas:

• If $G$ arises from a metric space $(X, d)$ with edges between points within radius $r$ (the “radius-$r$” graph $G[X, r]$), then $\mathcal{N}[G[X, r]]$ recovers the Čech complex $\check{C}_r(X)$. This links persistence in topological data analysis to combinatorial properties of graphs.
• The described relationship with $K_2\times G$ enables applications to independence complexes of Borsuk graphs and related graphs, leveraging results on Čech and Vietoris–Rips complexes.
• Via Alexander duality, minimal dominating sets in $G$ correspond to collapsing regions in the closed neighborhood complex of $\overline{G}$.
• The identification of the fundamental group with the GLMY group provides new perspectives on the multiplicativity of fundamental groups under graph products and enriches the study of path homology within topological combinatorics.

In summary, the closed neighborhood complex $\mathcal{N}[G]$ provides a nexus between classical topological combinatorics (neighborhood complexes), applied topology (Čech complexes), independence complex theory, and recent advances in path homology. Its homotopy, combinatorial, and group-theoretic invariants encode and unify several critical properties of the underlying graph and related constructions (Matsushita, 16 Nov 2025).