Closed Neighborhoods in Graphs

• Closed neighborhoods in graphs are defined as a vertex along with all its adjacent vertices, serving as fundamental units in domination, covering, and reconstruction studies.
• They underpin practical applications such as perfect codes, efficient domination, and algebraic representations through square-free monomial ideals and simplicial complexes.
• These structures lead to efficient algorithms and sharp extremal bounds in conflict-free colorings, reconstruction problems, and isolation techniques in complex networks.

A closed neighborhood in a graph is the set of all vertices adjacent to a given vertex, together with the vertex itself. Formally, for a graph $G=(V,E)$ and $v\in V$, the closed neighborhood is $N_G[v]=\{v\} \cup \{u \in V : uv \in E\}$. Closed neighborhoods serve as foundational building blocks in combinatorial, algebraic, and topological graph theory, underpinning topics such as domination, reconstruction, hypergraph coverings, monomial ideals, conflict-free colorings, and simplicial complexes.

1. Structural Roles and Definition of Closed Neighborhoods

The closed neighborhood of a vertex, and by extension union systems $N_G[S]=\bigcup_{v \in S} N_G[v]$, play central roles in domination theory, graph partitioning, and algebraic graph invariants. The concept is dual to the open neighborhood $N_G(v)$, but closed neighborhoods are uniquely suited for encoding covering properties due to their inclusion of the center vertex.

Throughout the literature, closed neighborhoods are deployed in various frameworks:

• Domination and Codes: Efficient closed dominating sets (perfect codes) are sets $P$ whose closed neighborhoods partition $V(G)$, i.e., $\bigcup_{v \in P} N[v]=V$ and $N[x]\cap N[y]=\emptyset$ for $x\neq y$. This structure enforces strict distance constraints among code vertices and corresponds to minimum-size dominating sets with disjoint closed neighborhoods (Klavzar et al., 2015).
• Isolation Numbers: Closed neighborhoods support the notion of $\mathcal{F}$-isolation—removal of all closed neighborhoods of a set $D$ eliminates all graphs in a forbidden family $\mathcal{F}$ from the surviving induced subgraph. The minimal cardinality of such sets, the $\mathcal{F}$-isolation number, leads to sharp extremal bounds and inductive proofs reliant on the coverage power of closed neighborhoods (Borg, 2021).
• Neighborhood Coverings: A neighborhood cover set comprises vertices whose closed neighborhoods cover all vertices and edges in $G$, leading to the parameter $\rho_n(G)$ and, dually, maximal sets of pairwise neighborhood-independent elements, measured by $\alpha_n(G)$ (Durán et al., 2016).

2. Algebraic Structures: Closed Neighborhood Ideals

Closed neighborhoods provide natural set systems for constructing square-free monomial ideals in polynomial rings $R = k[x_1, \ldots, x_n]$, with each generator corresponding to the monomial indexed by the closed neighborhood of a vertex: $$I_c(G) = (x_{N[v_1]}, x_{N[v_2]}, \ldots, x_{N[v_n]})$$ where $x_{N[v_i]} = \prod_{u \in N[v_i]} x_u$ (Nasernejad et al., 2021, Hien et al., 11 Jul 2025). Such ideals are instrumental in combinatorial commutative algebra, linking domination theory to Alexander duality and revealing deep connections between minimal vertex covers, domination ideals, and the structure of the generating set of closed neighborhoods.

Key results include:

• Normal torsion-freeness for trees: If $G$ is a tree, then all powers $I_c(G)^k$ have the same associated primes as $I_c(G)$, i.e., the ideal is normally torsion-free. This property fails for cycles, where additional embedded primes can appear in higher powers, though strong persistence still holds—i.e., $(I^{k+1}:I) = I^k$ for all $k$ (Nasernejad et al., 2021).
• Characterization via diameter and criticality: For the maximal homogeneous ideal $\mathfrak{m}$ to appear as an associated prime of $I_c(G)^2$, the graph must have diameter at most 6, with sharp characterizations for the diameter-2 and diameter-$d$ cases in terms of vertex-criticality and hypergraph covering criteria (Hien et al., 11 Jul 2025).

3. Extremal and Covering Properties

Closed neighborhoods underpin domination, covering, and independence parameters:

• Efficient closed domination (ECD): Existence of a partition of $V$ by closed neighborhoods corresponds to ECD sets, relevant for perfect codes. Deciding the existence of such sets is NP-complete, even for various restricted classes (Klavzar et al., 2015).
• Neighborhood-perfect graphs: These are graphs for which the minimum size of a closed-neighborhood cover ($\rho_n$) equals the maximum size of a neighborhood-independent set ($\alpha_n$) for all induced subgraphs. In several hereditary families—P$_4$-tidy graphs, tree-cographs—neighborhood-perfectness is characterized by forbidden induced subgraphs (e.g., $3 K_2$, 3-sun, $C_5$), and both recognition and optimization are achievable in linear time via modular decomposition (Durán et al., 2016).

4. Conflict-Free and Related Colorings on Closed Neighborhood Systems

Closed neighborhoods are central to the conflict-free coloring paradigm, where the goal is to color vertices such that in each closed neighborhood, some color appears exactly once:

• Conflict-free chromatic number ($\chi_{CN}(G)$): Asymptotically, for graphs of maximum degree $\Delta$, it is known that $\chi_{CN}(G) = \Theta(\log^2 \Delta)$ (Bhyravarapu et al., 2020). In planar graphs, three colors are always sufficient and sometimes necessary for conflict-free closed neighborhood colorings (Abel et al., 2017).
• Algorithmic resolutions: Efficient (polynomial-time or randomized polynomial-time) constructions exist for bounded-degree graphs, using inductive elimination schemes and hypergraph colorings based on the neighborhoods (Bhyravarapu et al., 2020).
• Further extensions: For claw-free and $K_{1,k}$-free graphs, $\chi_{CF}^{CN}(G)=O(k^2 \log \Delta)$ (Bhyravarapu et al., 2021). In planar and outerplanar graphs, tight bounds are established for both proper and improper variants of conflict-free and unique-maximum colorings with respect to closed neighborhoods (Fabrici et al., 2022).

5. Closed Neighborhood Complexes and Topological Invariants

The closed neighborhood complex $\mathcal{N}[G]$ is the abstract simplicial complex with faces comprising all finite subsets of $V(G)$ contained in some closed neighborhood. This construction enables the study of topological invariants:

• Relation to independence complexes: The suspension of $\mathcal{N}[\overline{G}]$ is homotopy equivalent to the independence complex of the canonical double cover $K_2 \times G$ (Matsushita, 16 Nov 2025).
• Alexander duality: For finite graphs, $\mathcal{N}[\overline{G}]$ is the combinatorial Alexander dual of the independence complex of the neighborhood hypergraph (Matsushita, 16 Nov 2025).
• Path homology: The fundamental group of $\mathcal{N}[G]$ is isomorphic to the path-homology fundamental group $\pi_1^{2}[G,v]$, providing a direct bridge between combinatorial and topological invariants (Matsushita, 16 Nov 2025).

Table 1 below highlights the interconnections between combinatorial and algebraic/topological invariants via closed neighborhoods.

Context	Closed Neighborhood Role	Key Result/Paper
Domination/Perfect codes	Partitions, minimum dominating sets	(Klavzar et al., 2015)
Algebraic invariants	Generators of square-free monomial ideals	(Nasernejad et al., 2021, Hien et al., 11 Jul 2025)
Conflict-free coloring	Hypergraph edges; unique color in each neighborhood	(Abel et al., 2017, Bhyravarapu et al., 2020)
Neighborhood-perfectness	Covers and independence, modular decomposition	(Durán et al., 2016)
Simplicial complexes	Simplices, Alexander dual of independence complexes	(Matsushita, 16 Nov 2025)

6. Reconstruction and Digital Convexity

Closed neighborhoods dictate reconstructibility phenomena:

• C$_4$-free graph reconstruction: Every C$_4$-free graph (and all girth ≥5) is uniquely determined up to isomorphism by its set (not just multiset) of closed neighborhoods. This extends to digital convexity, where the family of digitally convex vertex sets coincides with those reconstructible from closed neighborhoods (Borgwardt et al., 24 Oct 2025). The process is algorithmic and polynomial-time for C$_4$-free graphs.

7. Isolation, Deletion, and Extremal Techniques

Isolation numbers are minimization parameters for sets whose closed neighborhood deletions eliminate all copies of graphs from a forbidden family:

• Sharp bounds: For various forbidden families (graphs with ≥k edges or all cycles), there exist sharp upper bounds on the minimal size of closed-neighborhood deletion sets, typically of the form $\lfloor n/k' \rfloor$ with extremal equality cases specified by block constructions (Borg, 2021).
• Proof methodology: The closed-neighborhood deletion principle enables efficient inductive proofs and explains extremality via decomposition into special subgraphs (“blocks”).

• "Isolation of connected graphs" (Borg, 2021)
• "Closed neighborhood complexes of graphs" (Matsushita, 16 Nov 2025)
• "Conflict-free coloring on closed neighborhoods of bounded degree graphs" (Bhyravarapu et al., 2020)
• "Conflict-Free Coloring of Planar Graphs" (Abel et al., 2017)
• "Neighborhood covering and independence on two superclasses of cographs" (Durán et al., 2016)
• "Graphs that are simultaneously efficient open domination and efficient closed domination graphs" (Klavzar et al., 2015)
• "Dominating ideals and closed neighborhood ideals of graphs" (Nasernejad et al., 2021)
• "Associated primes of the second power of closed neighborhood ideals of graphs" (Hien et al., 11 Jul 2025)
• "Reconstruction of C_4-free graphs from the set of closed neighborhoods and digital convexity" (Borgwardt et al., 24 Oct 2025)
• "Proper conflict-free and unique-maximum colorings of planar graphs with respect to neighborhoods" (Fabrici et al., 2022)
• "Conflict-free coloring on open neighborhoods of claw-free graphs" (Bhyravarapu et al., 2021)

Closed neighborhoods thus function as a versatile, deeply structural concept in graph theory, tying together combinatorial, algebraic, algorithmic, and topological perspectives. Their study yields sharp extremal theorems, efficient algorithms in specific classes, bridge results between algebra and topology, and informs the design of robust graph invariants and coverings.