Optimal Contamination Sets in Grid Networks

• Optimal contamination sets are minimal seed nodes on grid networks that trigger complete spread under specified local propagation rules.
• The methodology includes deriving recurrence relations and leveraging combinatorial structures like forbidden-pattern words and Schröder numbers.
• The findings inform practical strategies for resource allocation in monitoring systems such as power grids and distributed sensor networks.

An optimal contamination set is a minimum cardinality subset of nodes (or vertices) on a grid graph (or more generally, a network), which, when declared as initially "contaminated" (or "active"), ensures that, under pre-specified propagation rules, the entire grid eventually becomes contaminated (active). The "power contamination problem" on grids is a dynamic variant of power domination, modeled by discrete-time local spreading rules that mimic the detection or infection process in practical systems such as electrical networks, biological populations, and distributed sensor grids. The combinatorial structure of optimal contamination sets is highly nontrivial, and their enumeration connects to classical objects in combinatorics, such as forbidden-pattern words and Schröder numbers.

1. Problem Statement and Local Propagation Rules

Consider a grid graph $G(n, m)$ with vertex set $V(n, m)$ representing an $n \times m$ rectangular lattice. A seed set $S \subseteq V(n, m)$ is selected as the initially contaminated set ($|S|$ as small as possible). Contamination then spreads according to the following discrete-time local rules (labeled (a)–(h) in the original paper):

• At each round, a contaminated cell can "infect" its neighbors, either directly or via combined effects, according to local combinatorial configurations.
• Key constraints require that no two consecutive columns or rows can remain uncontaminated at any step, and that the propagation cannot bypass such zones unless explicitly seeded.
• The goal is to find $S$ of minimal size such that, after finitely many rounds, every vertex in $V(n, m)$ becomes contaminated.

Define the power contamination number as: $$\gamma_{c}(G(n, m)) = \min \{|S| : S \subseteq V(n, m) \text{ leads to full contamination under the rules}\}$$ An optimal contamination set is a set $S^* \subseteq V(n, m)$ of cardinality $\gamma_{c}(G(n, m))$ that achieves full grid contamination.

2. Exact Values and Recurrence Relations

The paper (Mehiri et al., 16 Sep 2025) establishes explicit expressions and recurrences for $\gamma_{c}(G(n, m))$:

• For the 1-row grid:

$$\gamma_{c}(G(1, m)) = \left\lfloor m/2 \right\rfloor + 1$$

• For 2-row grids:

$\gamma_{c}(G(2, m)) = \begin{cases} \left\lceil m / 2 \right\rceil + 1 & \text{if %%%%10%%%% is odd} \ \left\lfloor m / 2 \right\rfloor + 1 & \text{if %%%%11%%%% is even} \end{cases}$

• For $n \ge 3$ and $m > n$, the paper provides the recurrence:

$$\gamma_{c}(G(n, m)) = \gamma_{c}(G(n, m-1)) + \tfrac{1}{2}(1+(-1)^m)$$

For $m$ even, the increment is 1; for $m$ odd, it is 0.

• For square grids of odd order:

$\gamma_{c}(G(2k+1, 2k+1)) = k+1$

These recurrences emerge from structural lemmas: e.g., that the contamination process cannot traverse two uncrossed columns or rows, mandating that optimal sets "break up" such obstacles.

3. Enumeration of Optimal Solutions and Integer Sequences

Beyond the size, the enumeration of all optimal contamination sets exhibits marked combinatorial richness. Let $\alpha_{n, m}$ denote the number of distinct $S^* \subseteq V(n, m)$ of size $\gamma_{c}(G(n, m))$ that achieve complete contamination:

Grid	Number of optimal sets	Related integer sequence
$G(1, m)$	$1$ (odd $m$); $m/2$ (even $m$)	N/A
$G(2, 2k)$	$k \cdot 2^k$	N/A
$G(2, 2k+1)$	A084857$(k)$	OEIS sequence A084857
$G(3, 2k+1)$	A193519$(k+1)$	OEIS sequence A193519
$G(2k+1, 2k+1)$	A006318$(k)$	Large Schröder numbers (A006318)

• For $G(2, 2k)$, optimal sets are counted by $k \cdot 2^k$ (the combinatorics encode different seed placements to ensure all rows/columns are covered).
• For $G(3, 2k+1)$, optimal sets correspond bijectively to ternary words of length $k+1$ over $\{1,2,3\}$ that contain at least one occurrence of the subword "13" or "31" (see OEIS A193519).
• For square grids $G(2k+1, 2k+1)$, the number of optimal sets is given by the large Schröder number $S_k$ (A006318), tied to the count of certain lattice paths or parenthesizations.

Notable recurrences include: $\alpha_{2k+1, 2k+1} = 3^{k+1} - u_{k+1}$ where $u_n$ counts ternary words of length $n$ avoiding both 13 and 31, showing the prevalence of forbidden pattern avoidance in characterizing contamination sets.

4. Structural Properties and Pattern Encoding

A crucial finding is that the structure of optimal contamination sets can often be encoded by combinatorial objects such as zig-zag patterns or pattern-avoiding words. For example:

• On odd-order square grids, place one seed on each row and each column with odd index, corresponding to a permutation of odd integers. This yields $S_k$ optimal configurations for $G(2k+1, 2k+1)$.
• On $G(3, 2k+1)$, read off the unique contaminated cell in each odd-indexed column; this forms a word $w = i_0i_1\cdots i_k$, $i_j \in \{1,2,3\}$, such that $w$ contains at least one of "13" or "31".
• In higher dimensions or more complex cases, optimal sets often correspond to solutions avoiding certain forbidden local patterns, reflecting a constraint satisfaction structure tied to the propagation rules.

5. Combinatorial Significance and Connections

The enumeration of optimal contamination sets reveals remarkable links between grid percolation processes and prominent combinatorial sequences:

• The large Schröder numbers count not only the optimal sets on odd square grids but also classical objects such as lattice paths above the axis with steps $(1,1),(1,-1),(2,0)$, certain rooted trees, and non-crossing over-parenthesizations.
• The OEIS sequences A084857 and A193519 provide a direct connection between contamination set enumeration and words with forbidden subwords—highlighting a deep relationship with pattern avoidance, an established area in combinatorics.

This interplay implies that power contamination, under suitable rules, provides a rich testing ground for bijective combinatorics, enumeration, and graph percolation.

6. Implications and Generalizations

The solution and enumeration of optimal contamination sets on grids have both theoretical and practical implications:

• For resource allocation in monitoring, the minimal contamination number $\gamma_c$ quantifies the minimum required investments to guarantee observability or containment under adversarial spreading.
• The structural results enable explicit construction of optimal sets for given $(n, m)$, facilitating algorithmic implementation in practical grid-based monitoring or control tasks (e.g., in power networks or distributed sensing).
• The bijective links to integer sequences suggest that similar percolation problems on other graphs (e.g., higher-dimensional grids, toroidal grids, or irregular lattices) may yield new combinatorial enumeration results and further links to the theory of forbidden patterns, lattice paths, or tiling problems.

As a prospective research direction, one may generalize these contamination rules and paper resulting enumeration problems in other network topologies. Exploration of randomized contamination processes, refined statistics (such as the maximum propagation time for optimal sets), or relationships with spectral properties of graphs are plausible next steps. The combinatorial framework developed here—for both the solution and rigorous enumeration of optimal contamination sets—opens a pathway to further mathematical investigation and interdisciplinary application.