Power Contamination Number in Grids

• Power contamination number is defined as the minimal seed set required to fully contaminate a grid graph using local rules similar to power domination.
• Exact formulas, such as ⌊m/2⌋+1 for even dimensions and ⌈m/2⌉+1 for odd cases, capture how grid size and parity influence the process.
• Enumeration methods link optimal seed sets to classical combinatorial sequences like the large Schröder numbers, highlighting deep mathematical connections.

Power contamination number denotes the minimal cardinality of a seed set that—under dynamic rules—achieves full contamination (i.e., coverage) of a graph, most notably grid graphs, via local propagation rules reminiscent of power domination processes. While "power domination number" (and its variants) traditionally arise in electrical network monitoring via device placement (see (Brimkov et al., 2017)), the power contamination number is defined by stricter dynamical rules that control the spread of contamination and is particularly studied in combinatorial settings such as grids (Mehiri et al., 16 Sep 2025).

1. Formal Definition and Propagation Rules

On an $n \times m$ grid graph $G(n,m)$, the power contamination number $\gamma_c(G(n,m))$ is the minimum size of a set $S$ such that, if the cells belonging to $S$ are initially contaminated, the entire grid becomes contaminated via a prescribed set of local geometric rules (labeled (a)–(h) in (Mehiri et al., 16 Sep 2025)). These typically involve conditions governing when an uncontaminated cell becomes contaminated based on the contamination status of its neighbors—analogous to zero forcing or bootstrap percolation, yet specialized for grid topology.

This parameter is distinct from the connected power domination number which additionally imposes connectivity constraints on the initial set (see (Brimkov et al., 2017)).

2. Exact Enumeration in Grid Graphs

The principal contribution of (Mehiri et al., 16 Sep 2025) is the explicit determination of $\gamma_c(G(n,m))$ for grid graphs via combinatorial arguments founded on propagation constraints. The main closed-form result is:

• For $n=1$ (a path), $n \geq 3$, or $n=2$ with $m$ even:

$$\gamma_c(G(n,m)) = \left\lfloor \frac{m}{2} \right\rfloor + 1$$

• For $n=2$ and $m$ odd:

$$\gamma_c(G(2,m)) = \left\lceil \frac{m}{2} \right\rceil + 1$$

These formulas are justified via propagation constraints—such as the necessity of polluting boundary cells and avoiding two consecutive columns being clean—and constructive proofs (notably, a Zig-Zag seeding strategy that achieves $\gamma_c(G(n,m))$ for all $n \geq 2$, excepting the $n=2, m$ odd case which necessitates an adjustment).

3. Recurrence Relations

The power contamination number for grids satisfies several recurrence relations established in (Mehiri et al., 16 Sep 2025):

• For row grids: $\gamma_c(G(1,m)) = \gamma_c(G(1,m-4)) + 2$ for $m \geq 4$, with base cases $\gamma_c(G(1,0)) = 1$, $\gamma_c(G(1,1)) = 1$, and $\gamma_c(G(1,2)) = \gamma_c(G(1,3)) = 2$.
• For $m > n \geq 3$:

$$\gamma_c(G(n,m)) = \gamma_c(G(n,m-1)) + \frac{1 + (-1)^m}{2}$$

This captures the parity dependence of added columns on required seed count.

Generalizations permit the removal of $p$ rows and $q$ columns:

$$\gamma_c(G(n,m)) = \gamma_c(G(n-p, m-q)) + \left\lceil \frac{q}{2} \right\rceil$$

for $m$ even and $q$ odd, or $\left\lfloor {q}/{2} \right\rfloor$ otherwise.

4. Enumeration of Optimal Contaminating Sets

Beyond calculating $\gamma_c$, (Mehiri et al., 16 Sep 2025) initiates the enumeration of all optimal seed sets $\alpha_{n,m}$. For specific families, these counts align with famed integer sequences. Notable results include:

• For $G(1,m)$, if $m$ is odd: $\alpha_{1,m} = 1$ (unique optimal solution); if $m$ even: $\alpha_{1,m} = m/2$.
• For $G(2,2k)$ (even): $\alpha_{2,2k} = k \cdot 2^k$.
• For $G(3,2k+1)$ (odd columns): Optimal solutions correspond to ternary words of length $k+1$ containing a "jump" (subword "13" or "31").
• For $G(2k+1,2k+1)$ (odd-sided square grids): $\alpha_{2k+1,2k+1}$ equals the large Schröder number $A006318(k)$, counting certain pattern-avoiding permutations.

These correspondences are established via encodings of the optimal configurations as words (for $n=3$) or permutations (for square grids), subject to constraints derived from the dynamic propagation process.

Table: Optimal Solution Enumeration in Small Grid Families

Grid Size	$\alpha_{n,m}$ Formula	Linked OEIS Sequence
$G(1,m)$	$1$ (if $m$ odd), $m/2$ (if $m$ even)	—
$G(2,2k)$	$k \cdot 2^k$	A036289
$G(3,2k+1)$	ternary words with forbidden subwords	A193519
$G(2k+1,2k+1)$	Large Schröder numbers $\alpha = A006318(k)$	A006318

5. Links to Classical Combinatorics and Integer Sequences

Enumeration results reveal deep connections with combinatorial structures:

• Words over a finite alphabet with forbidden subwords (see A193519).
• Large Schröder numbers, which count lattice paths or pattern-avoiding permutations (see A006318).
• The avoidance of two consecutively clean columns or rows establishes rigorous constraints on permissible seed patterns, paralleling motif avoidance in combinatorics.
• The Zig-Zag construction ensures optimal propogation, and the distinct combinatorial encodings (ternary words, permutations) permit leveraging enumeration results from classical combinatorics.

6. Significance, Related Problems, and Outlook

The resolution of the Ainouche–Bouroubi conjecture (Mehiri et al., 16 Sep 2025)—which previously posited a different value for $\gamma_c(G(n,m))$—clarifies the optimal seeding requirements for contamination in grids, settling fundamental questions for percolation-type processes on combinatorial structures. These results provide not only precise bounds for seed set minimization in dynamical models but also algorithmic avenues for constructing optimal sets and analyzing their enumeration in terms of integer sequences.

Future research may generalize these findings to irregular grids, higher dimensions, alternative contamination or domination rules, and the impact of topological variations, drawing further on connections to pattern avoidance and word enumeration in discrete mathematics.