Power Domination in Electric Networks

• Power domination is a graph-theoretic model that uses PMU placements and propagation rules to achieve full network observability.
• It encompasses various generalizations such as k-power domination, robust, and connected variants aimed at optimizing PMU placement strategies.
• The model links domination with zero forcing, yielding structural bounds, exact formulas, and cost-aware optimization methods in network monitoring.

Power domination is a graph-theoretic model of monitoring an electric power network with phasor measurement units (PMUs). Given a graph $G=(V,E)$ and an initial PMU placement $S\subseteq V(G)$, the process begins by observing $N[S]$ and then propagates observation whenever an observed vertex has exactly one unobserved neighbor; $S$ is a power dominating set if the process eventually observes all vertices, and the minimum cardinality of such a set is the power domination number $\gamma_P(G)$. The subject now includes deterministic, generalized, connected, restricted, polynomial, robustness-aware, stochastic-failure, and cost-benefit formulations, all motivated by variations of the PMU placement problem in electrical networks (Benson et al., 2015, Bjorkman et al., 2023, Morrison et al., 27 Jan 2026).

1. Classical formulation and generalizations

In its standard form, power domination is defined on a finite simple graph $G=(V,E)$ by the two-stage process

$PD(S)=N[S],$

followed by repeated propagation: if there exists $v\in PD(S)$ such that

$|N(v)\cap (V(G)\setminus PD(S))|=1,$

then $PD(S)$ is updated by adding $S\subseteq V(G)$0. A set $S\subseteq V(G)$1 is a power dominating set if the terminal set is $S\subseteq V(G)$2, and $S\subseteq V(G)$3 is the minimum size of such a set (Benson et al., 2015). Equivalent notation also appears as $S\subseteq V(G)$4, with $S\subseteq V(G)$5 power dominating precisely when $S\subseteq V(G)$6 (Bjorkman et al., 2023).

A standard generalization is $S\subseteq V(G)$7-power domination. For $S\subseteq V(G)$8, one starts from

$S\subseteq V(G)$9

and then iterates

$N[S]$0

A set $N[S]$1 is a $N[S]$2-power dominating set if $N[S]$3, and the minimum size is $N[S]$4. This interpolates between domination at $N[S]$5 and ordinary power domination at $N[S]$6 (Varghese et al., 2015, Ferrero et al., 2017).

The hypergraph literature contains two principal extensions. Chang and Roussel generalized the graph process by retaining a single-vertex propagation rule, while infectious power domination combines the domination step with Bergen et al.’s infection rule: if $N[S]$7, one first sets

$N[S]$8

then repeatedly allows a nonempty $N[S]$9 to infect an edge $S$0 if $S$1 and every unobserved vertex $S$2 with $S$3 contained in some edge must satisfy $S$4. The resulting minimum size is the infectious power domination number $S$5. Both hypergraph formulations reduce to ordinary power domination on $S$6-uniform hypergraphs (Bjorkman, 2019).

2. Zero forcing, domination, and product phenomena

A central structural fact is that power domination is domination followed by zero forcing. Precisely,

$S$7

which permits transfer of bounds and techniques between the two theories (Benson et al., 2015). In particular, if $S$8 denotes the zero forcing number and $S$9 the maximum degree, then every graph with an edge satisfies

$\gamma_P(G)$0

and this bound is tight (Benson et al., 2015). The $\gamma_P(G)$1-analogue sharpens the same principle: for a connected graph with $\gamma_P(G)$2,

$\gamma_P(G)$3

equivalently,

$\gamma_P(G)$4

again with tightness (Ferrero et al., 2017).

The relationship is not merely formal. In the $\gamma_P(G)$5-power domination setting, if $\gamma_P(G)$6 is a minimum $\gamma_P(G)$7-power dominating set chosen so that every vertex has at least $\gamma_P(G)$8 external $\gamma_P(G)$9-private neighbors, then one can explicitly build a $G=(V,E)$0-forcing set by retaining, for each $G=(V,E)$1, the vertices of $G=(V,E)$2 except $G=(V,E)$3 selected external private neighbors. This construction underlies the preceding inequality and makes the comparison constructive rather than purely existential (Ferrero et al., 2017).

Power domination also exhibits product behavior analogous to Vizing-type lower bounds. If $G=(V,E)$4 denotes the maximum number of parts in a failed power dominating partition of $G=(V,E)$5, then

$G=(V,E)$6

When $G=(V,E)$7 and $G=(V,E)$8, this yields

$G=(V,E)$9

For trees, $PD(S)=N[S],$0, so for any two trees $PD(S)=N[S],$1,

$PD(S)=N[S],$2

(Anderson et al., 2022). This suggests that, at least for classes where failed power dominating partitions are tight, Cartesian products preserve a strong multiplicative lower bound.

3. Restricted and connected variants

Restricted power domination fixes part of the PMU placement in advance. For $PD(S)=N[S],$3,

$PD(S)=N[S],$4

It satisfies

$PD(S)=N[S],$5

and more generally, for $PD(S)=N[S],$6,

$PD(S)=N[S],$7

A fundamental reduction attaches leaves to the required vertices: if $PD(S)=N[S],$8 is obtained by attaching $PD(S)=N[S],$9 leaves to every vertex of $v\in PD(S)$0, then

$v\in PD(S)$1

and

$v\in PD(S)$2

if and only if

$v\in PD(S)$3

This yields a linear-time algorithm on bounded-treewidth graphs and supports decomposition and integer-programming formulations for the restricted problem (Bozeman et al., 2017).

Connected power domination adds the requirement that the initial PMU set induce a connected subgraph. The minimum size is $v\in PD(S)$4. This variant is NP-hard in general, but it admits sharp structural decompositions. If $v\in PD(S)$5 is the set of cut vertices in the classes $v\in PD(S)$6, then every connected power dominating set of a connected nonpath graph contains $v\in PD(S)$7 (Brimkov et al., 2017). For trees and block graphs,

$v\in PD(S)$8

while cactus graphs satisfy the exact formula

$v\in PD(S)$9

where $|N(v)\cap (V(G)\setminus PD(S))|=1,$0 are the pendant paths and $|N(v)\cap (V(G)\setminus PD(S))|=1,$1 is the maximum size of a feasible omitted segment of cycle block $|N(v)\cap (V(G)\setminus PD(S))|=1,$2 (Brimkov et al., 2017).

A broader decomposition expresses connected power domination across nontrivial blocks $|N(v)\cap (V(G)\setminus PD(S))|=1,$3. If $|N(v)\cap (V(G)\setminus PD(S))|=1,$4 and $|N(v)\cap (V(G)\setminus PD(S))|=1,$5 is the number of nontrivial blocks containing $|N(v)\cap (V(G)\setminus PD(S))|=1,$6, then

$|N(v)\cap (V(G)\setminus PD(S))|=1,$7

This suggests that articulation structure is often decisive once connectivity is imposed (Brimkov et al., 2017).

4. Exact values on specific graph families and network classes

Several graph families admit exact formulas. For the triangular grid $|N(v)\cap (V(G)\setminus PD(S))|=1,$8 with hexagonal border,

$|N(v)\cap (V(G)\setminus PD(S))|=1,$9

(Bose et al., 2017). The proof combines an explicit periodic upper-bound construction with a lower bound based on a border argument and a shifting process that transforms arbitrary monitored sets into canonical staircase-shaped sets.

For maximal planar graphs of order $PD(S)$0,

$PD(S)$1

and the bound is tight for graphs on six vertices and for the triakis tetrahedron on ten vertices (Dorbec et al., 2017). The proof is constructive and proceeds through special-configuration elimination, greedy augmentation, and recursive handling of residual splitting structures.

For generalized power domination on WK-Pyramid networks $PD(S)$2, the $PD(S)$3-power domination number is determined for all positive $PD(S)$4 except the boundary case $PD(S)$5 with $PD(S)$6. The exact formula is

$PD(S)$7

For $PD(S)$8, only the upper bound

$PD(S)$9

is proved (Varghese et al., 2015).

For the corrected fractal cubic network $S\subseteq V(G)$00,

$S\subseteq V(G)$01

where $S\subseteq V(G)$02 is the resolving power domination number. The proof identifies $S\subseteq V(G)$03 disjoint open twin classes, giving the lower bound, and exhibits the set of all vertices ending in $S\subseteq V(G)$04 as an optimal power dominating set (Prabhu et al., 2024). A plausible implication is that recursive twin structure can simultaneously control monitoring and resolving requirements.

5. Polynomials, forts, and exact optimization methods

The power domination polynomial packages the counts of power dominating sets by size: $S\subseteq V(G)$05 where $S\subseteq V(G)$06 is the number of power dominating sets of size $S\subseteq V(G)$07 (Brimkov et al., 2018). It encodes $S\subseteq V(G)$08 as the multiplicity of the root $S\subseteq V(G)$09, has no positive real roots, and admits decomposition formulas. For disjoint unions,

$S\subseteq V(G)$10

while explicit formulas include

$S\subseteq V(G)$11

$S\subseteq V(G)$12

and, for stars $S\subseteq V(G)$13 with $S\subseteq V(G)$14,

$S\subseteq V(G)$15

The same paper studies $S\subseteq V(G)$16-uniqueness, power domination roots, and coefficient formulas such as

$S\subseteq V(G)$17

where $S\subseteq V(G)$18 is the number of isolates (Brimkov et al., 2018).

A separate optimization line is based on zero forcing forts. If $S\subseteq V(G)$19 is the set of forts and $S\subseteq V(G)$20 the fort neighborhoods, then

$S\subseteq V(G)$21

This yields an exact set-cover formulation: $S\subseteq V(G)$22 The resulting row-generation framework separates violated fort-neighborhood constraints using auxiliary integer programs, and the proposed method solved large benchmark instances, including the US Western Interconnection in $S\subseteq V(G)$23 seconds and PEGASE $S\subseteq V(G)$24 in $S\subseteq V(G)$25 seconds, after earlier infection-order formulations timed out (Smith et al., 2020).

The Power Domination Toolbox operationalizes related ideas in Python. It contracts components induced by degree-$S\subseteq V(G)$26 vertices while preserving $S\subseteq V(G)$27, exploits terminal forts and certain induced-$S\subseteq V(G)$28-based zero forcing forts, restricts the search to active vertices of degree at least $S\subseteq V(G)$29, orders candidate placements by a qualitative score, and parallelizes subset checking. On the IEEE $S\subseteq V(G)$30-bus system, the earlier JL-BW exhaustive search was terminated after a week, while PDT found $S\subseteq V(G)$31 in approximately $S\subseteq V(G)$32 seconds using $S\subseteq V(G)$33 threads (Koch et al., 2023).

6. Robustness, stochastic failure, and cost-aware formulations

Robust power domination replaces location failure by sensor failure and allows multiple PMUs at one vertex. In the multiset formulation, a $S\subseteq V(G)$34-robust power dominating set is a multiset $S\subseteq V(G)$35 such that for every submultiset $S\subseteq V(G)$36 with $S\subseteq V(G)$37, $S\subseteq V(G)$38 is a power dominating set. The foundational bounds are

$S\subseteq V(G)$39

and the parameter is monotone in $S\subseteq V(G)$40: $S\subseteq V(G)$41 If $S\subseteq V(G)$42, then $S\subseteq V(G)$43 (Bjorkman et al., 2023). The PMU-defect-robust refinement denotes the minimum size by $S\subseteq V(G)$44, proves

$S\subseteq V(G)$45

and obtains exact formulas for major families, including

$S\subseteq V(G)$46

for every block graph $S\subseteq V(G)$47, hence for every tree, and

$S\subseteq V(G)$48

In particular,

$S\subseteq V(G)$49

(Bjorkman et al., 2023).

Fragile power domination introduces random independent PMU failure before the domination step. If each PMU in a placement $S\subseteq V(G)$50 fails independently with probability $S\subseteq V(G)$51, the surviving multiset $S\subseteq V(G)$52 induces the random observed set $S\subseteq V(G)$53, and the central quantity is

$S\subseteq V(G)$54

This expected value is a polynomial in $S\subseteq V(G)$55 of degree at most $S\subseteq V(G)$56. Its low-degree coefficients characterize deterministic robustness: if $S\subseteq V(G)$57 has $S\subseteq V(G)$58 vertices and $S\subseteq V(G)$59 is a PMU placement, then $S\subseteq V(G)$60 is a $S\subseteq V(G)$61-PMU-defect-robust power dominating set if and only if

$S\subseteq V(G)$62

for some polynomial $S\subseteq V(G)$63. The model also isolates full-observation probabilities and yields explicit formulas for stars, generalized barbell graphs, and complete multipartite graphs (Bjorkman et al., 2023).

The fragile framework changes placement orderings. A particularly explicit illustration is the star $S\subseteq V(G)$64 with center $S\subseteq V(G)$65 and $S\subseteq V(G)$66: placing PMUs on all $S\subseteq V(G)$67 vertices gives

$S\subseteq V(G)$68

while the multiset placement $S\subseteq V(G)$69 gives

$S\subseteq V(G)$70

(Bjorkman et al., 2023). A later development characterizes when two graph-placement pairs have the same expected value polynomial, proves that all coefficients of degree at least $S\subseteq V(G)$71 can be prescribed arbitrarily by attaching gadgets, and gives an exact characterization of linearity of $S\subseteq V(G)$72 in terms of aggregated $S\subseteq V(G)$73-subset observation totals (Bjorkman et al., 19 Jul 2025). This suggests that stochastic robustness is governed by subset-level observability statistics rather than only by $S\subseteq V(G)$74.

A different extension introduces an explicit tradeoff between PMU cost and non-observance cost: $S\subseteq V(G)$75 where $S\subseteq V(G)$76 is the Observance Cost Ratio. A set is $S\subseteq V(G)$77-best if it minimizes this cost. The extreme regimes are exact: $S\subseteq V(G)$78 For fixed $S\subseteq V(G)$79, the maximum observance

$S\subseteq V(G)$80

induces

$S\subseteq V(G)$81

and the marginal observance

$S\subseteq V(G)$82

gives the threshold criterion

$S\subseteq V(G)$83

The model also proves a realizability theorem for useful PMU counts and derives fort-based thresholds on $S\subseteq V(G)$84 guaranteeing that full observation is optimal (Morrison et al., 27 Jan 2026).

Power domination has therefore evolved from a minimum-PMU covering question into a broader theory of domination-plus-propagation, with exact graph-family formulas, decomposition theorems, fort-based optimization, robustness notions based on deterministic failures or stochastic breakdown, and cost-sensitive criteria that can make incomplete observability optimal. The common core across these variants remains the same: an initial neighborhood observation step coupled to a propagation rule that transforms local uniqueness into global observability.