Domination Reduction in Graphs and Systems

• Domination reduction is a set of techniques that simplify domination parameters in combinatorial structures by reducing problem complexity and exploiting invariants.
• Algorithmic strategies such as kernelization, purification, and rule-based heuristics transform hard dominating set problems into more tractable forms.
• The approach extends to analytic, operator, and game-theoretic settings, yielding sharper bounds and improved reliability measures in complex systems.

Domination reduction encompasses a body of theory and algorithmic practice focused on simplifying, rescaling, or analyzing domination-type parameters in combinatorial and analytic structures—most prominently in graphs, hypergraphs, random structures, posets, function spaces, operator semigroups, and reliability systems. The underlying objective is to either transform a problem instance involving domination into a more tractable form, quantify the sensitivity of domination parameters under transformations, or exploit structure to obtain parameter reductions, invariants, or sharp bounds.

1. Foundational Notions in Domination and Reduction

The foundational concept is the dominating set: for a graph $G=(V,E)$, a set $D\subseteq V$ is dominating if every $v\in V\setminus D$ has a neighbor in $D$. This leads to the domination number $\gamma(G)$, the minimum size of such a set. Domination reduction arises in several forms:

• Reduction of problem size or structure via kernelization, transformation rules, or purification (e.g., (Inza et al., 27 Apr 2024, Geis et al., 17 Jun 2025)).
• Reduction via invariants such as domination polynomials, signed domination functions, or reliability polynomials—encoding combinatorial information (Dohmen et al., 2011, Beaton et al., 2017, Huseby, 25 Feb 2025).
• Parameter reduction: gauging how modifications (such as subdividing edges, removing vertices, or transforming graph classes) affect domination parameters (Dettlaff et al., 2013, Kaemawichanurat et al., 2022, Vemuri, 2019).
• Reduction in analytic or operator-theoretic settings, translating domination to relations over Dirichlet forms, semigroups, or random processes (Li et al., 12 Dec 2024).

Domination reduction is both a practical tool (shrinking NP-hard instances or improving heuristics) and a theoretical probe into the structure and complexity of domination phenomena in mathematics.

2. Structural Graph Reductions and Algorithms

Several papers develop, analyze, and improve explicit algorithmic reduction rules for the Dominating Set problem and its variants:

• Rule-based and Kernelization Reductions: The classic Rule1 of Alber et al. (which partitions neighborhood types and recursively removes dominated regions) is substantially optimized in (Geis et al., 17 Jun 2025) by reformulating the witness selection process and introducing linear-time algorithms. The method uses a canonical reference mapping $\rho(u) = \arg\max_{x\in N[u]} \text{val}(x)$ and global marking to avoid iteration. Analyses show that all reductions made by repeated applications of Rule1 can be anticipated and collapsed into a single linear-time sweep.
• Purification and Pruning: The two-phase approach of (Inza et al., 27 Apr 2024) applies a greedy dominating set algorithm followed by one of several sophisticated purification heuristics. These partition the candidate set into clusters (trees), compute outer and inner cover sets, and use a purification balance metric $PB(v,h) = \alpha |OCS(v,h)| + \beta |ICS(v,h)|$ to guide vertex removals. Reverse-order processing is also found effective. Experimental results, over 1300 benchmarks, revealed that these purification routines reduce dominating set size by a factor of 7 over upper bounds and find optimal solutions in 46.33% of cases.
• Subdivision Numbers: (Dettlaff et al., 2013) introduces the domination subdivision number $sd(G)$ (the minimum number of edge subdivisions increasing $\gamma(G)$) and domination multisubdivision number $msd(G)$ (the minimal number $k$ so subdividing some edge $k$ times increases $\gamma(G)$). The results show $msd(G)\leq 3$ for any $G$, and explicitly classify graphs (e.g., trees) for which $sd(G)=msd(G)$, as well as proving that finding $sd(G)$ is NP-complete even for bipartite graphs.
• Parameter Reductions: (Kaemawichanurat et al., 2022) establishes upper bounds relating $k$-isolation numbers $\iota_k(G)$ to the irredundance number $ir(G)$, generalizing $\gamma(G)<2 ir(G)$ to sharp inequalities for a range of $k$ values. The paper provides explicit extremal examples and constructions.

3. Domination Polynomials, Invariants, and Reliability

Graph polynomials encode domination-related structure and enable combinatorial reduction:

• Domination and Total Domination Polynomials: The domination polynomial $D(G,x)$, summing over dominating sets of all sizes, and the total domination polynomial $D_t(G,x)$, are central to the algebraic reduction of graph instances (Beaton et al., 2017, Hu et al., 2016). Reduction formulas—vertex and edge reductions—for $D_t(G,x)$ provide explicit recurrences decomposing the count of total dominating sets into simpler graphs (e.g., via deletion-contraction), critical for computations on paths, cycles, and trees.
• Domination Reliability: (Dohmen et al., 2011) introduces domination reliability as the probability that the set of operating vertices forms a dominating set in a random-failure network, establishing a close relationship between domination reliability and the domination polynomial through $DRel(G,p) = q^n D_G(p/q)$. The work provides explicit inclusion-exclusion and recursive formulas and shows an analogue of Whitney’s broken-circuit theorem reduces the size of inclusion-exclusion sums, akin to domination reduction at the formulaic level.
• Signed Domination Functions and Multistate Systems: In system reliability, the signed domination function $\delta$ counts (with alternating signs) the number of “formations” of minimal path sets. (Huseby, 25 Feb 2025) extends this to multistate systems and shows, via Möbius inversion, that the calculation of the signed domination function in such systems reduces to the corresponding binary system, thereby directly importing the combinatorial tools of binary system theory. Matroidal structure is leveraged for further reduction: in cases where the system is matroidal, $\delta(A)$ can be expressed in terms of the matroid $\beta$-invariant.

4. Domination in Non-Graphical Structures: Posets, Operator Theory, and Games

Domination reduction techniques extend beyond graphs:

• Order-Sensitive Domination in Posets: (Civan et al., 2020) defines order-sensitive dominating sets for posets as dominating sets in the comparability graph that additionally “sandwich” every non-extremal point. For graded constructions (height-3 or 4 posets derived from graphs), the order-sensitive domination number exactly realizes the Roman domination number and twice the domination number, respectively, of the original graph. Biclique partition numbers and clique partition numbers of associated graphs further reduce the computation.
• Reduction in Qualitative Games: (Patriche, 2013) generalizes the iterated elimination of strictly dominated strategies (IESDS) in classical games to qualitative games, introducing generalized dominance relations and paring structures. Under suitable conditions (e.g., “undominated dominators”), the IESDS routine reduces the strategy space uniquely and non-emptily, and the paper proves the independence from the order of strategy elimination.
• Operator-Theoretic Domination and Dirichlet Forms: (Li et al., 12 Dec 2024) identifies domination between semigroups or Dirichlet forms as an order relation—the “sandwiched” (intermediate) form lies between two extremals if $T^1_t f \leq T^2_t f$ for all $t \geq 0$ and $f \geq 0$. The paper characterizes domination reduction through the probabilistic killing transformation (Markov process killed with a multiplicative functional) and achieves explicit representations of the dominated Dirichlet form: $a^1(u,v) = a^2(u,v) + \sigma(u \otimes v)$. Robin boundary operators and non-local boundary problems thus admit complete analytic characterizations via this reduction.

5. Domination Reduction in Algorithmic Optimization and Heuristics

The NP-hardness or parameterized intractability of domination problems motivates heuristic and approximation strategies founded on reduction principles:

• Budgeted and Partial Domination: (Lamprou et al., 2019) develops improved approximation algorithms for the Budgeted Connected Dominating Set (BCDS) and Budgeted Edge-Vertex Domination (BEVD) by leveraging new tree decomposition methods and reduction to partial cover instances, with explicit performance ratios (e.g., $(1-e^{-7/8})/11$ for BCDS and $(1-1/e)$ for BEVD).
• Cross Entropy Method for Domination Variants: (Burdett et al., 2023) proposes an adaptation of the cross entropy (CE) method for domination and its many variants (total, 2-domination, secure domination). The CE approach is highly modular: define sampling/validation logic for the variant, then iteratively sample candidate sets and refine sampling probabilities based on elite solutions. High empirical performance is noted across standard and variant domination instances, including settings where CPLEX cannot compete at scale.
• Interval Graph Reductions: (Chiarelli et al., 2018) reduces weighted $k$-domination and total $k$-domination on proper interval graphs to shortest path computations in a DAG whose nodes represent partial solutions, drastically reducing the time complexity compared to previous state-of-the-art.

6. Reduction via Domination Relations and Graph Invariants

A further level of abstraction considers domination as a relation between pairs of objects:

• Graphon-based Domination and Dominating Graphs: (Conlon et al., 2023) defines a domination relation for graphs via homomorphism density inequalities: $H$ dominates $H'$ if $t_H(W)^{1/e(H)} \geq t_{H'}(W)^{1/e(H')}$ for all graphons $W$. This unifies and extends classical dominance notions, embedding Sidorenko-type conjectures as instances; the existence of a “percolating sequence” of reflections and relocations leading from a single edge to $H$ secures the domination property and thus enables structural reduction in extremal graph theory.

7. Implications, Limits, and Open Directions

Domination reduction, in all its guises, sharpens the understanding of domination-type invariants and enables reduction in complexity, parameter, or instance size. However, several limitations and challenges are emphasized:

• Sharp computational hardness persists: domination parameter evaluation, subdivision number computation, and equivalence class determination remain NP-hard in general (Dettlaff et al., 2013, Dohmen et al., 2011).
• Many reduction frameworks require careful tuning: e.g., the use of cluster structures, tie-breaking in rule-based reductions, or the selection of subgraph parameters for algebraic reduction.
• The interplay between local and global modifications is delicate: increasing local regularity (e.g., locally large shared tails in 3-tournaments) may not yield global reduction (Korándi et al., 2016), indicating that structure–parameter relationships are subtle and highly dependent on global configuration.

Open problems highlighted include:

• Extending domination reduction results to broader classes (e.g., higher-uniformity tournaments, hypergraphs, general non-local boundary problems).
• Fully classifying the reducing power of domination polynomials and reliability polynomials on complex graph classes.
• Developing practically robust, theoretically sound domination reduction frameworks for applied network optimization and algorithmic graph mining.

Domination reduction unites combinatorial, algebraic, analytic, and algorithmic strands, providing a unified lens for both theoretical exploration and practical problem-solving in the paper of covering, control, and reliability in complex systems.