Finite Dominating Set (FDS) in Graphs & Networks

• Finite Dominating Set (FDS) is defined as a set of vertices or candidate points ensuring every element is either in the set or adjacent/captured by one, covering both discrete graphs and continuous networks.
• Key insights include NP-hardness of underlying problems, geometric density bounds in triangulations, and uniform finite reduction techniques that preserve optimal coverage.
• Algorithmic approaches span hybrid tabu-search methods, metaheuristics, and exact formulations, effectively addressing both weighted and unweighted dominating set variants across various applications.

Finite Dominating Set (FDS) denotes two closely related constructions centered on domination. In finite graph theory, a dominating set $D\subseteq V$ is a subset of vertices such that every vertex is either in $D$ or adjacent to a vertex in $D$; in the context of finite triangulations on surfaces, an FDS is simply a dominating set of the finite embedded graph whose size is bounded in terms of the number of vertices [(Albuquerque et al., 2018); (Liu et al., 2010)]. In continuous network location, the term has a more specialized meaning: an FDS is a finite set of candidate points that dominates an infinite continuum of feasible locations in the sense that every feasible location is replaceable by some candidate covering a superset of routes, so an optimal solution is preserved within a finite candidate set (Nayeem et al., 14 Sep 2025). Across these usages, the unifying idea is finite reduction of a coverage relation.

1. Formal definitions and core variants

Let $G=(V,E)$ be a simple undirected graph. A subset $D\subseteq V$ is a dominating set if, for every vertex $i\in V$, either $i\in D$ or there exists $j\in D$ such that $\{i,j\}\in E$. If $N(i)=\{i\}\cup\{j\mid (i,j)\in E\}$ denotes the closed neighborhood, the condition is equivalently

$D$0

where $D$1 iff $D$2 (Albuquerque et al., 2018).

The unweighted optimization problem is the Minimum Dominating Set (MDS) problem: $D$3 The weighted variant assigns a weight $D$4 to each vertex and seeks a dominating set of minimum total weight, yielding the Minimum-Weight Dominating Set (MWDS) problem (Albuquerque et al., 2018). In the 0–1 formulation of MDS, one minimizes $D$5 subject to

$D$6

with the weighted objective replacing $D$7 by $D$8 in MWDS (Zouilekh et al., 2022, Albuquerque et al., 2018).

In finite triangulations on surfaces, the term FDS is used in the literal sense: a finite dominating set is a dominating set of a finite embedded graph whose size is studied asymptotically in $D$9, the number of vertices (Liu et al., 2010). In continuous location models, by contrast, the feasible region is infinite. For a network point $D$0, let $D$1 be the set of routes covered by $D$2, and let

$D$3

be its covered flow. A finite set $D$4 of candidate points is a dominating set if for every feasible location $D$5 there exists $D$6 with $D$7; an FDS is such a dominating set that is finite and guaranteed to contain an optimal solution (Nayeem et al., 14 Sep 2025).

2. Complexity and mathematical formulations

The classical Dominating Set problem is NP-hard, and the MDS decision problem is NP-complete by reduction from Set Cover. Its weighted analogue, MWDS, is also NP-hard. No polynomial-time exact algorithm is known, and even moderate-density graphs yield large covering matrices that challenge off-the-shelf solvers (Albuquerque et al., 2018, Zouilekh et al., 2022).

The standard covering formulations are direct. For MWDS, with binary variables $D$8,

$D$9

subject to

$G=(V,E)$0

The matheuristic study of MWDS explicitly notes that it does not add extra linear inequalities beyond this standard covering model, but instead intensifies search by fixing many $G=(V,E)$1 to zero in reduced subproblems (Albuquerque et al., 2018).

Parameterized and geometric complexity refine this basic picture. For one-dimensional geometric intersection graphs induced by translates of a fixed pattern $G=(V,E)$2, $G=(V,E)$3-Intersection Dominating Set is polynomially solvable when $G=(V,E)$4 contains at least one interval, and also when $G=(V,E)$5 is a point-pattern with all distance ratios rational. If $G=(V,E)$6 is a point-pattern with at least one irrational distance ratio, the problem is NP-complete but FPT parameterized by the solution size. In two and higher dimensions, Dominating Set is in W[1] for intersection graphs of constant-complexity semi-algebraic sets, while W[1]-hardness already holds for translates of any fixed simple polygon in $G=(V,E)$7, and W[2]-hardness holds when convex polygons are part of the input (Berg et al., 2017).

This landscape shows that “finite domination” does not correspond to a single algorithmic regime. The same domination predicate supports exact covering formulations, NP-hard optimization, FPT results in restricted geometric settings, and W-hierarchy hardness in higher-dimensional intersection models.

3. Structural bounds in finite triangulations on surfaces

For triangulations, FDS theory is primarily extremal. A dominating set $G=(V,E)$8 in a finite triangulation is studied through upper bounds on $G=(V,E)$9 relative to $D\subseteq V$0. The strongest results in the cited material concern triangulations whose degrees are mostly $D\subseteq V$1. There exists an absolute constant $D\subseteq V$2 such that every $D\subseteq V$3-vertex plane triangulation $D\subseteq V$4 with maximum degree $D\subseteq V$5 admits a dominating set $D\subseteq V$6 with

$D\subseteq V$7

and in fact $D\subseteq V$8. More generally, for any fixed surface $D\subseteq V$9, nonnegative integer $i\in V$0, and $i\in V$1, there exists

$i\in V$2

such that every $i\in V$3-vertex triangulation on $i\in V$4 having at most $i\in V$5 vertices of degree $i\in V$6 has a dominating set $i\in V$7 with

$i\in V$8

The final outcome is described as the first uniform $i\in V$9-density bound, up to lower-order additive constants, on domination in triangulations of arbitrary surfaces under the hypothesis that only $i\in D$0 vertices deviate from degree $i\in D$1; in the plane $i\in D$2 case, the $i\in D$3 asymptotic is best possible (Liu et al., 2010).

The proof machinery is geometric and topological. On orientable surfaces $i\in D$4, every $i\in D$5-vertex triangulation has a non-contractible cycle of length at most $i\in D$6; on non-orientable surfaces $i\in D$7, the bound becomes $i\in D$8. Cutting along such a cycle and capping the holes produces triangulations on surfaces of strictly smaller genus, while increasing the number of vertices by exactly $i\in D$9. This supports an inductive genus-reduction scheme (Liu et al., 2010).

A second ingredient maps planar triangulated disks with all interior degrees equal to $j\in D$0 onto the infinite $j\in D$1-regular triangular grid $j\in D$2. Pulling back the natural “every-seventh-vertex” pattern in $j\in D$3 yields a dominating set of density $j\in D$4. Cylinder substructures then sharpen the density. A $j\in D$5-cylinder can dominate its interior by $j\in D$6 vertices in the usual grid pattern, and when $j\in D$7, any $j\in D$8-cylinder contains a dominating set for its interior of size at most $j\in D$9, which is the step that “wins the density $\{i,j\}\in E$0” once the width is bounded (Liu et al., 2010).

These results place FDS in triangulations at the intersection of domination theory, discharging-style arguments, and surface topology. They also separate the finite-surface setting from general graphs: the geometry of the embedding directly controls dominating density.

4. Algorithmic methods for finite dominating sets

The algorithmic literature represented here spans matheuristics, population metaheuristics, approximation algorithms, and output-sensitive enumeration. For MWDS, the hybrid tabu-search/mathematical-programming method HTS-DS interleaves tabu search on the full graph with periodic reduced MIP solves. It uses the penalized objective

$\{i,j\}\in E$1

where $\{i,j\}\in E$2, $\{i,j\}\in E$3 is the number of non-dominated vertices, and $\{i,j\}\in E$4 is periodically ramped from $\{i,j\}\in E$5 to $\{i,j\}\in E$6 and reset to perform strategic oscillation. The full-graph search employs ADD, DEL, and restricted SWAP moves, a tabu list with aspiration, node elimination of redundant vertices, and perturbation every $\{i,j\}\in E$7 iterations. Reduced MIP subproblems keep free all vertices in the current best solution together with the most frequently used vertices in prior tabu-search solutions, and are solved for up to $\{i,j\}\in E$8 s. On 1,060 benchmark instances, HTS-DS matches all known optima on SMPI T1, improves or matches all known best-known solutions on LPI T1, has average CPU times $\{i,j\}\in E$9 s on SMPI and $N(i)=\{i\}\cup\{j\mid (i,j)\in E\}$0 s on LPI, and is reported faster than prior metaheuristics by a factor $N(i)=\{i\}\cup\{j\mid (i,j)\in E\}$1–$N(i)=\{i\}\cup\{j\mid (i,j)\in E\}$2; ablation experiments show measurable degradation without the reduced MIP, without SWAP, without perturbation, or under altered tabu-tenure or penalty-phase lengths (Albuquerque et al., 2018).

For unweighted MDS, HCSA-MDS combines Cuckoo Search’s global exploration through Lévy flights with one-point crossover and local intensification via filtering and repair. Candidate solutions are binary vectors $N(i)=\{i\}\cup\{j\mid (i,j)\in E\}$3, initialized as feasible solutions by a greedy heuristic. Filtering removes selected vertices when domination is preserved or fitness does not decrease; repair restores feasibility by adding the uncovered vertex of maximum degree, or its best neighbor, until all vertices are covered. After tuning, the reported parameters are $N(i)=\{i\}\cup\{j\mid (i,j)\in E\}$4, $N(i)=\{i\}\cup\{j\mid (i,j)\in E\}$5, $N(i)=\{i\}\cup\{j\mid (i,j)\in E\}$6, $N(i)=\{i\}\cup\{j\mid (i,j)\in E\}$7, and $N(i)=\{i\}\cup\{j\mid (i,j)\in E\}$8. On 42 random geometric graphs, HCSA-MDS improves the best-known solution in 32 cases and matches it in 10; on 21 synthetic graphs with known optima, it reaches the optimum in $N(i)=\{i\}\cup\{j\mid (i,j)\in E\}$9 of runs except two instances, where the success rate is $D$00. A Friedman plus Holm-adjusted Wilcoxon analysis reports statistical superiority in best, average, and worst solutions at $D$01 (Zouilekh et al., 2022).

Approximation methods appear in wireless-network settings. In disk-containment graphs with positive weights, an LP relaxation is “blown up” into a heavy fractional multiset and then sparsified through $D$02 rounds of randomized uniform sampling; with high probability, this yields a dominating set of total weight $D$03 in expected polynomial time. In directed communication graphs, a strongly dominating set can be approximated within $D$04 by combining a $D$05-approximate forward dominating set with a $D$06-approximate backward dominating set (Xu et al., 2012).

Enumeration addresses a different algorithmic question: output every dominating set, including non-minimal ones, exactly once. For $D$07-degenerate graphs, the reverse-search algorithm EDS-D enumerates all dominating sets in $D$08 time per solution using $D$09 space. For graphs of girth at least $D$10, EDS-G improves this to $D$11 amortized time per solution with the same space bound by exploiting the locally tree-like structure of high-girth neighborhoods (Kurita et al., 2018).

5. Finite dominating sets in continuous refueling-station location

A distinct FDS concept arises in the refueling station location problem for dedicated fleet routes on a general network. The network is a simple, connected, undirected graph $D$12; routes are directed closed walks

$D$13

with route length

$D$14

and only routes satisfying $D$15 and $D$16 are retained, because only these can be covered by a single station. Candidate station locations are all points on vertices and edge interiors, so the feasible set is infinite (Nayeem et al., 14 Sep 2025).

Coverage is governed by a driving range $D$17 and a one-way deviation limit $D$18. For a point $D$19 relative to a route $D$20, let $D$21 be the first vertex on the deviation route from $D$22 to $D$23, let $D$24 be the last vertex upon rejoining $D$25, and define

$D$26

Then $D$27 covers $D$28 iff

$D$29

and

$D$30

These are the two coverage constraints used to define the covered-routes set $D$31 (Nayeem et al., 14 Sep 2025).

The exact FDS is constructed by the Edge-Scanning (ES) algorithm. For each route $D$32 and edge $D$33, if $D$34, then every point on the segment $D$35 covers $D$36, so the recorded endpoints are $D$37. If $D$38, route coverage on that edge is represented by up to two reachable subsegments, one from each endpoint $D$39, with residual

$D$40

Each route-edge pair contributes at most four endpoints. The union $D$41 of all such endpoints is the key finite set. The Dominance theorem states that for every point $D$42 in the continuous network there exists an endpoint $D$43 with $D$44; the Finiteness theorem states that $D$45. The overall ES complexity is

$D$46

After grouping identical route-coverage sets $D$47, one obtains a reduced candidate set $D$48 and solves the set-covering MIP

$D$49

subject to

$D$50

This finite model is exact for the original continuous problem (Nayeem et al., 14 Sep 2025).

The reported numerical results illustrate the reduction. On the Sioux Falls network, with $D$51, $D$52, $D$53, $D$54, and $D$55, ES produces $D$56 endpoints and grouping reduces this to $D$57; the discrete-vertex approach needs $D$58 stations, whereas the FDS approach needs $D$59, with CPU time approximately $D$60 s. On the San Antonio network, with $D$61, $D$62, $D$63, and the same $D$64 and $D$65, ES gives $D$66, grouping gives about $D$67 candidates, the discrete approach again needs $D$68 stations, and the FDS approach needs $D$69, with solve time approximately $D$70 s. Sensitivity experiments show, for example, that on Sioux Falls, $D$71 and $D$72 produce $D$73 and allow one station to cover all five routes (Nayeem et al., 14 Sep 2025).

6. Related notions, terminology, and applications

Dominating sets and MWDS arise in social network analysis and influence propagation, wireless ad-hoc and sensor networks, coding theory, data mining, protein–protein interaction networks, and machine-learning, including feature selection and network controllability. In fleet operations, the FDS construction for refueling stations addresses dedicated public-transportation and distribution routes under deviation-permitted refueling (Albuquerque et al., 2018, Nayeem et al., 14 Sep 2025).

A recurrent terminological issue is that similar abbreviations denote different objects. In the finite-surface literature, an FDS is simply a dominating set of a finite graph whose size is bounded in terms of $D$74 (Liu et al., 2010). In continuous network location, an FDS is a finite candidate set that dominates an infinite continuum of feasible locations (Nayeem et al., 14 Sep 2025). In “Fair Domination in Graphs,” however, an FD-set means a fair dominating set: for some integer $D$75, a dominating set $D$76 such that every vertex outside $D$77 has exactly $D$78 neighbors in $D$79,

$D$80

The fair domination number is

$D$81

Every FD-set is a dominating set, so $D$82. For connected graphs of order $D$83 with no isolated vertex, $D$84, and there is an infinite family achieving equality. For maximal outerplanar graphs, $D$85. For a tree $D$86 of order $D$87, $D$88, with equality iff $D$89 is the corona of a tree (Caro et al., 2011).

This distinction matters conceptually. A dominating set need not be fair, and an FDS in the continuous-location sense need not itself be minimum; its role is to contain an optimal solution to a larger infinite problem. The literature therefore uses the same domination vocabulary for three separate purposes: finite extremal bounds in embedded graphs, optimization over finite or weighted graph instances, and exact finite reduction of continuous candidate spaces.