General Dominance (GD) Framework

• General Dominance (GD) is a framework where each vertex is assigned a Boolean signature representing its own state and those of its neighbors.
• It unifies classical domination problems such as dominating set, total dominating set, and (σ,ρ)-sets by formulating them as weighted counting problems.
• The framework leverages gadget construction and Holant reductions to establish #P-completeness and develop dichotomy theorems on restricted graph classes.

Searching arXiv for the specified paper and closely related work on domination-type counting and Holant to ground the article. General Dominance (GD), in the sense developed in "The Counting General Dominating Set Framework," is a signature-based framework for counting domination-type structures on graphs. It replaces the classical view of a dominating set as a subset $S\subseteq V$ satisfying a neighborhood condition by a more general weighted formulation in which each vertex $v$ carries a Boolean signature $f_v$ acting on the state of $v$ and the states of its neighbors. The resulting counting problem, denoted $\#\GDS(\mathcal F)$, computes a partition function over all Boolean vertex assignments, thereby subsuming $(\sigma,\rho)$-Set, dominating set, total dominating set, and related counting problems. Within this framework, the paper establishes equivalences with Holant, introduces a gadget-construction methodology tailored to $\#\GDS$, proves $\#P$-completeness for counting dominating sets and total dominating sets on 3-regular planar bipartite simple graphs, and derives dichotomy results in the uniform-signature regime (Zheng et al., 16 Mar 2026).

1. Formal definition of General Dominance

Let $G=(V,E)$ be a simple graph. A general dominating-set signature grid over a finite set of Boolean signatures $\mathcal F$ is an assignment

$v$0

where $v$1, the first input of $v$2 corresponds to the state of $v$3 itself, and the remaining $v$4 inputs correspond to the states of its neighbors. The convention is Boolean: $v$5 means “out of the set,” and $v$6 means “in the set.”

For each Boolean vertex-assignment

$v$7

the framework associates the weight

$v$8

The problem $v$9 then asks for the partition function

$f_v$0

This definition is the central abstraction of GD: a “general dominating set” is not merely a subset satisfying a single domination predicate, but a Boolean assignment weighted locally at every vertex by a signature that can depend simultaneously on the vertex state and the multiset of neighboring states. The data summarize this viewpoint explicitly: General Dominating Set simply means an assignment $f_v$1 weighted by $f_v$2 (Zheng et al., 16 Mar 2026).

2. Relation to $f_v$3-Set, dominating set, and total dominating set

The $f_v$4 framework strictly contains the classical $f_v$5-Set formalism. Given $f_v$6, a subset $f_v$7 is a $f_v$8-dominating set if, for each $f_v$9, the number of neighbors of $v$0 in $v$1 lies in $v$2, and for each $v$3, the number of neighbors of $v$4 in $v$5 lies in $v$6. This can be encoded by the vertex signature

$v$7

With these signatures, $v$8 exactly counts the number of $v$9-sets (Zheng et al., 16 Mar 2026).

The standard domination notions arise as immediate special cases.

Problem	$\#\GDS(\mathcal F)$0 choice	Consequence
Dominating set	$\#\GDS(\mathcal F)$1	Vertices outside the set must have at least one neighbor in the set
Total dominating set	$\#\GDS(\mathcal F)$2	Every vertex must have at least one neighbor in the set

The significance of this encoding is structural rather than merely notational. Classical domination-type problems are recovered by choosing signatures that depend only on the count of selected neighbors, but GD permits arbitrary Boolean signatures $\#\GDS(\mathcal F)$3. The synthesis given in the paper states that $\#\GDS(\mathcal F)$4 therefore unifies problems from $\#\GDS(\mathcal F)$5-domination, total domination, induced subgraphs of bounded degree, and beyond (Zheng et al., 16 Mar 2026).

3. Equivalence with Holant and factorization phenomena

A major feature of GD is its explicit connection to Holant. A Holant problem $\#\GDS(\mathcal F)$6 is defined on a bipartite graph $\#\GDS(\mathcal F)$7, with vertices in $\#\GDS(\mathcal F)$8 labeled by signatures from $\#\GDS(\mathcal F)$9 and vertices in $(\sigma,\rho)$0 labeled by signatures from $(\sigma,\rho)$1. The paper records two reductions that place $(\sigma,\rho)$2 within this landscape.

The first reduction states that for any signature set $(\sigma,\rho)$3 of arity at least $(\sigma,\rho)$4, there is a Boolean signature set $(\sigma,\rho)$5 such that

$(\sigma,\rho)$6

The proof sketch given in the data replaces each GDS vertex by a bipartite gadget that forces the vertex to have state $(\sigma,\rho)$7 internally and uses $(\sigma,\rho)$8 on new degree-2 vertices to propagate edge assignments.

The second reduction goes in the opposite direction: every non-bipartite $(\sigma,\rho)$9 instance is Turing-equivalent to a domain-4 Holant problem. This is done by encoding each vertex-state pair on an edge with one of four symbols and using a suitable $\#\GDS$0 signature to enforce consistency.

A further simplification occurs for uniform signatures,

$\#\GDS$1

meaning that $\#\GDS$2. In a bipartite graph $\#\GDS$3 with uniform families

$\#\GDS$4

Theorem 3.4 gives the factorization

$\#\GDS$5

The proof sketch in the data describes this as summing first over one side of the bipartite graph and extracting signatures insensitive to the states on the opposite side. This factorization is central in the later hardness proof for total dominating sets, where the partition function becomes a “powered” Holant expression (Zheng et al., 16 Mar 2026).

4. Gadget construction and interpolation in $\#\GDS$6

The gadget methodology for GD differs from the standard dangling-edge formalism of Holant. A $\#\GDS$7 gadget is organized around three vertex classes: external vertices $\#\GDS$8, where arbitrary GDS signatures will later be attached; bridging vertices $\#\GDS$9, which are internal but adjacent to $\#P$0; and internal vertices $\#P$1, which lie deeper inside the gadget.

The gadget’s associated function is called its gadgeture. If $\#P$2 has labels $\#P$3 and $\#P$4 has labels $\#P$5, then the gadgeture

$\#P$6

is defined by

$\#P$7

This is the local transfer object from which larger reductions are assembled (Zheng et al., 16 Mar 2026).

The paper includes two representative constructions. The first is a chain gadget $\#P$8 with one external vertex $\#P$9 and one bridging vertex $G=(V,E)$0 carrying $G=(V,E)$1. Its gadgeture satisfies the linear recurrence

$G=(V,E)$2

acting on the $G=(V,E)$3 vector of $G=(V,E)$4.

The second is the ladder gadget used to simulate Boolean gates such as OR and XNOR while preserving 3-regularity, planarity, and bipartiteness. Its initial gadgeture

$G=(V,E)$5

satisfies an affine-linear recurrence $G=(V,E)$6. The data then state that certain entries always coincide, so the state space collapses to a $G=(V,E)$7 recurrence matrix $G=(V,E)$8. By verifying that $G=(V,E)$9, that its eigenvalues $\mathcal F$0 are distinct, and that no nontrivial root-of-unity relation

$\mathcal F$1

holds, the authors invoke a multivariate polynomial-interpolation lemma of Cai–Lu–Xia. This makes it possible to extract exactly the coefficients corresponding to external assignments realizing XNOR or OR. The reduction then uses these extracted coefficients to implement the logical constraints of vertex cover inside a gadgetized dominating-set instance (Zheng et al., 16 Mar 2026).

5. Main $\mathcal F$2-completeness results

The framework yields two explicit hardness theorems on restricted graph classes. Theorem 1.1 states that $\mathcal F$3-Dominating Set is $\mathcal F$4-complete, where $\mathcal F$5 denotes 3-regular, planar, bipartite, simple graphs. The proof sketch begins from an instance of $\mathcal F$6-Vertex Cover, splits each original vertex into two external vertices, and connects them by a copy of the ladder gadget $\mathcal F$7 so that the pair must receive the same assignment and thereby simulate a single cover bit. For each original edge, one or two copies of another ladder gadget $\mathcal F$8 are inserted to enforce the OR-constraint that at least one endpoint is chosen. The resulting graph $\mathcal F$9 remains 3-regular, planar, bipartite, and simple. Its $v$00 partition function is a bivariate polynomial in the five distinct gadgeture values of $v$01 and the five of $v$02, over

$v$03

choices. Evaluating sufficiently many pairs $v$04 and applying polynomial interpolation recovers the original $v$05Vertex Cover count (Zheng et al., 16 Mar 2026).

Theorem 1.2 states that $v$06-Total Dominating Set is also $v$07-complete. Here total dominators correspond to the uniform signature

$v$08

By Corollary 3.5, on symmetric bipartite graphs one has

$v$09

which the summary describes as a powered Holant instance. The proof then reduces a known $v$10-hard case,

$v$11

by means of a simple interpolation gadget built from $v$12. Translating back through symmetric-bipartite embedding yields $v$13-hardness for $v$14 with $v$15 on symmetric bipartite graphs, and therefore on all $v$16 graphs (Zheng et al., 16 Mar 2026).

These theorems place natural domination problems in the same fine-grained complexity territory as other Holant and counting-CSP hardness results, but they do so through gadgets native to the GD formalism rather than by direct transfer from pre-existing frameworks.

6. Dichotomy results and tractable families

Beyond isolated hardness theorems, the paper gives a classification result in the uniform or signature-powered regime. Theorem 5.1, called the General Holant$v$17 Dichotomy, considers a ternary signature

$v$18

and the problem

$v$19

It states that this problem is in FP if and only if $v$20 belongs to one of the following families: the degenerate family $v$21 for some unary $v$22; the Gen-Eq family $v$23; or one of the affine families

$v$24

together with two additional planar-only families. Otherwise the problem is $v$25-hard (Zheng et al., 16 Mar 2026).

Corollary 5.2 translates this directly to uniform $v$26. If

$v$27

is a ternary signature arising from a $v$28 matrix $v$29, then

$v$30

is in FP exactly in the cases above, plus the two extra planar families, and is $v$31-hard otherwise.

The classification has two implications. First, it shows that the transfer from Holant to GD is not merely a hardness-transfer mechanism; it supports exact tractability boundaries. Second, it identifies the uniform-signature setting as a regime where the complexity landscape is sufficiently rigid to admit a full “if and only if” theorem, rather than a collection of isolated reductions.

7. Conceptual synthesis and terminological distinctions

The synthesis in the data characterizes GD as the natural signature-based extension of classical domination-type problems. In this formulation, the framework elevates the $v$32-Set concept, where one enforces only neighborhood counting constraints, to a fully Holant-style model in which each vertex carries an arbitrary Boolean signature $v$33. The paper further states that powerful Holant techniques—gadget constructions, linear-recurrence interpolation, and Galois-theoretic eigenvalue checks—carry over to $v$34, producing tight complexity-dichotomy results together with $v$35-hardness on 3-regular planar bipartite simple graphs. The listed future directions are bridging the gap between gadgetures and proper GDS signatures, extending dichotomies to larger signature-domains or higher arity, and exploring parameterized and approximation variants under the general dominance lens (Zheng et al., 16 Mar 2026).

The broader literature represented here uses the word “dominance” in distinct technical senses. In nonlinear control, Forni and Sepulchre define strict $v$36-dominance for the smooth autonomous system $v$37 by means of a quadratic differential storage $v$38 with $v$39 of inertia $v$40, and they show that bounded trajectories exhibit low-dimensional asymptotic behavior such as convergence to an equilibrium for $v$41 or to a simple attractor for $v$42 (Forni et al., 2017). In optimization and game theory, “gradient dominance” refers to the Polyak–Łojasiewicz condition

$v$43

and the 2026 general-sum game extension introduces the $v$44-sided PL condition

$v$45

as a basis for convergence guarantees of block-coordinate and adapted gradient methods toward Nash equilibria (Chao et al., 12 Feb 2026). Accordingly, within current technical usage, GD in the counting-complexity sense names a graph-signature framework for domination-type partition functions, rather than the control-theoretic or optimization-theoretic notions that use “dominance” in other ways.