Dominant Vertices in Graphs and Networks

• Dominant vertices are defined contextually, ranging from a universal vertex in graph domination to nodes that govern spectral measures and Boolean network dynamics.
• In triangulations, a dominant vertex uniquely concentrates curvature, reflecting its special role in controlling mesh geometry and topological properties.
• The term encapsulates diverse, context-specific notions of extremality, influencing coverage, dynamics, spectral behavior, and convex geometry across disciplines.

Searching arXiv for recent and relevant papers on “dominant vertices” across graph theory, Boolean networks, spectral graph theory, and mesh geometry. arXiv search query: "all:dominant vertices graph Boolean network spectral measure triangulation" “Dominant vertices” is not a uniform term across the literature. In classical graph domination, a dominating vertex is a vertex adjacent to all other vertices, equivalently the case $\gamma(G)=1$. In other research areas, the same phrase denotes vertices whose local spectral measures dominate all others, vertices or sets whose states determine the asymptotic dynamics of a Boolean network, the unique irregular interior vertex in an otherwise valence-$6$ triangulation of a disc, or vertices of intersection polytopes in the dominant chamber. A common pattern is extremality: a dominant object controls coverage, spectral type, dynamics, curvature concentration, or an extremal ray (Liu et al., 2010, Bruchez et al., 2022, España et al., 3 Sep 2025, Morvan, 2018, Besson et al., 2020).

1. Classical graph-theoretic meaning

In the standard graph-theoretic sense, a dominating set of a graph $G=(V,E)$ is a subset $S\subseteq V$ such that every vertex of $G$ not in $S$ has a neighbor in $S$. Equivalently, $$\gamma(G)=\min\{|S|:S\subseteq V(G),\forall v\in V(G)\setminus S,\exists u\in S\text{ s.t. }uv\in E(G)\}$$. A dominating vertex is the extreme case $|S|=1$: $v\in V$ is dominating iff $6$0, equivalently $6$1, and in this case $6$2. The same framework extends to $6$3-ruling sets, where $6$4 is a $6$5-ruling set iff $6$6; dominating sets are exactly $6$7-ruling sets (Koster, 2022).

The term should be distinguished from several nearby notions. A vertex may belong to all minimum dominating sets, to some minimum dominating sets but not all, or to no minimum dominating set. These classes are the paper’s $6$8, $6$9, and $G=(V,E)$0, respectively, where $G=(V,E)$1, $G=(V,E)$2, and $G=(V,E)$3. This is a different classification from being a dominating vertex: a vertex can be forced into every minimum dominating set without itself being universal (Bouquet et al., 2019).

Context	Meaning of “dominant” or “dominating” vertex	Characteristic condition
Graph domination	Universal vertex	$G=(V,E)$4, $G=(V,E)$5, $G=(V,E)$6
Spectral graph theory	Vertex whose local spectral measure dominates all others	$G=(V,E)$7 for all $G=(V,E)$8
Boolean networks	Singleton case of a dominant set	Every directed cycle meets the chosen vertex
Triangle meshes	Unique irregular interior vertex	Type $G=(V,E)$9: all interior valences $S\subseteq V$0 except one interior vertex of valence $S\subseteq V$1
Generalized Kostka cones	Vertex of $S\subseteq V$2 in the dominant chamber	$S\subseteq V$3 for $S\subseteq V$4

2. Dominating vertices inside triangulations and surface embeddings

In triangulations, the dominating-vertex case is highly constrained. A dominating set $S\subseteq V$5 satisfies $S\subseteq V$6, $S\subseteq V$7 or $S\subseteq V$8 is adjacent to a vertex in $S\subseteq V$9; the domination number is $G$0; and a dominating vertex is the special case $G$1. For plane triangulations, Matheson and Tarjan proved $G$2, constructed infinitely many plane triangulations requiring $G$3 vertices to dominate, and conjectured that $G$4 is the correct asymptotic upper bound for plane triangulations with at most finitely many exceptions. King and Pelsmajer proved that there exists $G$5 such that if $G$6 is a plane triangulation with $G$7 and $G$8, then $G$9. Liu and Pelsmajer sharpened this further: there exists a constant $S$0 such that any $S$1-vertex plane triangulation with maximum degree at most $S$2 has $S$3, and the proof yields the explicit constant $S$4. They also proved that for any surface $S$5, integer $S$6, and $S$7, there exists $S$8 such that if $S$9 is an $S$0-vertex triangulation on $S$1 and at most $S$2 vertices have degree $S$3, then $S$4, with $S$5 (Liu et al., 2010).

These asymptotic bounds place the $S$6 case at the far end of the domination spectrum. In a plane triangulation, a dominating vertex forces the graph to be the wheel $S$7, since the neighbors of the dominating vertex must induce a cycle to realize a triangulation. Wheel graphs $S$8 are therefore the canonical plane-triangulation examples with $S$9. In the bounded-degree regime $$\gamma(G)=\min\{|S|:S\subseteq V(G),\forall v\in V(G)\setminus S,\exists u\in S\text{ s.t. }uv\in E(G)\}$$0, a dominating vertex is impossible when $$\gamma(G)=\min\{|S|:S\subseteq V(G),\forall v\in V(G)\setminus S,\exists u\in S\text{ s.t. }uv\in E(G)\}$$1 because $$\gamma(G)=\min\{|S|:S\subseteq V(G),\forall v\in V(G)\setminus S,\exists u\in S\text{ s.t. }uv\in E(G)\}$$2. More generally, the existence of a dominating vertex on a surface requires a vertex of degree $$\gamma(G)=\min\{|S|:S\subseteq V(G),\forall v\in V(G)\setminus S,\exists u\in S\text{ s.t. }uv\in E(G)\}$$3, which is incompatible with the bounded-degree or near-regular triangulation regime analyzed in the near-$$\gamma(G)=\min\{|S|:S\subseteq V(G),\forall v\in V(G)\setminus S,\exists u\in S\text{ s.t. }uv\in E(G)\}$$4-regular theory (Liu et al., 2010).

The structural reason for the $$\gamma(G)=\min\{|S|:S\subseteq V(G),\forall v\in V(G)\setminus S,\exists u\in S\text{ s.t. }uv\in E(G)\}$$5 phenomenon is local $$\gamma(G)=\min\{|S|:S\subseteq V(G),\forall v\in V(G)\setminus S,\exists u\in S\text{ s.t. }uv\in E(G)\}$$6-regularity. The paper’s hexagonal tiling analogy notes that a near-$$\gamma(G)=\min\{|S|:S\subseteq V(G),\forall v\in V(G)\setminus S,\exists u\in S\text{ s.t. }uv\in E(G)\}$$7-regular triangulation locally resembles the infinite $$\gamma(G)=\min\{|S|:S\subseteq V(G),\forall v\in V(G)\setminus S,\exists u\in S\text{ s.t. }uv\in E(G)\}$$8-regular triangular grid $$\gamma(G)=\min\{|S|:S\subseteq V(G),\forall v\in V(G)\setminus S,\exists u\in S\text{ s.t. }uv\in E(G)\}$$9, in which a periodic pattern using every seventh vertex forms a dominating set $|S|=1$0. The finite proofs pull such patterns back from $|S|=1$1, compensate for exceptional vertices by a Steiner-tree construction, and control cylindrical substructures whose interiors are $|S|=1$2-regular. This suggests that a universal vertex is not the generic mechanism in triangulations; sparse, periodic domination is.

3. Minimum dominating sets, face-hitting partitions, and relaxed coverage

A frequent misconception is that a “dominant” vertex must be a universal vertex. In the minimum-domination literature, the central question is often whether a vertex belongs to all, some, or none minimum dominating sets. The paper on this question defines $|S|=1$3, $|S|=1$4, and $|S|=1$5, and gives characterizations via the effect of vertex deletion on $|S|=1$6. One notable test is the leaf-addition criterion: if $|S|=1$7 is obtained by adding a new leaf $|S|=1$8 adjacent only to $|S|=1$9, then $v\in V$0 if and only if $v\in V$1. Thus a vertex can be excluded from every minimum dominating set without any statement about universality or degree $v\in V$2 (Bouquet et al., 2019).

Plane embeddings add another layer. A face-hitting set of a plane graph $v\in V$3 is a set $v\in V$4 such that every face of $v\in V$5 contains at least one vertex of $v\in V$6. The main partition theorem states that for every plane (multi-)graph $v\in V$7 with no isolated vertices, no self-loops, and no $v\in V$8-faces, the vertex set admits a partition $v\in V$9 such that both $6$00 and $6$01 are dominating and face-hitting. As a corollary, every $6$02-vertex simple plane triangulation has a dominating set of size at most $6$03, where $6$04 is the maximum size of an independent set in the triangulation. The same paper states that Christiansen–Rotenberg–Rutschmann proved that every plane triangulation on $6$05 vertices has a dominating set of size at most $6$06, and that the corollary improves this bound when $6$07 for a maximal independent set or when $6$08 (Francis et al., 2024).

Relaxations of domination replace the one-hop condition by distance-$6$09 coverage. The reduction in “On Relaxation of Dominant Sets” constructs a graph $6$10 in which adjacency encodes distance at most $6$11 in the original graph, so that a dominating set in $6$12 is exactly a $6$13-ruling set in $6$14. In that language, a single vertex $6$15 is a $6$16-ruling set iff $6$17. The same reduction supports $6$18-ruling sets by applying a maximal independent set algorithm after closure, thereby separating the universal-vertex question from broader distance-dominance questions (Koster, 2022).

4. Spectral graph theory: dominant vertices as scalar spectral measures

For graphs of bounded degree, the phrase has a different meaning. Let $6$19 be a countable graph of bounded degree, and let $6$20 be the adjacency operator on $6$21. Each vector $6$22 defines a local spectral measure $6$23 on the spectrum $6$24, and each vertex $6$25 defines $6$26. A vertex $6$27 is dominant if, for all $6$28, the measure $6$29 is absolutely continuous with respect to $6$30, written $6$31. It then follows that $6$32 for all $6$33. In operator-theoretic language, $6$34 is a scalar spectral measure for $6$35 (Bruchez et al., 2022).

This notion is neither combinatorial domination nor universality of adjacency. It is a statement about which spectral null sets are visible from a vertex. In finite graphs, a vertex is dominant iff its local measure has positive mass on every eigenvalue, equivalently iff the minimal polynomial of $6$36 equals the minimal polynomial of $6$37, equivalently iff $6$38 are linearly independent, where $6$39 is the number of distinct eigenvalues. In infinite graphs, cyclicity provides a sufficient condition: any cyclic vector is dominant, and if $6$40 admits a cyclic vector then dominant $6$41 cyclic (Bruchez et al., 2022).

The paper emphasizes that all dominance regimes occur. In vertex-transitive graphs, $6$42 for all vertices, so every vertex is dominant; the same holds in walk-regular graphs. Finite paths $6$43 have dominant vertices precisely at positions $6$44 coprime to $6$45. In $6$46, when $6$47, all leaves are dominant and the center is not, while for $6$48 all vertices are dominant. There are also graphs without dominant vertices at all, including connected finite families $6$49, certain trees $6$50, and complements of $6$51. For infinite graphs, every vertex of the infinite ray is cyclic and dominant, while in infinite stars $6$52 with $6$53, all non-core vertices are dominant but no vertex is cyclic (Bruchez et al., 2022).

5. Boolean networks: dominant vertices as dynamical determinants

In Boolean-network theory, “dominant vertices” are not universal neighbors but nodes whose evolution determines the whole network’s dynamics after a transient time. Let $6$54 be a finite directed graph with state space $6$55 and synchronous update map $6$56. For $6$57, define $6$58, where $6$59 is the in-neighborhood. One says $6$60 determines $6$61 if $6$62, and defines the boundary $6$63. A set $6$64 is dominant if repeated addition of the boundary reaches the whole graph: $6$65, $6$66, and for some finite $6$67 one has $6$68 with strict inclusions $6$69. Equivalently, $6$70 is dominant iff every directed cycle of $6$71 contains at least one vertex of $6$72; when indegrees are all positive, dominant sets coincide with feedback vertex sets (España et al., 3 Sep 2025).

The dynamical consequence is explicit. If $6$73 is dominant with depth $6$74, and two trajectories satisfy $6$75 for $6$76, then $6$77 for all $6$78. The paper defines a reduced graph $6$79, a recurrence length $6$80, and an induced automata network $6$81 on the delay space $6$82 with shift-register form

$6$83

There is a factor map

$6$84

such that $6$85, and $6$86 is injective on attractors. Hence the original and induced systems are asymptotically equivalent on periodic dynamics (España et al., 3 Sep 2025).

The singleton case is especially close to the phrase “dominant vertex.” A clover network is a directed graph with distinguished hub $6$87 such that every cycle passes through $6$88; then $6$89 is dominant and minimal. For signed majority-rule dynamics, the induced map on the single dominant vertex has scalar delayed form

$6$90

where the coefficients $6$91 are cycle-sign sums. The paper derives topology-only bounds on periodic points, maximal prime period, number of attractors, basin sizes, and transient lengths in terms of $6$92, $6$93, and $6$94, and then numerically explores clover ensembles with a single dominant vertex (España et al., 3 Sep 2025).

6. Geometric and algebraic usages beyond domination theory

In discrete differential geometry, the term refers to a unique irregular vertex rather than to neighborhood coverage. The paper on triangle meshes studies triangulations of a closed topological disc $6$95 in which all interior vertices have valence $6$96, except for a single interior vertex whose valence is different from $6$97. A triangulation of type $6$98 is one in which every interior vertex has valence $6$99 except exactly one interior vertex $G=(V,E)$00, the irregular or dominant vertex, whose valence is $G=(V,E)$01. Under the equilateral Euclidean structure, this vertex carries cone angle $G=(V,E)$02 and discrete curvature $G=(V,E)$03. The main theorem states that if $G=(V,E)$04 is not a multiple of $G=(V,E)$05, then, up to equivalence, there exists at most one triangulation adapted to the given boundary and dominant data. By contrast, if $G=(V,E)$06 with $G=(V,E)$07, the paper exhibits non-isomorphic triangulations on $G=(V,E)$08 with the same boundary and a unique irregular vertex of valence $G=(V,E)$09 (Morvan, 2018).

This geometric notion concentrates all interior curvature at one vertex. The rigidity-versus-flexibility dichotomy is controlled by holonomy: when $G=(V,E)$10, the holonomy angle $G=(V,E)$11 is not an integer multiple of $G=(V,E)$12, and the developed boundary determines the apex uniquely; when $G=(V,E)$13, the holonomy is trivial modulo $G=(V,E)$14, allowing multiple gluings that preserve the boundary and dominant valence but alter the interior combinatorics (Morvan, 2018).

An algebraic use appears in the theory of generalized Kostka cones. For a connected reductive group $G=(V,E)$15, the generalized Kostka cone $G=(V,E)$16 is generated by pairs $G=(V,E)$17 where $G=(V,E)$18 are dominant integral weights and $G=(V,E)$19. Fixing $G=(V,E)$20, the intersection polytope $G=(V,E)$21 is the dominant-chamber slice of the Weyl polytope. Its vertices are exactly the points $G=(V,E)$22 determined by a partition $G=(V,E)$23, and they admit the averaging formula

$G=(V,E)$24

These dominant-chamber vertices induce the extremal rays of $G=(V,E)$25. In particular, the extremal rays are all of the form $G=(V,E)$26, where $G=(V,E)$27 is a fundamental weight and $G=(V,E)$28 is a vertex of $G=(V,E)$29; equivalently, for fixed $G=(V,E)$30, connected Levi subdiagrams containing node $G=(V,E)$31 enumerate the nontrivial vertices of $G=(V,E)$32 (Besson et al., 2020).

These two usages are terminologically distinct from graph domination. In the mesh setting, a dominant vertex is a singularity carrying the nonzero discrete curvature of an otherwise flat equilateral triangulation. In the generalized Kostka setting, dominant vertices are extremal points of a polytope in the dominant chamber, organized by Levi subgroups and inverse-transpose Cartan data. The shared vocabulary reflects dominance in curvature or convex geometry, not neighborhood coverage.

7. Conceptual relations and persistent ambiguities

Across these literatures, the phrase “dominant vertices” should be read contextually. In graph domination, it means a universal vertex and implies $G=(V,E)$33. In minimum-domination theory, the practically relevant trichotomy is often $G=(V,E)$34, $G=(V,E)$35, and $G=(V,E)$36, not the universal-vertex condition. In bounded-degree spectral graph theory, dominance is absolute continuity of spectral measures. In Boolean networks, dominant sets are cycle-hitting sets that determine asymptotic dynamics after a uniform transient. In triangulated-mesh geometry, the dominant vertex is the unique interior valence defect. In generalized Kostka cones, dominant vertices are vertices of $G=(V,E)$37 producing extremal rays (Bouquet et al., 2019, Bruchez et al., 2022, España et al., 3 Sep 2025, Morvan, 2018, Besson et al., 2020).

A useful way to separate these meanings is by the object being controlled. Classical domination controls vertices by adjacency; face-hitting refinements control faces as well as vertices; spectral dominance controls spectral null sets; Boolean dominance controls future orbits and attractors; mesh dominance controls curvature localization; and Kostka-cone dominance controls extremal convex data. This suggests that “dominant vertex” is less a single definition than a family of extremal notions indexed by the ambient theory.