Total Domination Number in Graph Theory

• Total domination number is the minimum size of a vertex set in a graph such that every vertex has a neighbor, ensuring complete coverage.
• Lower bounds based on degree sequences, maximum degree, and distance metrics provide robust methods for analyzing this invariant in various graph classes.
• Applications span diverse structures from toroidal meshes to zero-divisor graphs, employing algorithmic and structural techniques to tackle computational challenges.

A total dominating set in a graph is a subset of vertices such that every vertex has at least one neighbor in the set. The total domination number, denoted $\gamma_t(G)$, is the minimum cardinality of such a set in $G$. This invariant is central in domination theory and intersects with areas including graph products, algebraic graph theory (e.g., zero-divisor graphs of rings), extremal combinatorics, and the study of specific graph classes (e.g., planar graphs, meshes, Knödel graphs).

1. Definition and Basic Properties

Let $G=(V,E)$ be a simple undirected graph with no isolated vertices. A subset $S \subseteq V$ is a total dominating set (TDS) if every vertex $v\in V$ has a neighbor in $S$, i.e.,

$\forall\,v\in V,\;\; N(v) \cap S \neq \emptyset.$

The total domination number is

$$\gamma_t(G) = \min\Bigl\{ |S| : S \subseteq V,\; N(v) \cap S \neq \emptyset\;\forall\,v\in V \Bigr\}.$$

By definition, $\gamma_t(G) \geq 2$ for connected graphs with minimum degree at least 1, and always $\gamma(G) \leq \gamma_t(G)$, where $G$0 is the (standard) domination number (Davila, 2024).

For digraphs $G$1 (no loops, simple), total domination requires every vertex to have an in-neighbor in $G$2: $G$3 The total domination number in digraphs, $G$4, is analogously defined, with the existence of a TDS equivalent to $G$5 (Blázsik et al., 2024).

2. Lower Bounds and Extremal Cases

Structural lower bounds for $G$6 are foundational for applications and complexity analyses. Principal lower bounds include:

• Degree-sequence bound: For $G$7 with degree sequence $G$8,

$G$9

• Maximum-degree bound:

$G=(V,E)$0

where $G=(V,E)$1 is the maximum degree.

• Diameter and radius bounds: For connected $G=(V,E)$2,

$G=(V,E)$3

These bounds generalize from the total (distance) $G=(V,E)$4-domination framework; for $G=(V,E)$5, all reduce to bounds for $G=(V,E)$6 (Davila, 2024).

Tightness: These bounds are attained in several key families:

• Paths $G=(V,E)$7: $G=(V,E)$8 and equality in the diameter bound.
• Even cycles $G=(V,E)$9: $S \subseteq V$0.
• Regular graphs, including cycles, realize the $S \subseteq V$1 bound.

However, such bounds can be weak for star-like or highly connected graphs; further refinements might combine local (degree) and global (distance) parameters or exploit neighborhood overlaps (Davila, 2024).

3. Exact Results in Notable Graph Classes

Toroidal Meshes and Cubic Knödel Graphs

• Toroidal meshes $S \subseteq V$2 (Cartesian product of cycles): For $S \subseteq V$3, the total domination number is determined exactly:
  • $S \subseteq V$4: $S \subseteq V$5
  • $S \subseteq V$6: $S \subseteq V$7 follows a sharp residue-dependent formula (see (Hu et al., 2011)).
• For general $S \subseteq V$8, the asymptotic is $S \subseteq V$9 with explicit block constructions providing upper bounds and the regular-graph argument providing the lower bound (Hu et al., 2011).
• Cubic Knödel graphs $v\in V$0 (bipartite, 3-regular, $v\in V$1 even $v\in V$2): The total domination number is given by a piecewise function in $v\in V$3 mod 10:

$v\in V$4

where $v\in V$5 (Mojdeh et al., 2018).

Hypercubes and Prisms

For hypercubes $v\in V$6 and their prisms, the following holds:

• For bipartite $v\in V$7, $v\in V$8.
• For hypercubes, $v\in V$9, and explicit values can be recursively computed (Azarija et al., 2016).

For non-bipartite $S$0, $S$1 can be strictly less than $S$2; the gap can be arbitrarily large (Azarija et al., 2016).

Planar Graphs and Near-Triangulations

For near-triangulations $S$3 (biconnected planar graphs with all faces except possibly the outer being triangles), it is shown that

$S$4

for all $S$5 except in two exceptional graphs of order 12 (Claverol et al., 2020).

4. Total Domination in Algebraic Graphs

In zero-divisor graphs of commutative rings $S$6, the total domination number often coincides with the standard domination number. The main result is: $S$7 The sole exception is $S$8, where $S$9 and $\forall\,v\in V,\;\; N(v) \cap S \neq \emptyset.$0 (Anderson et al., 3 Jun 2025). This equality is established by analyzing girth, loop structure, and universal annihilators within the ring.

5. Orientations and Extremal Constructions

For a simple undirected $\forall\,v\in V,\;\; N(v) \cap S \neq \emptyset.$1, orientable total domination numbers are defined as follows:

• $\forall\,v\in V,\;\; N(v) \cap S \neq \emptyset.$2: The maximum total domination number over all valid orientations (each vertex has indegree at least one).
• $\forall\,v\in V,\;\; N(v) \cap S \neq \emptyset.$3: The minimum over valid orientations.

Key results include:

• $\forall\,v\in V,\;\; N(v) \cap S \neq \emptyset.$4 if and only if $\forall\,v\in V,\;\; N(v) \cap S \neq \emptyset.$5 is a disjoint union of cycles.
• All connected graphs $\forall\,v\in V,\;\; N(v) \cap S \neq \emptyset.$6 in specific families $\forall\,v\in V,\;\; N(v) \cap S \neq \emptyset.$7 can satisfy $\forall\,v\in V,\;\; N(v) \cap S \neq \emptyset.$8, and the difference between $\forall\,v\in V,\;\; N(v) \cap S \neq \emptyset.$9 and $$\gamma_t(G) = \min\Bigl\{ |S| : S \subseteq V,\; N(v) \cap S \neq \emptyset\;\forall\,v\in V \Bigr\}.$$0 can be as large as $$\gamma_t(G) = \min\Bigl\{ |S| : S \subseteq V,\; N(v) \cap S \neq \emptyset\;\forall\,v\in V \Bigr\}.$$1 by suitable orientation choices (Blázsik et al., 2024).

6. Algorithmic and Structural Techniques

The computation of $$\gamma_t(G) = \min\Bigl\{ |S| : S \subseteq V,\; N(v) \cap S \neq \emptyset\;\forall\,v\in V \Bigr\}.$$2 is combinatorially complex for general graphs. Proof techniques include:

• Double counting and neighborhood overlap analysis: Exploited in regular graphs and product graphs (Hu et al., 2011, Mojdeh et al., 2018).
• Inductive and decomposition approaches: Used in planar and near-triangulation settings through vertex deletion, edge contraction, and reduction to smaller subgraphs (Claverol et al., 2020).
• Hypergraph transversals: Critical for establishing equivalences with hitting set problems, particularly in product and bipartite graphs (Azarija et al., 2016).

7. Open Problems and Extensions

Significant open questions remain, especially in product graphs. Determining $$\gamma_t(G) = \min\Bigl\{ |S| : S \subseteq V,\; N(v) \cap S \neq \emptyset\;\forall\,v\in V \Bigr\}.$$3 exactly for $$\gamma_t(G) = \min\Bigl\{ |S| : S \subseteq V,\; N(v) \cap S \neq \emptyset\;\forall\,v\in V \Bigr\}.$$4 is unresolved, with known bounds off by $$\gamma_t(G) = \min\Bigl\{ |S| : S \subseteq V,\; N(v) \cap S \neq \emptyset\;\forall\,v\in V \Bigr\}.$$5 depending on residue classes (Hu et al., 2011).

Directions for further research include:

• Combining local and global graph parameters to yield tighter lower bounds, e.g., refining $$\gamma_t(G) = \min\Bigl\{ |S| : S \subseteq V,\; N(v) \cap S \neq \emptyset\;\forall\,v\in V \Bigr\}.$$6 by accounting for higher-order neighborhood overlap (Davila, 2024).
• Investigation of weighted and fractional total domination parameters, with analogous lower bounds expected via combinatorial arguments.
• Structural bounds for orientable total domination numbers as explicit functions of edge density and other global invariants (Blázsik et al., 2024).

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Summary Table: Key Lower Bounds for $$\gamma_t(G) = \min\Bigl\{ |S| : S \subseteq V,\; N(v) \cap S \neq \emptyset\;\forall\,v\in V \Bigr\}.$$7 in Simple Graphs

Bound Type	Formula	Sharpness Example
Degree-sequence	$$\gamma_t(G) = \min\Bigl\{ |S| : S \subseteq V,\; N(v) \cap S \neq \emptyset\;\forall\,v\in V \Bigr\}.$$8	Paths, cycles
Max degree	$$\gamma_t(G) = \min\Bigl\{ |S| : S \subseteq V,\; N(v) \cap S \neq \emptyset\;\forall\,v\in V \Bigr\}.$$9	Regular graphs
Diameter	$\gamma_t(G) \geq 2$0	Paths, even cycles
Radius	$\gamma_t(G) \geq 2$1	Varies

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The total domination number serves as a robust quantitative invariant, deeply linked to combinatorial structure, graph classes, algebraic constructions, and product operations. The breadth of exact results, sharp bounds, and open questions underscores its foundational role in modern domination theory.