Total Dominator Chromatic Number

• Total dominator chromatic number is a graph parameter for isolate-free graphs that combines proper coloring with the requirement that each vertex has all neighbors in at least one color class.
• Foundational work established basic bounds, exact values for classical families, and NP-completeness, while later studies refined its complexity and behavior under various graph operations.
• The parameter's analysis under graph modifications provides practical insights into structural properties and motivates ongoing research into related open problems.

The total dominator chromatic number is a graph parameter for isolate-free graphs that combines proper coloring with total domination. In the standard notations $\chi_{td}(G)$ and $\chi_d^t(G)$, it is the minimum number of color classes in a proper coloring of a graph $G$ such that each vertex is adjacent to every vertex of at least one color class. Kazemi’s foundational treatment established the basic bounds, extremal characterizations, exact values for several classical families, and NP-completeness, while later work reinterpreted the parameter through compelling colorings, refined the complexity landscape, and analyzed its behavior under a range of graph operations and derived constructions (Kazemi, 2013, Bachstein et al., 2021, Henning et al., 2023).

1. Definitions and equivalent formulations

Let $G=(V,E)$ be a finite simple graph with no isolated vertices. A proper $k$-coloring is a map $f:V\to\{1,\dots,k\}$ such that $f(u)\neq f(v)$ whenever $uv\in E$. If the color classes are $V_i=f^{-1}(i)$, then a total dominator coloring is a proper coloring with the additional property that for every vertex $v\in V$ there exists at least one color class $\chi_d^t(G)$0 satisfying $\chi_d^t(G)$1. In later formulations this is stated as the existence, for each vertex $\chi_d^t(G)$2, of a color class other than its own whose entire class lies in $\chi_d^t(G)$3 (Kazemi, 2013, Henning et al., 2023).

Two auxiliary notions recur in structural arguments. For a color class $\chi_d^t(G)$4, the common neighborhood is

$\chi_d^t(G)$5

and a vertex may be a private neighbor of $\chi_d^t(G)$6 with respect to a coloring if it lies in the common neighborhood of $\chi_d^t(G)$7 but in no common neighborhood of any other color class. These notions are used repeatedly in minimality proofs, especially in private-neighbor arguments and class-merging obstructions (Kazemi, 2013, Kazemi, 2013).

A later conceptual reformulation places total dominator coloring inside the framework of $\chi_d^t(G)$8-compelling colorings. If a rainbow committee is a choice of exactly one vertex from each color class, then $\chi_d^t(G)$9 is the minimum number of colors in a proper coloring for which every rainbow committee is a total dominating set. Equivalently,

$G$0

where $G$1 is the property “forms a total dominating set” and $G$2 is the property “the induced subgraph has no isolated vertex” (Bachstein et al., 2021). This viewpoint makes the parameter formally comparable with other domination-type chromatic numbers.

2. Fundamental bounds and extremal structure

The basic inequalities are immediate but central: $G$3 for an isolate-free graph of order $G$4. The lower bound reflects the simultaneous coloring and total-domination requirements; the upper bound is realized by the trivial coloring assigning a distinct color to each vertex. A stronger general upper bound is

$G$5

where the minimum is taken over all minimum total dominating sets $G$6, and in particular

$G$7

For $G$8-partite graphs one further has $G$9 (Kazemi, 2013, Henning et al., 2023).

For connected graphs of order $G=(V,E)$0, the extremal range is

$G=(V,E)$1

Moreover, $G=(V,E)$2 if and only if $G=(V,E)$3 is a complete bipartite graph, while $G=(V,E)$4 if and only if $G=(V,E)$5. Another structural equality occurs when $G=(V,E)$6 has a universal vertex: if $G=(V,E)$7, then $G=(V,E)$8. Kazemi also noted that there is no monotonicity of $G=(V,E)$9 versus $k$0 under $k$1 (Kazemi, 2013).

The parameter also behaves nontrivially under disjoint union. If $k$2 has $k$3 connected components $k$4, each without isolates, then

$k$5

Within the compelling-coloring framework, one further has

$k$6

so the total dominator chromatic number sits above both the ordinary chromatic number and the dominator chromatic number (Kazemi, 2013, Bachstein et al., 2021).

3. Exact values for standard families

Representative exact formulas recorded in the literature include the following (Kazemi, 2013, Alikhani et al., 2015, Jalilolghadr et al., 2020).

Family	Exact value
$k$7	$k$8
Complete $k$9-partite $f:V\to\{1,\dots,k\}$0	$f:V\to\{1,\dots,k\}$1
$f:V\to\{1,\dots,k\}$2	$f:V\to\{1,\dots,k\}$3 if $f:V\to\{1,\dots,k\}$4, and $f:V\to\{1,\dots,k\}$5 otherwise
$f:V\to\{1,\dots,k\}$6	$f:V\to\{1,\dots,k\}$7 if $f:V\to\{1,\dots,k\}$8 is even, $f:V\to\{1,\dots,k\}$9 if $f(u)\neq f(v)$0 is odd
$f(u)\neq f(v)$1 for $f(u)\neq f(v)$2	$f(u)\neq f(v)$3 if $f(u)\neq f(v)$4, and $f(u)\neq f(v)$5 if $f(u)\neq f(v)$6
$f(u)\neq f(v)$7 for $f(u)\neq f(v)$8	$f(u)\neq f(v)$9 if $uv\in E$0, and $uv\in E$1 if $uv\in E$2
Friendship graph $uv\in E$3	$uv\in E$4
Ladder $uv\in E$5	$uv\in E$6 if $uv\in E$7 is odd, $uv\in E$8 if $uv\in E$9 is even
Chain triangular cactus $V_i=f^{-1}(i)$0	$V_i=f^{-1}(i)$1
Ortho-chain square cactus $V_i=f^{-1}(i)$2	$V_i=f^{-1}(i)$3
$V_i=f^{-1}(i)$4	$V_i=f^{-1}(i)$5 if $V_i=f^{-1}(i)$6, and $V_i=f^{-1}(i)$7 if $V_i=f^{-1}(i)$8

The path formula is one of the most frequently reused templates in subsequent work, appearing again in studies of graph operations, coronas, and subdivisions. The wheel value is particularly simple because the universal center compresses the domination side of the constraint. For complements of paths and cycles, the values collapse to essentially $V_i=f^{-1}(i)$9 outside small exceptional orders.

The cycle case is recorded in the supplied summaries in two modular forms. Kazemi’s exposition states that if $v\in V$0 with $v\in V$1, then $v\in V$2 for $v\in V$3, $v\in V$4 for $v\in V$5 with $v\in V$6 or $v\in V$7, and $v\in V$8 for $v\in V$9, with $\chi_d^t(G)$00 and $\chi_d^t(G)$01. A later summary states the cycle formula as $\chi_d^t(G)$02 for $\chi_d^t(G)$03 and $\chi_d^t(G)$04 for $\chi_d^t(G)$05, again with $\chi_d^t(G)$06 treated exceptionally (Kazemi, 2013, Alikhani et al., 2015). Both summaries place cycles among the earliest families for which exact modular behavior was derived.

Trees form a major structural subclass. Early results expressed $\chi_d^t(G)$07 in terms of the set $\chi_d^t(G)$08 of support vertices and $\chi_d^t(G)$09: $\chi_d^t(G)$10, with equality when every nonleaf has a leaf neighbor, and also when $\chi_d^t(G)$11. For $\chi_d^t(G)$12, the value is $\chi_d^t(G)$13 if two leaves sit at distance $\chi_d^t(G)$14, and $\chi_d^t(G)$15 otherwise; for $\chi_d^t(G)$16 with center edge $\chi_d^t(G)$17, the values $\chi_d^t(G)$18, $\chi_d^t(G)$19, and $\chi_d^t(G)$20 arise according to the status of $\chi_d^t(G)$21 and $\chi_d^t(G)$22 as support vertices (Kazemi, 2013). Later work completed the trichotomy $\chi_d^t(G)$23: $\chi_d^t(G)$24 if and only if $\chi_d^t(G)$25, and $\chi_d^t(G)$26 is characterized by the existence of a minimum TD-set $\chi_d^t(G)$27 satisfying three explicit conditions involving private neighbors, forbidden pairs in $\chi_d^t(G)$28, and an induced independent set outside a specified neighborhood (Henning et al., 2023).

4. Complexity and algorithmic status

The decision problem TDCD asks whether an isolate-free graph $\chi_d^t(G)$29 admits a total dominator coloring with at most $\chi_d^t(G)$30 colors. It is NP-complete in general. One standard reduction adds a universal vertex $\chi_d^t(G)$31 and shows $\chi_d^t(G)$32, reducing from CHROMATIC NUMBER. Later work strengthened the hardness picture by proving that TDCD remains NP-complete on split graphs, connected bipartite graphs, and planar graphs (Kazemi, 2013, Henning et al., 2023).

The restricted-class hardness results are tied to class-specific inequalities. On split graphs, if $\chi_d^t(G)$33 is the clique part, then

$\chi_d^t(G)$34

and the dominator-coloring and total-dominator-coloring thresholds coincide at $\chi_d^t(G)$35. On bipartite graphs,

$\chi_d^t(G)$36

and the hardness proof uses the fact that a polynomial-time algorithm for $\chi_d^t(G)$37 would imply a $\chi_d^t(G)$38-approximation for $\chi_d^t(G)$39. On planar graphs the corresponding bound is

$\chi_d^t(G)$40

These results connect the parameter directly to approximation hardness for total domination (Henning et al., 2023).

Positive algorithmic results are known for several hereditary or recursively structured graph classes. Trees are polynomial-time solvable, with a linear-time dynamic-programming algorithm on rooted trees. For connected cographs,

$\chi_d^t(G)$41

and for disconnected cographs with $\chi_d^t(G)$42 components,

$\chi_d^t(G)$43

both computable in $\chi_d^t(G)$44 time via cotrees. Chain graphs satisfy $\chi_d^t(G)$45, with $\chi_d^t(G)$46 if and only if $\chi_d^t(G)$47, $\chi_d^t(G)$48 if and only if $\chi_d^t(G)$49, and $\chi_d^t(G)$50 if and only if $\chi_d^t(G)$51; a chain ordering yields linear-time computation in $\chi_d^t(G)$52 time (Henning et al., 2023). Within the compelling-coloring framework, deciding $\chi_d^t(G)$53 is NP-complete for fixed $\chi_d^t(G)$54, whereas testing $\chi_d^t(G)$55 can be done in polynomial time by an $\chi_d^t(G)$56-style routine (Bachstein et al., 2021).

5. Graph operations and derived constructions

A substantial portion of the later literature studies how $\chi_d^t(G)$57 changes under graph operations. For the join of connected graphs,

$\chi_d^t(G)$58

For the corona, if $\chi_d^t(G)$59 is connected of order $\chi_d^t(G)$60, then $\chi_d^t(G)$61, and more generally

$\chi_d^t(G)$62

For the neighborhood corona,

$\chi_d^t(G)$63

and for $\chi_d^t(G)$64-gluing,

$\chi_d^t(G)$65

(Alikhani et al., 2015, Ghanbari et al., 2017).

Kazemi’s Mycielski study proved the sharp dichotomy

$\chi_d^t(G)$66

The increment is $\chi_d^t(G)$67 exactly for graphs in Class 1, namely those admitting a $\chi_d^t(G)$68-coloring with a color class of empty private neighborhood; otherwise the increment is $\chi_d^t(G)$69 (Kazemi, 2013).

Local graph modifications also admit quantitative bounds. If $\chi_d^t(G)$70 is a non-bridge in a connected graph $\chi_d^t(G)$71, then

$\chi_d^t(G)$72

If $\chi_d^t(G)$73 is a non-cut vertex, then

$\chi_d^t(G)$74

For edge contraction,

$\chi_d^t(G)$75

and for vertex contraction $\chi_d^t(G)$76,

$\chi_d^t(G)$77

The neighborhood-sparsification operation $\chi_d^t(G)$78, obtained by deleting all edges among neighbors of $\chi_d^t(G)$79, satisfies

$\chi_d^t(G)$80

(Ghanbari et al., 2016).

Derived graph constructions have produced additional exact families. For the $\chi_d^t(G)$81-subdivision $\chi_d^t(G)$82, one has monotonicity in $\chi_d^t(G)$83 together with general path-based bounds, including

$\chi_d^t(G)$84

where $\chi_d^t(G)$85 (Alikhani et al., 2018). For middle graphs and central graphs, exact values are known for paths, cycles, wheels, complete graphs, complete multipartite graphs, stars, double-stars, and friendship graphs; for example,

$\chi_d^t(G)$86

(Kazemnejad et al., 2021, Kazemnejad et al., 2018).

6. Related parameters, stability, and open problems

The parameter has inspired total-coloring analogues. If $\chi_d^t(G)$87 is the total graph of $\chi_d^t(G)$88, then the total dominator total chromatic number satisfies

$\chi_d^t(G)$89

This identity underlies exact formulas for cycles, paths, wheels, complete bipartite graphs, and complete graphs in the total-coloring setting, and imports the total-dominator-coloring machinery directly to total graphs (Kazemi et al., 2019, Kazemi et al., 2020).

Sensitivity questions have also been formalized. The TDC-stability $\chi_d^t(G)$90 is the minimum number of vertices whose removal alters $\chi_d^t(G)$91, and the TDC-bondage number $\chi_d^t(G)$92 is the minimum number of edges whose removal alters it. For paths $\chi_d^t(G)$93, both parameters equal $\chi_d^t(G)$94; for cycles $\chi_d^t(G)$95, the values depend on $\chi_d^t(G)$96; for friendship graphs $\chi_d^t(G)$97, both again equal $\chi_d^t(G)$98. The same work gives the Nordhaus–Gaddum-type lower bounds

$\chi_d^t(G)$99

(Ghanbari et al., 2017).

Specialized graph families continue to expand the catalogue of exact values. For $G$00, Jalilolghadr and Behtoei proved

$G$01

using decompositions of $G$02 into starlike and triangular classes together with a nonexistence argument involving Steiner triple systems of order $G$03 (Jalilolghadr et al., 2020). For circulant graphs $G$04 with $G$05, $G$06, and $G$07, the graph is isomorphic to $G$08, and a piecewise exact formula is given in terms of $G$09 (Kazemi et al., 2019).

Several open problems remain central. Kazemi asked for the determination of $G$10 for trees of diameter at least $G$11, Nordhaus–Gaddum-type bounds for $G$12 and $G$13, characterization of graphs with $G$14 for each fixed $G$15, and characterization of graphs satisfying equalities such as $G$16, $G$17, and

$G$18

(Kazemi, 2013). Later work added class-specific questions, including whether the $G$19 test for the tree condition $G$20 can be improved to truly linear time, and classification problems for middle graphs such as $G$21 or $G$22 (Henning et al., 2023, Kazemnejad et al., 2021). These problems indicate that, despite a substantial exact and algorithmic literature, the total dominator chromatic number remains structurally incomplete even on natural sparse and transformed graph classes.