Orientable Total Domination Numbers

• Orientable total domination numbers are parameters that quantify the minimum and maximum sizes of total dominating sets across all valid graph orientations where every vertex has at least one incoming edge.
• The study identifies specific graph families (F1, F2, F3) for which the upper orientable total domination number equals |V|-1, highlighting key degree constraints and cycle-related structures.
• Extremal gap analysis reveals that the difference between the upper and lower orientable total domination numbers can reach |V|-4, providing insights for optimizing domination in graph orientations.

An orientable total domination number is a parameter that measures, for a given undirected graph, the extremal values (minimum and maximum) of the total domination number under all valid orientations—that is, all assignments of directions to edges where every vertex has indegree at least one. This concept lies at the intersection of domination theory, extremal graph orientation, and structural characterizations, and has been the focus of recent rigorous characterizations and extremal constructions.

1. Definitions and Fundamental Concepts

Let $G=(V,E)$ be a finite, simple, undirected graph. An orientation of $G$ is a digraph $D=(V,A)$ formed by replacing each edge $uv\in E$ with exactly one of the arcs $uv$ or $vu$. An orientation is valid if every vertex has indegree at least 1: $\forall v\in V:~ d_-^D(v)\ge1.$ Given a digraph $D$, a set $S\subseteq V$ is a total dominating set if for every $v\in V$,

$G$0

where $G$1 is the set of in-neighbors of $G$2. The total domination number $G$3 is the minimum size of such $G$4.

For an undirected graph $G$5, the lower and upper orientable total domination numbers, denoted $G$6 and $G$7, are the minimum and maximum, respectively, of $G$8 over all valid orientations $G$9 of $D=(V,A)$0: $D=(V,A)$1 These parameters are only defined for graphs in the class $D=(V,A)$2, consisting of graphs where every component contains at least one cycle—ensuring the existence of a valid orientation and hence a total dominating set (Blázsik et al., 2024, Anderson et al., 2023).

2. Main Structural Characterization: $D=(V,A)$3

A comprehensive structural result identifies all graphs that achieve the second-largest possible value of $D=(V,A)$4. The main theorem (Blázsik et al., 2024) can be stated as follows:

Let $D=(V,A)$5 be connected. Then,

$D=(V,A)$6

where the families $D=(V,A)$7, $D=(V,A)$8, $D=(V,A)$9 are defined as:

• $uv\in E$0: Connected graphs with exactly one degree-1 vertex $uv\in E$1, every other vertex of degree at least 2, and a unique path $uv\in E$2 ($uv\in E$3); all non-path edges are incident from $uv\in E$4 to (zero or more) vertex-disjoint cycles.
• $uv\in E$5: Connected graphs of minimum degree at least 2, consisting of $uv\in E$6 vertex-disjoint cycles plus one additional vertex $uv\in E$7, where all extra edges are incident to $uv\in E$8 and $uv\in E$9 is adjacent to at least one vertex of each cycle (with $uv$0).
• $uv$1: Formed from graphs in $uv$2 by adding one or two edges incident to the unique leaf $uv$3 under precise adjacency constraints, ensuring valid orientation is preserved.

If $uv$4 is disconnected, $uv$5 holds if and only if all but one component are cycles and the remaining component belongs to $uv$6 (Blázsik et al., 2024).

Auxiliary degree constraints for any extremal orientation $uv$7 realizing $uv$8 include $uv$9 for all $vu$0, at most one vertex with $vu$1, and a restriction on the union of out-neighborhoods: for any $vu$2, $vu$3. These properties drive the proof by constraining the orientation structure.

3. Extremal Gaps and Additive Discrepancy

The structure of $vu$4, $vu$5, and $vu$6 allows for the existence of graphs where the upper and lower orientable total domination numbers are maximally separated. Explicit constructions yield graphs $vu$7 such that: $vu$8 so the gap is $vu$9 and the ratio $\forall v\in V:~ d_-^D(v)\ge1.$0. A canonical example uses $\forall v\in V:~ d_-^D(v)\ge1.$1 disjoint directed 3-cycles, a universal vertex $\forall v\in V:~ d_-^D(v)\ge1.$2 adjacent to all cycle vertices, and appropriate orientations: one realizing $\forall v\in V:~ d_-^D(v)\ge1.$3 (by orienting all cycles and edges towards $\forall v\in V:~ d_-^D(v)\ge1.$4), another achieving $\forall v\in V:~ d_-^D(v)\ge1.$5 by reversing cycle orientations except for three arcs entering $\forall v\in V:~ d_-^D(v)\ge1.$6 (Blázsik et al., 2024).

This demonstrates that for simple graphs, the largest possible additive difference between the extremal orientable total domination numbers is $\forall v\in V:~ d_-^D(v)\ge1.$7.

4. Connection to Classical Total Domination and