Hereditarily Linear Delaunay Graphs

• Hereditarily Linear Delaunay Graphs are Delaunay-type structures defined via families of linear orders and bounded Dushnik–Miller dimensions, forming the basis of TD-Delaunay complexes.
• They extend the planar theory by using regular simplices in Hₙ and coordinatewise extremality to generalize triangular distance concepts to higher-dimensional spaces.
• The framework employs a TD-Delaunay system solved via Farkas’ lemma and multi-flow techniques, illustrating that a bounded order-dimension is necessary but insufficient for geometric realizability in dimensions four and above.

Searching arXiv for papers relevant to Hereditarily Linear Delaunay Graphs and TD-Delaunay complexes. Hereditarily Linear Delaunay Graph, in the context of TD-Delaunay graphs and TD-Delaunay complexes, can be understood as a Delaunay-type structure whose geometry is encoded by families of linear orders and bounded Dushnik–Miller dimension. The relevant formal framework is the higher-dimensional theory of TD-Delaunay complexes, where TD stands for triangular distance. In that framework, every TD-Delaunay complex of $H_d \simeq \mathbb{R}^{d-1}$ has Dushnik–Miller dimension at most $d$; the converse holds for $d=2$ and $d=3$, but the conjectured extension to larger $d$ is disproved already for $d=4$ (Gonçalves et al., 2018).

1. Geometric origin in triangular-distance Delaunay theory

TD-Delaunay graphs are a variation of classical Delaunay triangulations obtained from a specific convex distance function. For a compact convex shape $S$ with interior point $c$, the associated convex or Minkowski distance between two points $p,q$ is the smallest $\lambda$ such that a translate of $d$0 centered at $d$1 contains $d$2. When $d$3 is an equilateral triangle, this yields the triangular distance.

In the plane, TD-Delaunay graphs were introduced by Chew. They are defined by empty homothetic equilateral triangles instead of empty disks. The planar theory has two structural features that motivate the later order-theoretic viewpoint: every triangulation is the TD-Delaunay graph of a set of points in $d$4, and conversely every TD-Delaunay graph is planar. The paper also emphasizes that these graphs are plane spanners and were historically important in computational geometry.

This planar equivalence is the source of the “linear” interpretation. A plausible reading is that planar TD-Delaunay geometry admits an exact combinatorial description, and that higher-dimensional analogues might also be characterized by purely order-theoretic data. The paper develops that program for simplicial complexes rather than only graphs.

2. Higher-dimensional triangular distance and positive simplices

The higher-dimensional generalization is obtained by replacing the equilateral triangle with a regular simplex in a $d$5-dimensional hyperplane of $d$6. The ambient hyperplane is defined by

$d$7

which is naturally identified with $d$8.

For $d$9, the regular simplex

$d=2$0

is introduced. Such an $d=2$1 is called positive if

$d=2$2

The canonical simplex is $d=2$3 with $d=2$4.

The key geometric fact is that the positive regular simplices in $d=2$5 are exactly the positive homothetic copies of the canonical simplex. Thus the higher-dimensional analogue of triangular distance is realized through empty positive simplices in $d=2$6. This shifts the Delaunay condition from Euclidean empty balls to a simplex-based convex distance model.

3. TD-Delaunay complexes

Given a finite point set $d=2$7 in general position, the TD-Delaunay complex $d=2$8 is the abstract simplicial complex whose faces are those subsets $d=2$9 for which there exists a positive regular simplex $d=3$0 such that $d=3$1 and no point of $d=3$2 lies in the interior of $d=3$3.

For a subset $d=3$4, the coordinatewise maximum vector is defined by

$d=3$5

The paper proves the criterion

$d=3$6

This gives a direct combinatorial-geometric test for membership in the complex.

The construction generalizes the graph case. In dimension $d=3$7, TD-Delaunay graphs are ordinary graph objects on planar point sets. In higher dimensions, the theory is expressed in terms of simplicial complexes. This is the precise setting in which a hereditary description by linear orders becomes possible: faces are determined by coordinatewise extremality and empty-simplex conditions rather than by pairwise adjacency alone.

4. Linear orders, Dushnik–Miller dimension, and $d=3$8

The combinatorial counterpart of the geometric theory is order dimension. For a poset $d=3$9, the Dushnik–Miller dimension is the smallest number of linear extensions whose intersection is the poset. For an abstract simplicial complex $d$0, its Dushnik–Miller dimension is the dimension of its inclusion poset $d$1, written $d$2 (Gonçalves et al., 2018).

Several recalled facts locate this invariant within geometric graph theory. One has $d$3 if and only if $d$4 is just a vertex, and $d$5 if and only if $d$6 is a union of paths. Schnyder’s theorem says a graph $d$7 is planar if and only if $d$8. The paper also recalls the theorem of Bayer et al. and Ossona de Mendez: if $d$9, then $d=4$0 has a straight-line embedding in $d=4$1.

To connect order dimension with TD-Delaunay geometry, the paper uses Schnyder-style representations. A $d=4$2-representation $d=4$3 is a set of $d=4$4 linear orders on the vertex set $d=4$5. For a subset $d=4$6, a vertex $d=4$7 dominates $d=4$8 in order $d=4$9 if every $S$0 satisfies $S$1. It dominates $S$2 in $S$3 if it dominates $S$4 in at least one of the orders. Then

$S$5

This $S$6 is always a simplicial complex.

The key theorem of Ossona de Mendez used in the paper is that $S$7 if and only if there exists a $S$8-representation $S$9 on $c$0 such that $c$1. Thus linear-order data form the order-theoretic side of the theory.

For a point set $c$2, one defines linear orders $c$3 by coordinate comparison,

$c$4

and writes $c$5. The paper proves the structural identity

$c$6

In particular, any TD-Delaunay complex in $c$7 has Dushnik–Miller dimension at most $c$8. This is the strongest formal basis for reading the theory as a “hereditarily linear” Delaunay framework: faces are exactly those subsets selected by coordinatewise dominance across $c$9 linear orders.

5. The conjectured converse and its failure in dimension four

The natural converse was conjectured independently by Mary and by Evans–Felsner–Kobourov–Ueckerdt: every $p,q$0-representation $p,q$1 should produce a TD-Delaunay complex $p,q$2 in $p,q$3. Equivalently, complexes of Dushnik–Miller dimension at most $p,q$4 would be exactly the TD-Delaunay complexes in $p,q$5.

The status before the counterexample was sharply dimension-dependent. For $p,q$6, the statement is true. For $p,q$7, the statement is also true. For $p,q$8, it was conjectural. In graph language, the case $p,q$9 corresponds to planar graphs, matching Schnyder’s theorem and the known TD-Delaunay characterization of planar graphs.

To decide when a representation is geometric, the paper introduces a matrix $\lambda$0 and the system

$\lambda$1

called the TD-Delaunay system. A simplicial complex $\lambda$2 is a TD-Delaunay complex of $\lambda$3 if and only if there exists a $\lambda$4-representation $\lambda$5 such that

$\lambda$6

and the corresponding TD-Delaunay system has a solution.

The obstruction theory uses Farkas’ lemma. If the TD-Delaunay system has no solution, then there is a nonzero dual certificate, reformulated as a multi-flow. From each order $\lambda$7, one obtains a directed graph $\lambda$8 by orienting each edge according to $\lambda$9. A multi-flow is a collection of flows $d$00 on these digraphs such that all vertices have equal divergence in every order. The key proposition is an alternative: either the TD-Delaunay system has a solution, or $d$01 admits a nonzero multi-flow, and not both (Gonçalves et al., 2018).

The main counterexample is a $d$02-representation on

$d$03

given by

$d$04

for which the simplicial complex $d$05 has Dushnik–Miller dimension $d$06 but is not a TD-Delaunay complex of $d$07.

The proof strategy is explicit. First, any $d$08-representation realizing $d$09 must be equivalent to the displayed one up to order permutation and relabeling of small elements. Second, for every such representation, a nonzero multi-flow is constructed. Third, by the Farkas alternative, a nonzero multi-flow implies that the TD-Delaunay system has no solution. Therefore $d$10 is not TD-Delaunay. The consequence is precise: Dushnik–Miller dimension $d$11 is strictly weaker than TD-Delaunay realizability in $d$12 for $d$13.

6. Rectangular Delaunay complexes and the scope of the linear characterization

The paper also compares TD-Delaunay complexes with rectangular Delaunay complexes. A rectangular Delaunay complex $d$14 is defined from axis-parallel empty rectangles in $d$15. It is shown that $d$16 corresponds to a special $d$17-representation where two orders are reversals of each other and similarly for the other pair. Therefore $d$18 has Dushnik–Miller dimension at most $d$19. More strongly, every rectangular Delaunay complex is also a TD-Delaunay complex of $d$20 (Gonçalves et al., 2018).

This comparison clarifies the range of the higher-dimensional theory. TD-Delaunay complexes strictly contain rectangular Delaunay complexes, but not all $d$21-dimensional order-dimension complexes are TD-Delaunay. The linear-order formalism is therefore exact on one side and incomplete on the other: every TD-Delaunay complex yields a bounded-dimension order representation, yet bounded order dimension alone does not characterize the geometric class once $d$22.

In graph language, the planar case remains exceptional. The original planar theorem can be read as saying that planar graphs are precisely the graphs of order dimension at most $d$23, and that these are exactly the subgraphs of TD-Delaunay graphs in the plane. The higher-dimensional generalization preserves the forward implication but breaks the converse. Interpreted through the phrase “hereditarily linear Delaunay graph,” this suggests a separation between a hereditary or order-theoretic condition and a geometric Delaunay-type realization condition: the former is necessary for TD-Delaunay realizability, but from dimension $d$24 onward it is not sufficient.