Stacked Graphs in Planar Triangulations

• Stacked Graphs are the 1-skeleton of a stacked triangulation, formed by recursively subdividing a triangle and also known as planar 3-trees or Apollonian networks.
• Their recursive structure allows for precise edge guarding, achieving bounds like ⌊2n/7⌋ guards for n vertices, which outperforms general triangulation bounds.
• The concept extends to higher dimensions and connects with threshold phenomena in random simplicial complexes, while remaining distinct from similarly named graph models.

A stacked graph, in the graph-theoretic sense used for planar triangulations, is the adjacency structure—or $1$-skeleton—of a stacked triangulation in dimension $2$. Starting from the triangle $K_3$, one repeatedly selects an inner face, inserts a new vertex inside it, and connects that vertex to the three vertices on the face boundary. The resulting graphs are also known as planar $3$-trees or Apollonian networks, and in the edge-guarding literature they coincide with the stacked triangulations that form the $3$-degenerate triangulations (Lubetzky et al., 2021, Jungeblut et al., 2020). In higher dimensions, the same recursive idea defines stacked triangulations of a simplex via repeated interior subdivisions, so the term “stacked graph” usually denotes the graph obtained by forgetting the higher-dimensional faces in the two-dimensional case (Lubetzky et al., 2021).

1. Definition in simplicial and graph-theoretic terms

A stacked triangulation of the $d$-simplex $\mathbf{o}=\{1,\ldots,d+1\}$, for $d \geq 2$, is a triangulation obtained by repeatedly subdividing a $d$-simplex into $d+1$ new ones via a new vertex in its interior (Lubetzky et al., 2021). In the language used there, a bistellar $2$0-move replaces a $2$1-simplex $2$2 by the $2$3 simplices $2$4, where $2$5 is the new interior vertex and $2$6.

For $2$7, the construction specializes to repeated subdivision of a triangle into three smaller triangles by a new interior vertex. The resulting object is a plane triangulation, and the stacked graph is its $2$8-skeleton: the graph consisting only of its vertices and edges (Lubetzky et al., 2021). This distinction matters. A stacked triangulation is a simplicial complex carrying its triangular faces, whereas a stacked graph records only adjacency.

The same family is described in the planar graph literature by an equivalent recursive formulation. One begins with $2$9; if $K_3$0 is any inner face of a stacked triangulation $K_3$1, then a new vertex $K_3$2 may be added inside $K_3$3 and connected to $K_3$4, subdividing $K_3$5 into three new triangular faces. The resulting graph is again a stacked triangulation (Jungeblut et al., 2020).

2. Equivalent descriptions and structural viewpoint

The literature attached to stacked triangulations supplies several synonymous descriptions. In the planar case, these graphs are called planar $K_3$6-trees or Apollonian networks (Jungeblut et al., 2020). The same source identifies them as the $K_3$7-degenerate triangulations, placing them inside a broader degeneracy-based hierarchy of planar graphs.

Their recursive construction gives them a particularly rigid internal organization. Every new vertex is introduced inside a pre-existing triangular face and is immediately adjacent to the three boundary vertices of that face. This yields a nested sequence of face refinements rather than an arbitrary triangulation process. In higher dimensions, the analogous sequence of bistellar $K_3$8-moves plays the same role, successively replacing one simplex by $K_3$9 smaller simplices (Lubetzky et al., 2021).

This recursive structure is reflected in both extremal and probabilistic results. In inductive proofs about edge guarding, one isolates a “minimal” triangle that separates off a chunk of vertices, removes the corresponding subgraph, applies induction to the remainder, and then reintroduces the deleted part with a bounded number of extra guards (Jungeblut et al., 2020). A plausible implication is that the usefulness of stacked graphs in such arguments comes less from planarity alone than from the strong recursive control provided by face subdivision.

3. Edge guarding and extremal bounds

One of the most explicit quantitative theories for stacked graphs concerns edge guarding in plane graphs. For a plane graph $3$0, a face $3$1 is guarded by an edge $3$2 if at least one vertex from $3$3 lies on the boundary of $3$4. An edge guard set is a set of edges such that every face, including the outer face, is guarded by at least one edge from the set (Jungeblut et al., 2020).

For stacked triangulations on $3$5 vertices, the main bound is: $3$6 edge guards are always sufficient for $3$7, and for every sufficiently large $3$8 there exist stacked triangulations for which at least

$3$9

edge guards are necessary (Jungeblut et al., 2020). The upper and lower bounds therefore differ only by a small additive constant.

The lower-bound construction is explicit. It starts from a base stacked triangulation with $3$0 faces, and in each face three new vertices are inserted, forming a new triangle distinct from the base. The resulting graph has

$3$1

vertices, and the construction is arranged so that no single edge can guard more than one of these new triangles. Hence at least $3$2 edge guards are necessary, which yields the lower bound when rewritten in terms of $3$3 (Jungeblut et al., 2020).

The upper bound is proved inductively. A suitable subgraph is removed, the smaller stacked triangulation is guarded by the induction hypothesis, and the deleted part is reinserted while adding only a bounded number of extra guards. The analysis ensures that the number of newly added guards per newly introduced vertices does not exceed the ratio $3$4, leading to the global bound (Jungeblut et al., 2020). The proof is constructive and can be turned into an efficient linear-time algorithm for finding such a guard set.

The same work places stacked triangulations in context by observing that, for general triangulations, the best upper bound is $3$5, and no matching construction is known. Stacked triangulations therefore constitute a non-trivial subclass of triangulations for which one can do strictly better in both upper and lower bounds (Jungeblut et al., 2020).

4. Threshold phenomena in random simplicial complexes

Stacked triangulations also arise in probabilistic topology. Lubetzky and Peled study when a random Linial–Meshulam complex $3$6 contains the faces of a stacked triangulation of the $3$7-simplex $3$8, with internal vertices labeled in $3$9 (Lubetzky et al., 2021). In that model, the $d$0-skeleton is complete and each $d$1-simplex is included independently with probability $d$2.

Their main result identifies the threshold for the appearance of stacked triangulations: $d$3 where $d$4 is the growth rate of the Fuss–Catalan numbers of order $d$5 (Lubetzky et al., 2021). The number of combinatorially distinct stacked triangulations of a $d$6-simplex with $d$7 unlabeled internal vertices is

$d$8

and asymptotically

$d$9

(Lubetzky et al., 2021).

The threshold statement has both subcritical and supercritical forms. In the subcritical regime, $\mathbf{o}=\{1,\ldots,d+1\}$0, the probability that a fixed $\mathbf{o}=\{1,\ldots,d+1\}$1-simplex has a stacked contraction is $\mathbf{o}=\{1,\ldots,d+1\}$2. In the supercritical regime, $\mathbf{o}=\{1,\ldots,d+1\}$3, every $\mathbf{o}=\{1,\ldots,d+1\}$4-simplex admits a stacked contraction with high probability (Lubetzky et al., 2021).

This problem is equivalent, in the language of bootstrap percolation in hypergraphs, to the threshold for $\mathbf{o}=\{1,\ldots,d+1\}$5, the $\mathbf{o}=\{1,\ldots,d+1\}$6-uniform clique on $\mathbf{o}=\{1,\ldots,d+1\}$7 vertices (Lubetzky et al., 2021). The proof uses a second moment argument in the supercritical regime and Kalai’s algebraic shifting in the subcritical regime.

5. Distinct objects with similar names

A persistent source of ambiguity is that several unrelated notions use the adjective “stacked” or the noun “stack.” The graph-theoretic stacked graph should be distinguished from both stacked-book graphs and stack-based graph layouts.

Term	Definition	Representative result
Stacked graph	$\mathbf{o}=\{1,\ldots,d+1\}$8-skeleton of a stacked triangulation; in $\mathbf{o}=\{1,\ldots,d+1\}$9, a planar $d \geq 2$0-tree / Apollonian network	Edge guarding bound $d \geq 2$1 (Jungeblut et al., 2020)
Stacked-book graph	Cartesian product $d \geq 2$2 of a star graph and a path graph	For $d \geq 2$3 and even $d \geq 2$4, $d \geq 2$5 (Adefokun et al., 2019)
Stack in a linear layout	A page in which, under a total vertex order, no two edges in the same part cross	Stack number $d \geq 2$6, queue number $d \geq 2$7, mixed number $d \geq 2$8 are layout parameters (Katheder et al., 2024)

A stacked-book graph is therefore not a triangulation-derived object at all. It is the Cartesian product of $d \geq 2$9 and $d$0, and it has been studied through radio labeling and induced matching parameters rather than recursive face subdivision (Adefokun et al., 2019, Adefokun et al., 2022). Likewise, in linear-layout theory, a stack is one component of an edge partition with pairwise non-crossing edges relative to a vertex order; this gives rise to parameters such as $d$1, $d$2, and $d$3, which concern graph layouts rather than stacked triangulations (Katheder et al., 2024, Alam et al., 2021).

6. Modern computational usage of “stacked” in graph-based models

In recent machine learning literature, “stacked” often refers not to a graph family but to an architectural composition. The paper “Stacked Graph Filter” studies graph convolutional networks from a graph signal processing viewpoint and proposes learning graph filters by stacking filter layers with learnable polynomial parameters (NT et al., 2020). In that setting, the claim is that any polynomial filter of order $d$4 can be represented by stacking $d$5 simple filter layers, and that the resulting model relaxes the low-frequency, or equivalently high-homophily, assumptions common in existing vertex classification models (NT et al., 2020).

A related but broader use appears in robust stacking frameworks for graph data with multifaceted node features. One such framework combines arbitrary models intended for IID data with graph-aware propagation, bagging, and multiple stacking layers, using out-of-fold predictions to mitigate label leakage and overfitting (Chen et al., 2022). Here again, “stacked” describes how models are composed, not a graph class.

These usages are terminologically adjacent but conceptually separate from stacked graphs in the classical combinatorial sense. The older meaning concerns a recursively defined triangulation-derived graph family; the newer one concerns how graph filters, graph neural modules, or graph-aware predictors are layered during training and inference (NT et al., 2020, Chen et al., 2022).