Graph Tubings: Nested Complexes & Polyhedra

• Graph tubings are collections of connected vertex subsets (tubes) that follow strict compatibility rules, forming the faces of nested complexes and graph associahedra.
• They unify classical combinatorial structures by linking graph-associahedral constructions, compatibility fans, and lattice-theoretic properties across different graph families.
• Variants such as marked tubings, Δ-graph tubings, and hypercube-graph tubings extend the framework to diverse applications including cosmological geometry and kinematic flow analysis.

Graph tubings are collections of graph-theoretic tubes subject to compatibility rules. In the classical graph-associahedron setting, a tube of a graph $G$ is a non-empty subset of vertices inducing a connected subgraph, and a tubing is a set of pairwise compatible proper tubes; the resulting tubings form the faces of the graphical nested complex, while maximal tubings index the vertices of the associated graph associahedron (Manneville et al., 2015, Barnard et al., 2018). Later work has retained this core idea but modified the meaning of tubes or compatibility in several directions, including marked tubings, $\Delta$-graph and hypercube-graph tubings, lattice-theoretic tubing posets, and several constructions in cosmological geometry and kinematic flow (0807.4159, Almeter, 2022, Capuano et al., 20 May 2025).

1. Classical graph-associahedral tubings

In the standard graph-associahedral convention, if $G$ has vertex set $V$, a tube is a non-empty subset $\tube\subseteq V$ such that $G[\tube]$ is connected. The inclusion-maximal tubes are the connected components $\kappa(G)$, and all others are proper. Two tubes are compatible if they are either nested,

$\tube\subseteq \tube' \quad\text{or}\quad \tube'\subseteq \tube,$

or disjoint and non-adjacent, meaning that $\tube\cup\tube'$ is not a tube. A tubing is then a set of pairwise compatible proper tubes, and the collection of all tubings is the nested complex $N(G)$ (Manneville et al., 2015).

A closely related formulation says that two tubes are compatible precisely when they do not intersect and are not adjacent, where adjacency means disjoint tubes whose union is a tube. In that language, intersecting but non-nested tubes are incompatible, and disjointness alone is not sufficient for compatibility. This point is basic but often obscured in informal discussions (0807.4159).

The nested complex is the simplicial complex whose vertices are proper tubes and whose faces are tubings. If

$\Delta$0

then $\Delta$1 is an $\Delta$2-dimensional simplicial sphere, and every maximal tubing has exactly $\Delta$3 tubes. Two maximal tubings are adjacent when they differ by exactly one tube; such an exchange is a flip, and the exchanged tubes are called exchangeable (Manneville et al., 2015).

2. Nested complexes, graph associahedra, and compatibility fans

Graph associahedra are polytopal realizations of nested complexes. With

$\Delta$4

projection $\Delta$5, and

$\Delta$6

the nested fan is

$\Delta$7

This is a complete simplicial fan realizing $\Delta$8, and it is the normal fan of the graph associahedron $\Delta$9 (Manneville et al., 2015).

A different realization uses a compatibility degree between tubes. For two tubes $G$0,

$G$1

This degree is generally asymmetric. It satisfies

$G$2

and

$G$3

Fixing a maximal tubing

$G$4

the compatibility vector of a tube $G$5 is

$G$6

The cones generated by compatibility vectors of the tubes in a tubing form a complete simplicial fan

$G$7

and there is a dual version $G$8 built from the reversed degree. These compatibility fans realize the same nested complex as the graph associahedron (Manneville et al., 2015).

For paths, the compatibility degree takes values in $G$9 and coincides with the type $V$0 cluster compatibility degree. For cycles, the same construction recovers the type $V$1 compatibility degree, while the dual degree recovers type $V$2. This is one of the main points at which graph tubings and finite-type cluster combinatorics meet (Manneville et al., 2015).

3. Maximal tubings as ordered and lattice-theoretic objects

Maximal tubings can be ordered by orienting flips. In one standard construction, the vertices of the graph associahedron are maximal tubings, and the orientation comes from the linear functional

$V$3

The resulting poset $V$4 has maximal tubings as elements and cover relations given by directed flips. There is a canonical surjection

$V$5

from weak order on permutations to maximal tubings. This map is a lattice quotient map if and only if $V$6 is filled. More generally, if $V$7 is right-filled, $V$8 is a lattice and $V$9 is a meet-semilattice map; if $\tube\subseteq V$0 is left-filled, $\tube\subseteq V$1 is a lattice and $\tube\subseteq V$2 is a join-semilattice map. Complete graphs recover weak order, and path graphs recover the Tamari lattice (Barnard et al., 2018).

For filled connected graphs, and in particular for complete graphs, path graphs, and lollipop graphs, the maximal tubing poset can also be described as a quotient or subposet of weak order via $\tube\subseteq V$3-equivalence classes of permutations. In the lollipop graph

$\tube\subseteq V$4

with a complete graph on $\tube\subseteq V$5 and a path on $\tube\subseteq V$6, the lattice $\tube\subseteq V$7 interpolates between weak order and Tamari. Its elements are represented by $\tube\subseteq V$8-312 avoiding permutations, and its maximal chains admit descriptions in terms of commuting shuffles, balanced tableaux, and $\tube\subseteq V$9-row-shiftable tableaux (Duetsch, 2024).

The cycle graph was a notable open case because it is not filled. It is now known that the poset of maximal tubings of the cycle graph $G[\tube]$0 is a lattice, and more strongly a semidistributive and congruence uniform lattice. Its global order is characterized by

$G[\tube]$1

and there is a cut map

$G[\tube]$2

to path-graph tubings that is a lattice quotient map. This places cycle-graph tubings alongside weak order and Tamari as a third major lattice-theoretic family arising from graph associahedra (Adenbaum et al., 10 Oct 2025).

4. Variants and generalizations of tubing data

One major extension replaces ordinary tubes by marked tubes. In this setting a tube may be thin, thick, or broken. Two marked tubes are compatible if they do not intersect, are not adjacent, and satisfy the additional rule that if $G[\tube]$3 where $G[\tube]$4 is not thick, then $G[\tube]$5 must be thin. Marked tubings form the face poset of the graph multiplihedron $G[\tube]$6, a convex polytope of dimension $G[\tube]$7 for a graph with $G[\tube]$8 nodes. Codimension-$G[\tube]$9 faces correspond to marked tubings with exactly $\kappa(G)$0 non-broken tubes, and the facets decompose as products such as

$\kappa(G)$1

depending on whether one is in a lower or upper facet (0807.4159).

Another extension changes the ambient combinatorics. For a simplicial complex $\kappa(G)$2, a $\kappa(G)$3-graph $\kappa(G)$4 has tubes defined as faces of $\kappa(G)$5 inducing connected subgraphs of $\kappa(G)$6. A tubing is then a strongly compatible set of such tubes: pairwise weak compatibility is required, and in addition the union of the tubes must again be a face of $\kappa(G)$7. This leads to $\kappa(G)$8-nested complexes, $\kappa(G)$9-nestohedra, and $\tube\subseteq \tube' \quad\text{or}\quad \tube'\subseteq \tube,$0-graph associahedra obtained by truncating a simple polyhedron $\tube\subseteq \tube' \quad\text{or}\quad \tube'\subseteq \tube,$1 along faces $\tube\subseteq \tube' \quad\text{or}\quad \tube'\subseteq \tube,$2 indexed by tubes. In the hypercube case, tubes and tubings must avoid dashed pairs $\tube\subseteq \tube' \quad\text{or}\quad \tube'\subseteq \tube,$3, and the tubing complex of a hypercube graph is a flag complex (Almeter, 2022).

A different variant appears in the study of the dual cosmological polytope. There, a tube is a connected, non-empty, not necessarily induced subgraph, and a tubing is a set of tubes in which every pair is either disjoint or nested in the sense of edge-set inclusion. In particular, disjoint tubes need not be non-adjacent. This broader notion is related to ordinary graph-associahedral tubings by a bijection between tubings of $\tube\subseteq \tube' \quad\text{or}\quad \tube'\subseteq \tube,$4 containing each singleton and GA-tubings of the line graph $\tube\subseteq \tube' \quad\text{or}\quad \tube'\subseteq \tube,$5 (Birkemeyer et al., 4 Mar 2026).

These variants show that “graph tubings” is not a single immutable definition but a family of closely related combinatorial devices. The main invariants across these settings are connectedness, compatibility by exclusion of certain overlaps, and the use of maximal compatible families as face labels, basis labels, or triangulation data.

5. Tubings in cosmological geometry, canonical forms, and kinematic flow

In the cosmological-polytope setting, tubes can index facets directly. For a graph $\tube\subseteq \tube' \quad\text{or}\quad \tube'\subseteq \tube,$6 without isolated vertices, the cosmological polytope

$\tube\subseteq \tube' \quad\text{or}\quad \tube'\subseteq \tube,$7

has facets indexed by tubes $\tube\subseteq \tube' \quad\text{or}\quad \tube'\subseteq \tube,$8 through normals

$\tube\subseteq \tube' \quad\text{or}\quad \tube'\subseteq \tube,$9

The dual cosmological polytope has vertices

$\tube\cup\tube'$0

and maximal tubings index simplices

$\tube\cup\tube'$1

that triangulate the dual polytope. Uniquely completable almost maximal tubings index a second triangulation after coning over $\tube\cup\tube'$2. Both triangulations yield explicit tubing expansions for the canonical form (Birkemeyer et al., 4 Mar 2026).

A complementary application occurs in the asymptotic analysis of cosmological integrals. There the singularities of a process are associated to subgraphs, and the Newton polytope of the denominator is a special class of nestohedron. Its facets are identified by overlapping tubings: the specific tubing determines the divergent direction, while the number of overlapping tubings determines the degree of divergence. Compatible facets obtained from tubings also determine the sectors in a sector decomposition of the integration domain (Benincasa et al., 2024).

Several recent works use tubings as the basic language of canonical differential equations and symbol alphabets. One construction introduces $\tube\cup\tube'$3-tubings, built by cutting some edges, taking the connected components as roots, and then choosing $\tube\cup\tube'$4-tubings on those components. The functions

$\tube\cup\tube'$5

form a canonical differential-equation basis, and region variables

$\tube\cup\tube'$6

attached to regions of a tubing appear as the $\tube\cup\tube'$7 coefficients of the system (Capuano et al., 20 May 2025).

A related marked-graph formalism defines tubes as connected subgraphs of a marked graph, forbids a tube containing only a cross, and identifies complete tubings with basis functions. In this language, merger of adjacent tubes reproduces the source structure of the differential equations for arbitrary tree graphs and loop integrands, while reversing the process leads to tubing-splitting rules equivalent to the kinematic flow at tree level (Baumann et al., 21 Apr 2025, Ke et al., 18 May 2026).

Tubings also control symbol alphabets. For $\tube\cup\tube'$8-site chain and loop graphs, tube variables on marked graphs map to polygon chords, complete tubings become polygon triangulations, and the symbol alphabets are realized as subsets of $\tube\cup\tube'$9 and $N(G)$0 cluster variables with cluster adjacency (Paranjape et al., 9 Mar 2026). In flat-space wavefunction coefficients, binary tubings built from connected subgraphs and unary tubings built from connected induced subgraphs lead respectively to the wavefunction coefficient and the amplitube, while cut tubings organize an expansion of the wavefunction into products of amplitubes and decorated orientations (Glew, 17 Mar 2025). In integrated de Sitter correlators, the correlator alphabet can be strictly smaller than the corresponding wavefunction alphabet, with the missing letters admitting an interpretation in terms of tubing data (Chowdhury et al., 29 Apr 2026).

6. Representative graph families and model cases

Several graph families serve as canonical testing grounds. For the complete graph $N(G)$1, every nonempty proper subset is a tube, compatibility reduces to nesting, tubings are chains, and the nested complex is the simplicial permutahedron. For the path $N(G)$2, the nested complex is the simplicial associahedron; tubes correspond to diagonals of an $N(G)$3-gon, and the compatibility degree becomes

$N(G)$4

For the cycle $N(G)$5, the nested complex is the simplicial cyclohedron, and tubes correspond to centrally symmetric pairs of diagonals in a regular $N(G)$6-gon. For the star $N(G)$7 with center $N(G)$8 and leaves $N(G)$9, the tubes are either single leaves or connected sets containing $\Delta$00; with initial maximal tubing

$\Delta$01

the compatibility fan is obtained from the coordinate hyperplane fan by barycentric subdivision of the positive orthant (Manneville et al., 2015).

In marked-tubing geometry, paths recover the classical multiplihedra, complete graphs recover permutohedra, and edgeless graphs give a family combinatorially equivalent to

$\Delta$02

In hypercube-graph geometry, a path on positive vertices gives the $\Delta$03 associahedron, a positive cycle gives the halohedron, and the double complete graph $\Delta$04 gives the type $\Delta$05 permutahedron (0807.4159, Almeter, 2022).

These examples show that graph tubings unify several classical polyhedral and combinatorial families while also supporting substantial variation in definition. The standard theory centers on connected induced subgraphs and nested/non-adjacent compatibility, but many later constructions preserve only the deeper principle: connected graph pieces, organized into maximal compatible families, control faces, fans, lattices, canonical forms, or differential-equation bases.