Pedigree Polytope Overview

• Pedigree polytope is a family of polytopes defined from Hamiltonian cycles where each vertex encodes a tour’s construction via successive vertex insertions.
• It provides a graph-theoretic framework featuring a pedigree graph that characterizes adjacency, revealing an asymptotically almost complete 1-skeleton.
• Its triangle-based recursive formulation enables efficient membership verification, with potential implications for polynomial-time TSP optimization.

Searching arXiv for papers on the pedigree polytope and closely related results. The pedigree polytope is a family of polytopes associated with Hamiltonian cycles on the complete graph $K_n$, introduced by Arthanari as an extension of the classical symmetric Traveling Salesman Problem polytope in which vertices remain in bijection with tours, but the combinatorial encoding records how a tour can be built by successive insertions of vertices into edges. Two distinct but compatible viewpoints dominate the literature: a graph-theoretic one, in which the 1-skeleton of the pedigree polytope is studied via an associated “pedigree graph” built from two evolving cycles (Makkeh et al., 2016), and a triangle-coordinate or multistage-insertion viewpoint, in which pedigrees are represented by sequences of triangles and membership in the convex hull is analyzed through layered networks and flow formulations (Arthanari, 11 Jul 2025, Arthanari, 2 Jun 2026).

1. Basic objects: tours, insertions, and pedigrees

For fixed $n$, the underlying combinatorial objects are Hamiltonian cycles on the node set $[n]:=\{1,\dots,n\}$. In the symmetric TSP model, an undirected cycle on $[n]$ is a graph-theoretic cycle that goes through all $n$ nodes exactly once, and the number of such cycles is

$\frac{(n-1)!}{2}.$

The classical symmetric TSP polytope $TSP(n)$ is the convex hull of the incidence vectors of these cycles (Makkeh et al., 2016).

A pedigree refines this by recording a constructive history. In the cycle-evolution formulation, an infinite cycle is a sequence

$A=(c_n)_{n\ge 3}\in \prod_{n=3}^\infty [n],$

where $A_3$ is the unique 3-cycle on $\{1,2,3\}$, and for $n$0, $n$1 is obtained from $n$2 by inserting the new node $n$3 into the $n$4-th edge of $n$5. Thus the pedigree concept is tied to a step-wise growth process in which node $n$6 is always created by subdividing a single existing edge (Makkeh et al., 2016).

The triangle formulation encodes the same process more rigidly. For each $n$7, let

$n$8

A pedigree is a sequence of $n$9 triangles

$[n]:=\{1,\dots,n\}$0

such that each $[n]:=\{1,\dots,n\}$1 has a generator in the preceding sequence and the common edges $[n]:=\{1,\dots,n\}$2 are all distinct (Arthanari, 11 Jul 2025). Equivalently, a pedigree may be represented by its sequence of common edges $[n]:=\{1,\dots,n\}$3, provided the generator and distinctness conditions hold.

The literature repeatedly emphasizes a 1-1 correspondence between pedigrees and Hamiltonian cycles. Insertion of vertex $[n]:=\{1,\dots,n\}$4 into an edge $[n]:=\{1,\dots,n\}$5 replaces $[n]:=\{1,\dots,n\}$6 by $[n]:=\{1,\dots,n\}$7 and $[n]:=\{1,\dots,n\}$8; conversely, shrinking an $[n]:=\{1,\dots,n\}$9-tour back to the 3-tour by removing vertices in reverse order recovers a unique pedigree. This is the structural reason the pedigree polytope has one vertex per tour even though it lives in a different ambient space (Arthanari, 11 Jul 2025).

2. Polyhedral definition and relation to the TSP polytope

Arthanari’s pedigree polytope for $[n]$0 cities is the convex hull of pedigree encodings, and the papers treat it as an extension of the symmetric TSP polytope without hidden vertices. Concretely, there is a linear projection from the pedigree polytope onto $[n]$1 which is onto and bijective on vertices; therefore, vertices of the pedigree polytope correspond bijectively to tours on $[n]$2 (Makkeh et al., 2016).

In the triangle-incidence model, one introduces a $[n]$3–$[n]$4 variable $[n]$5 for each triangle $[n]$6, indicating whether that triangle occurs in the pedigree. Omitting the base triangle $[n]$7, the ambient dimension is

$[n]$8

The set $[n]$9 consists of the characteristic vectors of pedigrees for $n$0, and the pedigree polytope is $n$1 (Arthanari, 2 Jun 2026). The dimension result quoted in the membership papers is

$n$2

(Arthanari, 2 Jun 2026, Arthanari, 11 Jul 2025).

This extension relationship has an immediate graph-theoretic consequence. Since the vertex sets of the pedigree polytope and $n$3 are in natural bijection, the 1-skeleton of $n$4 is a spanning subgraph of the 1-skeleton of the pedigree polytope: every TSP adjacency remains an adjacency in the pedigree polytope, but the latter can contain additional edges (Makkeh et al., 2016). The graph of the pedigree polytope is therefore combinatorially denser than the graph of the TSP polytope, although the two share the same factorial vertex count $n$5.

A further distinction is symmetry. The TSP polytope graph is vertex-transitive and hence regular, because permutations of cities induce affine automorphisms sending any tour to any other. The pedigree polytope breaks this symmetry: its graph is not vertex-transitive and, in the words of the 2016 papers, “not even regular” (Makkeh et al., 2016, Makkeh et al., 2016).

3. Combinatorial adjacency via the pedigree graph

The central combinatorial tool for understanding the pedigree polytope’s 1-skeleton is the pedigree graph $n$6, associated with two infinite cycles $n$7 and $n$8, often described as Alice’s cycle and Bob’s cycle (Makkeh et al., 2016).

For each $n$9, $\frac{(n-1)!}{2}.$0 is built inductively from $\frac{(n-1)!}{2}.$1. Its vertex set is a subset of $\frac{(n-1)!}{2}.$2. The defining condition is

$\frac{(n-1)!}{2}.$3

where $\frac{(n-1)!}{2}.$4 and $\frac{(n-1)!}{2}.$5 are the unordered pairs of neighbors of $\frac{(n-1)!}{2}.$6 in the respective evolving cycles, equivalently the edges into which $\frac{(n-1)!}{2}.$7 was inserted (Makkeh et al., 2016). If Alice and Bob insert node $\frac{(n-1)!}{2}.$8 into the same edge, then $\frac{(n-1)!}{2}.$9 does not appear in the pedigree graph.

When $TSP(n)$0 is a pedigree-graph vertex, edges from $TSP(n)$1 to earlier vertices are added by four rules: type-1 and type-2, each in an “$TSP(n)$2 to $TSP(n)$3” and “$TSP(n)$4 to $TSP(n)$5” version. Type-1 edges arise when insertion-edge pairs match exactly, for example

$TSP(n)$6

Type-2 edges connect $TSP(n)$7 to the larger endpoint of its insertion edge when a disjointness condition involving the other cycle holds (Makkeh et al., 2016). The graph is undirected, although the rule labels retain directional terminology.

Arthanari’s adjacency theorem identifies polytope adjacency with pedigree-graph connectivity: $TSP(n)$8 (Makkeh et al., 2016). This yields a purely combinatorial characterization of adjacency between pedigree vertices. The same source states that, unlike the TSP polytope where deciding adjacency is coNP-complete, adjacency in the pedigree polytope is polynomially decidable using this criterion (Makkeh et al., 2016).

The papers also distinguish carefully between the pedigree graph $TSP(n)$9 and the 1-skeleton of the pedigree polytope. The pedigree graph depends on a pair of tours and has bounded local degree; the polytope graph ranges over all tours $A=(c_n)_{n\ge 3}\in \prod_{n=3}^\infty [n],$0 adjacent to a fixed $A=(c_n)_{n\ge 3}\in \prod_{n=3}^\infty [n],$1, so its degree is of order $A=(c_n)_{n\ge 3}\in \prod_{n=3}^\infty [n],$2 rather than $A=(c_n)_{n\ge 3}\in \prod_{n=3}^\infty [n],$3. Conflating these two graphs is a common source of confusion, and the distinction is explicit in the exposition of the 2016 work (Makkeh et al., 2016).

4. Asymptotic structure of the 1-skeleton

The principal graph-theoretic result is that the graph of the pedigree polytope is asymptotically almost complete. If $A=(c_n)_{n\ge 3}\in \prod_{n=3}^\infty [n],$4 denotes the 1-skeleton of the pedigree polytope for $A=(c_n)_{n\ge 3}\in \prod_{n=3}^\infty [n],$5 cities, then the minimum degree satisfies

$A=(c_n)_{n\ge 3}\in \prod_{n=3}^\infty [n],$6

as $A=(c_n)_{n\ge 3}\in \prod_{n=3}^\infty [n],$7 (Makkeh et al., 2016). Equivalently,

$A=(c_n)_{n\ge 3}\in \prod_{n=3}^\infty [n],$8

The corresponding probabilistic formulation fixes one cycle $A=(c_n)_{n\ge 3}\in \prod_{n=3}^\infty [n],$9 and draws $A_3$0 uniformly at random from all cycles on $A_3$1. Then for every $A_3$2 there exists $A_3$3 such that for all $A_3$4,

$A_3$5

uniformly over the choice of $A_3$6 (Makkeh et al., 2016). Since pedigree-graph connectivity is equivalent to polytope adjacency, almost every other tour is adjacent to any fixed tour when $A_3$7 is large.

The proof is organized through an “adjacency game” in which Alice chooses insertion edges adversarially and Bob chooses uniformly at random. The analysis tracks two state variables: $A_3$8, the number of common cycle-edges of Alice and Bob at time $A_3$9, and $\{1,2,3\}$0, the number of connected components of the pedigree graph (Makkeh et al., 2016). Alice’s choices are classified as c-moves if she inserts into a common edge and d-moves otherwise. The transition analysis shows that c-moves deplete common edges, whereas d-moves have a probability proportional to $\{1,2,3\}$1 of decreasing the number of connected components.

A key estimate is the control of isolated pedigree-graph vertices. If $\{1,2,3\}$2 denotes the event that node $\{1,2,3\}$3 is added as an isolated vertex and $\{1,2,3\}$4, then

$\{1,2,3\}$5

for any strategy of Alice (Makkeh et al., 2016). The same paper states

$\{1,2,3\}$6

which makes isolated births increasingly rare. The combination of rare isolated vertices, depletion of common edges, and repeated opportunities for Bob to merge components yields eventual connectivity with probability tending to $\{1,2,3\}$7.

The extended abstract presents the same conclusion in compressed form: the quotient of minimum degree over number of vertices tends to $\{1,2,3\}$8, even though the pedigree polytope graph is not symmetric and not regular (Makkeh et al., 2016). This asymptotic density is the defining global property of the pedigree polytope’s 1-skeleton in the 2016 literature.

5. Triangle space, MI-relaxation, and the membership problem

A different line of work focuses on the membership problem for the pedigree polytope: given $\{1,2,3\}$9, decide whether $n$00. In the triangle-coordinate representation, this is denoted $n$01 (Arthanari, 11 Jul 2025, Arthanari, 2 Jun 2026).

The starting point is the multistage insertion formulation. For each triangle $n$02, let $n$03 indicate whether that triangle is chosen. The core constraints are one triangle per stage,

$n$04

together with inequalities enforcing that common edges are used at most once and that generators are available (Arthanari, 11 Jul 2025). Replacing $n$05 by $n$06 yields the MI-relaxation $n$07 with feasible set $n$08 (Arthanari, 2 Jun 2026).

The relation to the symmetric TSP is encoded by the objective coefficients

$n$09

so that the total insertion increment equals the tour length (Arthanari, 2 Jun 2026). The papers state that every integer solution of $n$10 corresponds to a Hamiltonian tour, and the slack vector in the edge constraints coincides with the edge-incidence vector of the resulting $n$11-tour (Arthanari, 2 Jun 2026, Arthanari, 11 Jul 2025). Thus the pedigree polytope is a tour polytope in triangle variables rather than in edge variables.

A structural tool in this line is the stem property: if $n$12, then each prefix $n$13 lies in $n$14 (Arthanari, 11 Jul 2025). This allows recursive treatment of membership from stage $n$15 up to stage $n$16. The same paper defines $n$17 as the set of convex representations of $n$18 in terms of pedigrees of size $n$19, and then formulates the forbidden-arc transportation problem $n$20 to determine whether a convex decomposition at stage $n$21 can be extended to stage $n$22 (Arthanari, 11 Jul 2025).

This suggests a layered inductive structure that is absent in the edge-space formulation of the TSP polytope: membership is not tested against a static family of inequalities alone, but by tracking compatibility of successive insertion choices across levels.

6. Layered networks, multicommodity flow, and later claimed algorithmic consequences

The membership papers construct a layered network $n$23, where $n$24 is a network whose layers correspond to insertion stages, $n$25 is a set of rigid pedigrees, and $n$26 are their weights (Arthanari, 2 Jun 2026). Restricted networks $n$27 are defined for links $n$28 representing possible consecutive insertion choices, and their maximum flows $n$29 determine capacities in a stagewise transportation problem $n$30 (Arthanari, 2 Jun 2026, Arthanari, 11 Jul 2025).

The 2025 and 2026 papers both state that $n$31 feasibility is necessary but not sufficient for membership, motivating a multicommodity flow problem $n$32 on the layered network (Arthanari, 11 Jul 2025, Arthanari, 2 Jun 2026). In the 2026 formulation, the total flow objective is

$n$33

and the maximum possible flow is

$n$34

(Arthanari, 2 Jun 2026). The central necessary-and-sufficient condition is stated as

$n$35

under the stated inductive assumptions (Arthanari, 2 Jun 2026).

The same papers further state that the resulting linear programs are combinatorial LPs in Tardos’s sense, with $n$36-valued constraint matrices, and that the dimension of $n$37 is polynomial in $n$38 (Arthanari, 2 Jun 2026). The 2026 paper gives the complexity claim

$n$39

for checking whether $n$40 has $n$41, and concludes that $n$42, indeed strongly polynomial (Arthanari, 2 Jun 2026).

These later works also make broader claims. The 2025 article argues that strongly polynomial-time membership for pedigree polytopes implies efficient linear optimization over the pedigree polytope, and that the multistage insertion formulation then yields a polynomial-time algorithm for the symmetric TSP, from which it concludes $n$43 (Arthanari, 11 Jul 2025). The 2026 paper presents a Lean 4 formalization of parts of this chain and states that the sufficiency of $n$44 for membership in $n$45 is machine-verified, while also noting that the full biconditional’s necessity direction is “proved in the book; Lean formalisation left as future work,” and that the final $n$46 chain uses six external published results as axioms (Arthanari, 2 Jun 2026). A plausible implication is that the membership line of research should be distinguished from the 2016 graph-theoretic results: the former concerns a triangle-space oracle problem and layered flow constructions, whereas the latter concerns adjacency in the 1-skeleton.

7. Position within the literature

Across the literature, the pedigree polytope is characterized by three persistent features. First, it is a TSP-related extension without hidden vertices, so it retains the tour set while changing the ambient combinatorics (Makkeh et al., 2016). Second, its graph-theoretic behavior is unexpectedly dense: although the graph is not vertex-transitive and not regular, its minimum degree is asymptotically equal to the total number of vertices (Makkeh et al., 2016, Makkeh et al., 2016). Third, its triangle-based recursive structure supports membership formulations that are much more explicit than those usually available for the TSP polytope (Arthanari, 11 Jul 2025, Arthanari, 2 Jun 2026).

The 2016 work leaves open graph-theoretic questions such as whether other non-complete polytope graphs can be asymptotically almost complete, and whether analogous combinatorial adjacency criteria might exist closer to the TSP polytope itself (Makkeh et al., 2016). The 2025 and 2026 membership papers instead position the pedigree polytope as a vehicle for studying polyhedral membership, separation, and optimization through layered network structure (Arthanari, 11 Jul 2025, Arthanari, 2 Jun 2026).

Taken together, these strands make the pedigree polytope a distinctive object in polyhedral combinatorics: it is simultaneously an extension of the symmetric TSP polytope, a source of a nontrivial polynomial adjacency criterion, an example of a 1-skeleton that is asymptotically almost complete, and the basis of an active line of work on the membership problem in triangle space.