Papers

We study generic $d$-dimensional rigidity in sparse random graphs. Our main result is that for every $d\ge 2$, the Erdős–Rényi random graph $G\sim G(n,c/n)$ undergoes a $d$-rigidity phase transition at the known, explicit, $d$-orientability threshold $c_d$: If $c<c_d$, then $G$ is asymptotically almost surely (a.a.s.) independent in the generic $d$-rigidity matroid. Moreover, in this regime $G$ has no linear-size rigidity components: it contains no induced $d$-rigid subgraphs with more than $3$ vertices, and the largest clique in its $d$-rigidity closure has size at most $o(\sqrt n)$. If $c>c_d$, then $G$ is a.a.s. not independent in the generic $d$-rigidity matroid, and we give a sharp asymptotic estimate for its rank. In addition, the $d$-rigidity closure of $G$ has a giant clique of linear size, which contains all but at most $o(n)$ vertices of the $((d+1)+d)$-core of the graph. More generally, we compute, up to a $1+o(1)$ factor, the generic $d$-rigidity rank of random graphs with a given degree distribution. For example, we show that the uniform $n$-vertex $k$-regular graph a.a.s. has rank $\min(k/2,d)n+o(n).$ Our approach is to estimate the rigidity rank of a random graph from its Galton–Watson local weak limit, using a parameter that we call local flexibility.

An edge cut C of a graph G is tight if |C \M| = 1 for every perfect matching M of G. Barrier-cuts and 2-separation cuts, also referred to as ELP-cuts, are two important types of tight cuts in matching covered graphs. Edmonds, Lovasz and Pulleyblank [Brick decompositions and the matching rank of graphs, Combinatorica 2(3) (1982) 247-274] proved that if a matching covered graph has a non-trivial tight cut, then it also has a non-trivial ELP-cut. In confirmation of a conjecture proposed by Carvalho, Lucchesi and Murty, Chen et. al. [Laminar tight cuts in matching covered graphs, J. Comb. Theory, Ser. B, 150 (2021) 177-194] showed that if C is a non-trivial tight cut of a matching covered graph G, then G has at least one C-sheltered non-trivial barrier or a 2-separation cut that is laminar with C. In this paper, we present a complete characterization of non-trivial tight cuts in matching covered graphs, from which the result of Chen et. al. can be derived directly. Moreover, we show that the lower bound of the number of C-sheltered non-trivial barrier or a 2-separation cut that is laminar with C in the result of Chen et. al. is sharp.

While motivated by structural problems in mathematical music theory, this article introduces a novel combinatorial framework that advances the classification of cyclic cubic bipartite graphs. We extend the classical study of Levi graphs by endowing their vertices with an internal algebraic anatomy – specifically, treating them not as empty geometric nodes, but as defined subsets of a cyclic base space Z_n. This internal structure allows us to formalize and classify a highly restricted class of graph isomorphisms: those strictly induced by global affine bijections f(x) = ax+b (mod n) operating directly on the underlying base set. By applying this framework to generalized tone networks (Tonnetze) unrolled via the Chinese Remainder Theorem in composite dimensions – specifically the classic 12-TET (3x4) and the decaphonic 10-TET (2x5) – we reveal absolute geometric anchors for these spaces, namely the (9,4) and (6,5) systems respectively. We completely classify the topological orbits of these structured graphs, proving a fundamental architectural dichotomy: while the isomorphic landscape of 12-TET splits into an orientation-preserving family and an orientation-reversing chiral mirror (providing a rigorous foundation for musical Negative Harmony), the 10-TET space is unconditionally orientation-preserving. Finally, we demonstrate that these abstract combinatorial properties manifest as rigidly coherent, parallel auditory universes through explicit structural voice-leading maps and acoustic physical modeling synthesis.

Korsky, Saffat and Aiylam introduced a growth constant $c(G)$ for integer-valued $h$-Lipschitz functions on a finite graph $G$ and proved that, for $G=G(n,d/n)$, \[ \frac{1}{2d}+O(d^{-2})\le \log c(G)\le \frac{4\log^2 d}{d}+O(d^{-1}) \] with high probability. We sharpen the random-graph part of their result; as $n\to\infty$ and then $d\to\infty$, we prove \[ \log c(G)=\frac{\pi^2}{6d}+o(d^{-1}) \] with high probability. Additionally, we derive bounds on $\log c(Q_d)$ where $Q_d$ is the $d$-dimensional hypercube graph: \[ \frac{\pi^2}{6d}+o(d^{-1}) \le \log{c(Q_d)}\le \left(\frac{3}{4} + o(1)\right)\frac{\log d}{d}. \]

Given a signing $\sigma\colon E(K_{n,n})\to\{-1,+1\}$ of the complete bipartite graph, when does $K_{n,n}$ admit a $1$-factorization in which every perfect matching has discrepancy bounded below by a positive absolute constant? Unlike the complete-graph case resolved by Ai, He, Im, and Lee, the bipartite setting carries an unavoidable obstruction: any balanced one-sided signing – one whose edge signs depend on a single bipartition class, with the two labels split as evenly as possible – forces every perfect matching to have discrepancy at most $1/n$. We prove that this is essentially the only obstruction: For every $\varepsilon>0$ there exists $c=c(\varepsilon)>0$ such that, for all sufficiently large $n$, every signing of $K_{n,n}$ either (i) admits a $1$-factorization in which every perfect matching has discrepancy at least $c$, or (ii) is $\varepsilon$-close, in normalized Hamming distance, to a balanced one-sided signing. A key ingredient is a spectral stability argument forcing the sign matrix to be close to a balanced one-sided pattern when both the overall discrepancy and the density of local switching patterns are small.

We introduce flag positroid pipe dreams (FPPs), whose role in the study of complete flag positroids is analogous to the role of Le-diagrams in the study of positroids. We develop the combinatorics of these diagrams and highlight some of their properties. FPPs are in bijection with intervals in the Bruhat order of the symmetric group, and the number of elbows in an FPP is the dimension of the corresponding Richardson cell in the decomposition of the nonnegative flag variety. We show how complete flag positroids can be built rank by rank via FPPs, and how the Le-diagrams of the positroid constituents of the flag can be obtained from the FPP via a simple standardization operation. Using partial FPPs, we give an alternative proof of a conjecture of Benedetti, Chavez, and Tamayo on the problem of characterizing elementary positroid quotients via cyclic shifts of decorated permutations, in the nonnegatively representable case. Our proof partially addresses a problem of Chen et al. regarding an explicit characterization of the cyclic shift operators purely in terms of decorated permutations. We show that the poset of nonnegatively representable elementary positroid quotients is self-dual. The maximal chains of this poset are in bijection with FPPs.

We study length-filtered BFACF graphs of self-avoiding polygons in the simple cubic lattice, with merge heights recording the least length cap needed to connect configurations. We determine the complete minimal-layer merge trees for $4_1$ (four components at length 30 merging at 32) and $6_3$ (twelve components at length 40, two at 42, and one at 44). For one trefoil chirality, we certify six selected nonminimal components born at length 26 and merging with the minimal branch by length 28 or 30; these do not exhaust the full filtration. We characterize eventual connectivity of capped graphs by bounded birth levels of strong local-minimum components. A 64-edge Hamiltonian lattice unknot with no nonincreasing BFACF moves has exact merge height 66 with the elementary square, verified by an explicit move certificate. Repeated two-edge splicing produces frozen unknots of length $64m$ for every $m\ge1$. We also construct an 88-edge frozen trefoil by splicing the 64-edge unknot with a minimal 24-edge trefoil; iteration and reflection yield frozen trefoils of both chiralities with $64m+24$ edges. Hence the unknot and both trefoil chiralities have infinitely many nonminimal strong local-minimum births, and their capped BFACF graphs are not eventually connected. These results concern standard discrete BFACF moves, not continuous ropelength barriers; exact excess heights for the longer families and extensions to arbitrary knot types remain open.

The concept of weakly regular residuated mappings was investigated for lattices recently by S. Radeleczki and L. Veres. We modify this concept for posets. We define so-called operator-residuated mappings in posets, show their important properties and point out differences between them and residuated mappings defined in the usual way. We modify the concept of a lattice homomorphism for posets where suprema need not exist and show its relation to residuated mappings and induced ideals. Finally, we present several examples of weakly regular residuated mappings in posets and show how such mappings may be constructed.

In this article we describe a new inductive approach to compute the chromatic polynomial of simple graphs and the characteristic polynomial of central hyperplane arrangements.

Consider a symmetric function $\mathcal{C}(x,y)$ on $[0,1]\times[0,1]$ which is twice continuously differentiable up to the boundary, and which satisfies $ \mathcal{C}(x,y)=\mathcal{C}(1-x,1-y)$. Let $A^{(n)} = \big(a^{(n)}_{i,j}\, :\, i,j \in [n]\big)$ be the matrix with entries $a^{(n)}_{i,j}\, =\, \exp(-\mathcal{C}(i/n,j/n))$. Soumik Pal conjectured the asymptotics $$\operatorname{perm}\big(A^{(n)}\big)/n!\sim \exp\big(n \Lambda[\mathcal{C}]\big)/ \sqrt{\mathcal{D}[\mathcal{C}]}$$ as $n \to \infty$ for known functionals that arise naturally in the context of entropy regularized optimal transport. The functional $\Lambda[\mathcal{C}]$ is the known large deviation rate function, already proved rigorously by Sumit Mukherjee. It is $\int_{0}^1 \int_0^{1} (\alpha(x)+\beta(y))\, dx\, dy$ where $\alpha(x)+\beta(y)$ is chosen such that $\rho(x,y) := \exp(-\mathcal{C}(x,y)-\alpha(x)-\beta(y))$ has uniform marginals. The algebraic term $\mathcal{D}[c]$ is given by Peter McCullagh's formula for doubly stochastic matrices: $\operatorname{det}_F(I+J-T^*T)$, the Fredholm determinant, where $I$ is the identity on $L^2([0,1])$, $Jf(x) \equiv \int_{0}^1 f(z)\, dz$ (for all $x$) and $Tf(x) = \int_0^1 \rho(x,y) f(y)\, dy$. We prove the conjecture for functions $\mathcal C$ that are constant on blocks, exploiting a well-known Ross Pinsky's combinatorial decomposition of permutations in blocks.

For a split graph $S$, the combinatorics of 2-switches on $S$ is faithfully encoded by the factor graph $\Phi(S)$, a multigraph whose induced cycles have length at most $4$. In this paper we address the following question: for which $n \in \mathbb{N}$ is there a split graph $S$ whose factor graph contains an $n$-simple triangle, that is, a triangle all of whose edges have multiplicity $n$? We show that the answer is governed by a purely arithmetic condition, the $\Delta$ property, relating the differences and sums of complementary divisors of $n$, and thereby establish a two-way bridge between Graph Theory and Number Theory.

The shortest path multiplicity, here denoted by $\mu$, is an important metric of complex networks. For real networks $\mu$ is high and it correlates with the network community structure. Since local network evolution induces network communities, it is possible that a high shortest path multiplicity is the natural expectation of local evolution rules. Here I demonstrate, by means of numerical simulations and exact results, that this is indeed the case. For random graphs with arbitrary degree distributions $p_k$, $\langle\mu\rangle\sim \langle k(k-1)\rangle / (\langle k\rangle e)$, growing with the network size when $p_k\sim k^{-\gamma}$ and $\gamma\leq3$. For networks generated by stochastic local rules, $\langle\mu\rangle$ increases with the network size and it does so faster than what is observed in their randomized versions. Furthermore, the number of communities increases with the network size and the correlation with $\langle \mu\rangle$ follows.

Tonks' projection from the permutohedron to the associahedron and the Loday–Ronco map both send permutations to planar binary trees. We give a syntactic account of these maps in the equational calculus of the free non-symmetric, non-unital operad on one binary generator. The vertex restriction of Tonks' projection is obtained by evaluating the head-insertion encoding on the reversed permutation, while the Loday–Ronco map is obtained by evaluating the decreasing encoding. We also give a local operadic proof that Tonks' vertex map is order-preserving from the weak Bruhat order to the Tamari order.

Let $K$ be a number field, and $\varphi_{1},\ldots,\varphi_{g}\in K(t)$ be finitely many rational maps, each of degree at least $2$. We first show that for generic finite sets $\mathcal{A}_{1},\ldots,\mathcal{A}_{g}$ consisting entirely of points that are not $\varphi_{i}$-periodic, there exists a set of primes $\mathfrak p$ of $K$ of positive density such that for each $\mathcal{A}_{i}$ and every $\alpha\in\mathcal{A}_i$, $\alpha$ is not $\varphi_i$-periodic modulo $\mathfrak p$. The notion of genericity used here is defined in terms of the associated arboreal fields and is sharper than those previously used in the literature. Leveraging our proof in the generic case, we then show that the same conclusion holds for most expected cases of non-generic sets $\mathcal{A}_{i}$. Finally, we apply our result to confirm the dynamical Mordell–Lang conjecture for coordinate-wise actions of a class of maps that includes rational maps that are generic in this sense.

Let $G$ be a graph and $r \ge 1$. A vertex subset is $r$-independent if every connected component of its induced subgraph has size at most $r$. The family of all such subsets forms a simplicial complex, the $r$-independence complex $\Ind_r(G)$, generalizing the classical independence complex. Recent work has focused on shellability and vertex decomposability of these complexes. For chordal graphs, $\Ind_r(G)$ has the homotopy type of a wedge of spheres for all $r$, and some chordal subfamilies are known where these complexes are not even sequentially Cohen-Macaulay. Thus, determining chordal graph classes and values of $r$ for which $\Ind_r(G)$ is sequentially Cohen-Macaulay, shellable, or vertex decomposable remains an active area. Existing methods, based on chordal hypergraphs or special graph properties, do not extend to arbitrary chordal graphs. In this paper, we show that for every tree $T$ and every integer $r \ge 1$, the complex $\Ind_r(T)$ is vertex decomposable, resolving a conjecture \cite[Conjecture 3.15]{PD23chordal} of Abdelmalek et al. Our approach gives a structural description of shedding vertices via rooted subtrees and uses it to prove vertex decomposability recursively.

For a full-rank integral lattice $\mathcal{L}\subset\mathbb{R}^n$, Regev and Stephens-Davidowitz proved that \[N_{=k}(\mathcal{L}):=|\{y\in\mathcal{L}:\lVert y\rVert^2=k\}|\le 2\binom{n+2k-2}{2k-1}.\] We classify the equality cases. For $n\ge2$, equality holds if and only if either $k=1$ and $\mathcal{L}\cong\mathbb{Z}^n$, or $n=8$, $k=2$, and $\mathcal{L}\cong E_8$. For $n=1$, equality holds exactly when $\mathcal{L}$ represents $k$. The proof shows that equality is rigid. Saturation of the shell bound forces the normalized norm-$k$ shell to be an antipodal tight spherical $(4k-1)$-design. The associated Delsarte–Goethals–Seidel annihilator polynomial gives an arithmetic root condition, which isolates $E_8$ at $k=2$, rules out $k=3$, and combines with the Bannai–Damerell/Bannai theorem and an elementary circle argument to exclude all remaining cases in dimension at least $2$.

We improve upon the Johnson-type bounds of Hayashi–Yasunaga and Liu–Tjuawinata–Xing for insertion–deletion codes by encoding each local list into a binary constant-weight code. The resulting local list-size bound is tight over sufficiently large alphabets. Combining this bound with an averaging argument and the constant-weight McEliece–Rodemich–Rumsey–Welch bound yields an asymptotic rate bound that strictly improves Yasunaga's Elias-type bound throughout the nontrivial range.

For a graph $G$, let $\tau(G)$ denote the number of spanning trees. We show that for every fixed $0 < c < 1/4$, the number of distinct values of $\tau(G)$, as $G$ ranges over simple graphs on $n$ vertices, is at least $\exp(c n \log n)$ for all sufficiently large $n$. This is optimal up to the choice of the constant $c$ and resolves a conjecture of Chan-Kontorovich-Pak regarding a problem of Sedláček from the late 1960s.

In this paper, we study the structure of the complete asymptotic expansion of the probability that a large combinatorial object is connected or consists of a given number of connected components. For rapidly growing labeled families of structures, the coefficients involved in these expansions are possibly negative integers. Using species theory, we interpret these coefficients as the difference between the counting sequences of two derivative species of structures. In particular, we show that this difference can be viewed as the counting sequence of the virtual species obtained with the help of an "anti-$\mathrm{SEQ}$" operator applied to the initial family of structures. Applications include $P$-angulated discrete surfaces, quadratic square-tiled surfaces, and non-orientable graph encoded manifolds, which were not reachable with our previous methods. Moving on to the weighted species, we establish the whole structure of the asymptotic expansion of the probability that a graph is connected in the Erdős-Rényi model $G(n,p)$. Here, the asymptotic coefficients are polynomials in $\frac{p}{1-p}$ and can be described both in terms of simple graphs and irreducible tournaments with ties. We also provide general asymptotic results for sequence and cycle decomposition, as well as the complete asymptotic expansion of the probability that a random labeled tournament with ties is irreducible.

This manuscript studies a special case of the Hurwitz enumeration problem: for branched covers from genus g compact Riemann surface to the Riemann sphere, with three branch points, and require the branching data at one of the branch points to be of the partition (m 1^{n-m}), obtain a formula of Hurwitz number. The Hurwitz enumeration problem can be transformed into enumeration of a class of unrooted hypermaps. We first provide a enumeration formula for rooted hypermaps, thereby obtaining the weighted Hurwitz numbers. Next give the quantitative relationship between the enumeration of unrooted hypermaps and that of rooted hypermaps in the general setting. Finally, combining these two results, we obtain a formula for the unrooted hypermaps or the Hurwitz numbers in the special setting.

The Szemerédi Regularity Lemma, in combination with the Blow-up Lemma, form the Regularity Method, a fundamental tool in graph embeddings, albeit restricted to very large and dense graphs. We propose an alternative vertex-partitioning framework that remains effective even as the density tends to zero and without requiring astronomically large vertex sets. This approach, while narrower in scope, extends regularity-type techniques to relatively small graphs previously inaccessible to the Regularity Method. As an application, we use this novel vertex-partitioning method for bipartite packing problems.

The Wiener index of a connected graph is defined as the sum of distances between all its unordered pairs of vertices. Characterising graphs on $n$ vertices with a fixed diameter that maximise the Wiener index is a long-standing open problem. This problem has been resolved fully for trees on $n$ vertices with diameter $d \in \{1,2,3,4,n-3,n-2,n-1\}$ while partial results are available for $d=5$ and $6$. In this context, a conjecture proposed by DeLaViña and Waller has remained open for the last 18 years. In this paper, we establish a necessary condition for a tree to attain the maximum Wiener index among all trees on $n$ vertices with a given diameter. Using this condition, we characterise the maximal trees for diameter $n-4$ and $n-5$. Furthermore, we prove the DeLaViña Waller conjecture for the classes of graphs having $0,1,2,3$ or $n-4$ cut vertices.

This paper considers the properties of Tribonacci numbers on identities, matrices, and determinants. In the first front part, we obtain several symmetric identities of Tribonacci numbers by a matrix-based approach and binomial inversion technique. In the core section of the latter half, we present a determinant representation of Tribonacci numbers in a slightly modified Toeplitz–Hessenberg form derived from Bell polynomials.

Combinatorial counting problems pervade artificial intelligence, statistics, and discrete mathematics. Whether the task is enumerating subsets, multisets, permutations, partitions, or compositions under structural and arithmetic constraints, solving it remains a stubbornly manual exercise. Closed-form derivations are powerful but brittle, while naive encodings to propositional model counting or constraint satisfaction destroy the exchangeability that makes counting tractable in the first place. We present Cofola (COmbinatorial counting LAnguage with First-Order logic), a typed declarative language whose primitives are the combinatorial objects that recur in everyday counting questions, including sets, bags, tuples, sequences, circles, partitions, and compositions, together with natural relational and arithmetic constraints over them. A denotational semantics maps every Cofola program to a well-defined combinatorial counting problem, and a three-phase compilation pipeline (preprocessing, decomposition, and symmetry-preserving encoding) reduces this problem to a weighted first-order model counting (WFOMC) instance augmented with coefficient-extraction constraints. To stay inside known domain-liftable fragments whenever possible, the encoding groups indistinguishable entities, breaks the symmetry of unordered groupings lexicographically, and encodes sequences and circles via order axioms. On a suite of representative combinatorial counting problems, ranging from textbook math problems to multi-object scenarios that the closest prior framework cannot express, Cofola produces concise specifications and a uniform solving pipeline that is practical end-to-end.

A transitive tournament is an acyclic orientation of a complete graph. We study decompositions and packings of the transitive tournament \(TT_n\) into connected two-arc motifs. The three motifs considered are chains, colliders, and forks, which are also fundamental local configurations in directed acyclic graphs. We first construct decompositions of \(TT_n\) into mixtures of these motifs whenever such decompositions exist. We then consider the corresponding pure packing problem for each individual motif. For \(H\) equal to a chain, a collider, or a fork, we determine the maximum number of arc-disjoint copies of \(H\) in \(TT_n\). These results give a precise extremal description of two-arc motif packings in transitive tournaments and suggest further questions on motif decompositions in broader classes of directed acyclic graphs.

Submodular optimization has become a fundamental paradigm for data selection, retrieval, summarization, and representation learning due to its ability to model coverage, diversity, and representativeness. However, classical submodular objectives optimize only the selected subset and do not explicitly preserve structural information between the selected subset and the remaining data. In many modern machine learning applications, including train/validation/test splitting, benchmark construction, and robust subset selection, the quality of a selection depends critically on preserving balanced structure across both the selected subset and its complement. In this work, we introduce Complement Submodular Information (CSI), a new class of complement-aware submodular objectives that quantify shared structural information between a subset and its complement. Our framework induces complement-aware variants of several classical submodular functions including Facility Location, Graph Cut, LogDet, Saturated Coverage, Set Cover, Probabilistic Set Cover, and Feature Based Functions. We analyze the theoretical properties of CSI objectives and show that they exhibit approximate monotonicity under bounded curvature conditions, leading to near-$(1-1/e)$ greedy approximation guarantees. Empirically, CSI objectives consistently outperform standard submodular objectives on robust hidden-slice-aware subset selection. In particular, CSI objectives significantly improve preservation of coherent rare/tail semantic structure while simultaneously suppressing noisy and isolated outliers, leading to substantially improved downstream predictive performance. Synthetic experiments further illustrate how different CSI instantiations capture complementary notions of representativeness, diversity, connectivity, and balanced neighborhood preservation.

We show that for any finite partition of $\mathbb{N}$ there is an infinite sequence whose finite sums are monochromatic and such that infinitely many of the products with a fixed number of factors are monochromatic – though not necessarily belonging to the same color class as the finite sums. We are able to build these infinite configurations in parallel by refining arbitrary partitions of $\mathbb{N}$. We apply these techniques to prove that many complex infinite sum-product configurations are guaranteed to be monochromatic for arbitrary finite colorings of $\mathbb{N}$.

We disprove a long-standing open conjecture due to Simon stating that all skeleta of simplices are extendably shellable. In particular, for every $d \geq 3$ we provide a pure $d$-dimensional shellable simplicial complex which is not shelling completable.

Motivated by Kontsevich's graph complexes, this paper gives a systematic study of matroid complexes. We construct deletion and contraction bicomplexes on the vector space spanned by matroid classes equipped with ground-set orientations, organizing the several naturally arising variants into a single unified framework. We show that direct sum and restriction-contraction make this space into a connected graded Hopf algebra extending Schmitt's matroid Hopf algebra, and use the resulting dg-algebra structure to prove broad acyclicity results. We compute the total, simple, loopless, regular, binary, and ternary matroid complexes through ground-set size $9$, and the connected quotient of the simple loopless regular complex through ground-set size $15$. These computations detect nontrivial homology and lead to a conjectural description in terms of odd-wheel matroids.

We introduce the Morse ensemble polynomial $\ME_K(z_0,\ldots,z_d)$ of a finite simplicial complex $K$, defined as the generating function $\ME_K = \sum_M \prod_i z_i^{c_i(M)}$ over all acyclic matchings $M$ on the face poset of $K$, where $c_i(M)$ counts critical $i$-simplices. This polynomial records the complete critical-vector distribution over all acyclic matchings, equivalently over all discrete gradient vector fields arising from discrete Morse functions on $K$, and is an isomorphism invariant of simplicial complexes. In dimension one, this invariant recovers the Chari–Joswig graph formula for the $f$-vector of the discrete Morse complex in a two-variable Morse-vector normalization: $\ME_G=z_1^{m-n}\det(z_0z_1\,I_n+L_G)$ for a connected graph $G$. The main new contributions are higher-dimensional and structural. First, we prove a Top-Face Recursion for adding a top-dimensional simplex, with a non-liftable correction term $F(K,\sigma,\tau)$. The vanishing and leading obstruction of this correction term are controlled by the top incidence graph: an incidence-separation criterion detects exactly when $F=0$, a leading obstruction layer is governed by shortest obstruction paths, and a tree-like (incidence-forest) regime of the top incidence structure gives a correction-free higher-dimensional recursion, including stacked balls as a concrete class. Second, we introduce the independence ME polynomial $\Phi(G):=\ME_{\mathrm{Ind}(G)}$, a graph invariant which strictly refines the graph-level Morse ensemble, separates examples not distinguished by $T_G$ and $I(G;t)$, and records collapse-level information of $\mathrm{Ind}(G)$ through coefficients such as $[z_0]\Phi(G)$.