Minimal D-Decomposition Tree

• The minimal D-decomposition tree is a tree-structured partition of a network where clusters are irreducible relative to a specific structural criterion, such as d-convexity in Bayesian networks.
• It factorizes probability distributions by using minimal d-decomposers, supporting localized computations and parallel inference without relying on full moralization or triangulation.
• This framework extends to graphs, hypergraphs, density models, and coding theory, optimizing structures by minimizing separator redundancy, bag count, or tree depth.

Searching arXiv for the term and closely related decomposition literature to ground the article in cited papers. A minimal D-decomposition tree denotes a tree-structured decomposition in which the decomposition units are irreducible relative to a chosen structural criterion $D$. In the most explicit current usage, for Bayesian networks it is a reduced tree of d-convex clusters whose edge intersections are minimal d-decomposers; adjacent literatures use closely aligned notions under other names, including proper tree decompositions induced by minimal triangulations, minimal tree projections, minimum monotone-tree sets, minimum decomp trees of maxmin trees, minimal tree realizations of codes, and minimum-depth elimination trees (Heng et al., 6 Jul 2026, Ravid et al., 2017, Greco et al., 2012, Magee et al., 2023, Huang et al., 2023, 0711.1383, Trimble, 2020).

1. Directed formulation for Bayesian networks

For a DAG $\mathcal{G}=(V,E)$, the defining ingredient is directed convexity. If $A \subseteq V$, the induced subgraph $\mathcal{G}_A$ is directed convex, or d-convex, if for every pair of non-adjacent vertices $u,v \in A$, there is no path $l_{uv}$ between $u$ and $v$ such that

$$A \cap V^o(l_{uv}) \subseteq V^c(l_{uv}) \subseteq An_{\mathcal{G}}(\{u,v\}),$$

where $V^o(l_{uv})$ is the set of internal vertices of the path, $\mathcal{G}=(V,E)$0 is the set of collider vertices on the path, and $\mathcal{G}=(V,E)$1 is the ancestor set of $\mathcal{G}=(V,E)$2 or $\mathcal{G}=(V,E)$3. Such a path is an inducing path, or information path, over $\mathcal{G}=(V,E)$4 (Heng et al., 6 Jul 2026).

A directed decomposition of a Bayesian network $\mathcal{G}=(V,E)$5 is specified by mutually disjoint sets $\mathcal{G}=(V,E)$6 satisfying $\mathcal{G}=(V,E)$7, such that $\mathcal{G}=(V,E)$8 d-separates $\mathcal{G}=(V,E)$9 and $A \subseteq V$0 in $A \subseteq V$1, and such that $A \subseteq V$2 is d-convex in $A \subseteq V$3. The set $A \subseteq V$4 is then called a directed decomposer, or d-decomposer. If both $A \subseteq V$5 and $A \subseteq V$6 are non-empty, the decomposition is proper. A minimal d-decomposer is a d-decomposer that is also a minimal d-separator, meaning that there exist vertices $A \subseteq V$7 for which $A \subseteq V$8 is a minimal $A \subseteq V$9-d-separator (Heng et al., 6 Jul 2026).

Given a reduced tree $\mathcal{G}_A$0 whose nodes are subsets $\mathcal{G}_A$1, $\mathcal{G}_A$2 is a minimal d-decomposition tree if, for every edge $\mathcal{G}_A$3, the intersection

$\mathcal{G}_A$4

is a minimal d-decomposer in $\mathcal{G}_A$5, and each node $\mathcal{G}_A$6 cannot be further decomposed by any minimal d-decomposer of $\mathcal{G}_A$7. The minimality here is structural rather than width-based: separators are minimal d-separators and clusters are irreducible under the same separator notion (Heng et al., 6 Jul 2026).

2. Collapsibility, factorization, and construction

The central reason for imposing d-convexity is collapsibility. If $\mathcal{G}_A$8 is d-convex in $\mathcal{G}_A$9, then

$u,v \in A$0

so the induced submodel on $u,v \in A$1 is exactly the marginal model of the global Bayesian network on $u,v \in A$2. For a single decomposition $u,v \in A$3, this yields the factorization

$u,v \in A$4

and, recursively over a minimal d-decomposition tree $u,v \in A$5 with clusters $u,v \in A$6,

$u,v \in A$7

This gives a directed analogue of clique-separator factorization, but without moralization or triangulation as a primary representation layer (Heng et al., 6 Jul 2026).

Construction proceeds in two stages. First, one builds a minimal d-separator tree. Second, one tests every separator $u,v \in A$8 for d-convexity. The d-convexity test reduces inducing-path detection to a path search in a moral graph: for non-adjacent $u,v \in A$9, the existence of an inducing path is equivalent to the existence of a path in $l_{uv}$0 whose internal vertices all lie in $l_{uv}$1. The resulting DCTest runs in $l_{uv}$2, and the overall construction of a minimal d-decomposition tree runs in $l_{uv}$3, where $l_{uv}$4 and $l_{uv}$5 (Heng et al., 6 Jul 2026).

The same structure supports local and parallel computation. Parameter estimation can be performed independently on every cluster $l_{uv}$6 and every separator $l_{uv}$7, after which the global model is reconstructed by the product-over-clusters and division-over-separators formula. For inference, leaf clusters that do not contain unique query or evidence variables can be pruned; if a single cluster remains, variable elimination suffices, otherwise belief propagation is run on the pruned tree. Reported experiments show faster exact inference than junction-tree belief propagation for low-dimensional queries, with the advantage diminishing as query size grows (Heng et al., 6 Jul 2026).

3. Graph and hypergraph analogues

In graph decomposition theory, the closest analogue is the proper tree decomposition. For a graph $l_{uv}$8, a tree decomposition is proper if it is not strictly subsumed by another decomposition via bag splitting or bag removal without bag growth. A fundamental equivalence states that a tree decomposition $l_{uv}$9 of $u$0 is proper if and only if it is a clique tree of a minimal triangulation of $u$1. Thus a minimal decomposition tree, in this sense, is a clique tree of a minimal triangulation, and every such decomposition is non-redundant at the level of bags and separators (Ravid et al., 2017).

That framework also supports optimization and enumeration. The relevant costs are split-monotone bag costs, including width,

$u$2

fill-in,

$u$3

weighted variants, hypertree-width-style bag costs, and composite costs such as

$u$4

Under the poly-MS assumption, or under fixed width bound $u$5, one can compute an optimal minimal triangulation and enumerate all minimal triangulations, hence all proper tree decompositions, in increasing order of cost with polynomial delay (Ravid et al., 2017).

A related hypergraph formulation is given by tree projections. If $u$6, a tree projection is an acyclic hypergraph $u$7 such that

$u$8

A minimal tree projection is one minimal under the partial order $u$9. Every minimal tree projection is reduced, has the same node set as $v$0, preserves $v$1-components, and admits a normal-form join tree that is both $v$2-connected and an $v$3-component tree. The same paper gives a Captain–Robber game characterization, with monotone and non-monotone winning strategies having equal power. This suggests that, when $v$4 denotes a structural decomposition method such as treewidth or generalized hypertree width, a minimal D-decomposition tree is naturally realized as a normal-form join tree of a minimal tree projection (Greco et al., 2012).

4. Function-valued trees, density decompositions, and maxmin trees

For densities on finite metric trees, minimality takes the form of minimizing the number of unimodal summands. A tame density $v$5 on a finite metric tree $v$6 admits a decomposition

$v$7

into unimodal components, and the smallest such $v$8 is the unimodal category $v$9. The constructive algorithm repeatedly finds a mode-forced vertex, performs the sweeping operation $$A \cap V^o(l_{uv}) \subseteq V^c(l_{uv}) \subseteq An_{\mathcal{G}}(\{u,v\}),$$0, subtracts the remainder $$A \cap V^o(l_{uv}) \subseteq V^c(l_{uv}) \subseteq An_{\mathcal{G}}(\{u,v\}),$$1, and continues until zero. The resulting decomposition is minimal, meaning that $$A \cap V^o(l_{uv}) \subseteq V^c(l_{uv}) \subseteq An_{\mathcal{G}}(\{u,v\}),$$2. The running time is $$A \cap V^o(l_{uv}) \subseteq V^c(l_{uv}) \subseteq An_{\mathcal{G}}(\{u,v\}),$$3, or $$A \cap V^o(l_{uv}) \subseteq V^c(l_{uv}) \subseteq An_{\mathcal{G}}(\{u,v\}),$$4 if every unimodal component has uniformly bounded support size $$A \cap V^o(l_{uv}) \subseteq V^c(l_{uv}) \subseteq An_{\mathcal{G}}(\{u,v\}),$$5 (Baryshnikov et al., 2018).

For density graphs with cycles, the analogue is the minimum M-Tree Set: a smallest collection of rooted monotone subtrees whose vertex-wise densities sum to the observed density graph. The decision version is NP-complete, and the same is true for the complete, strong, and full variants. On trees, however, a greedy monotone-sweeping algorithm computes a minimum M-Tree Set exactly, and on density cactus graphs there is a 3-approximation for minimum SM-Tree Sets. This establishes a distinct meaning of “minimal decomposition tree”: not an irreducible separator tree, but a minimum-cardinality family of monotone trees reconstructing a graph-defined density (Magee et al., 2023).

For maxmin trees, the paper introduces what it explicitly calls the minimum decomp tree. If $$A \cap V^o(l_{uv}) \subseteq V^c(l_{uv}) \subseteq An_{\mathcal{G}}(\{u,v\}),$$6, where $$A \cap V^o(l_{uv}) \subseteq V^c(l_{uv}) \subseteq An_{\mathcal{G}}(\{u,v\}),$$7 is the minimum, the minimum decomp tree is obtained by connecting $$A \cap V^o(l_{uv}) \subseteq V^c(l_{uv}) \subseteq An_{\mathcal{G}}(\{u,v\}),$$8 to the minimum in each subtree $$A \cap V^o(l_{uv}) \subseteq V^c(l_{uv}) \subseteq An_{\mathcal{G}}(\{u,v\}),$$9 rather than to a descent. The tree is rooted at $V^o(l_{uv})$0, and its subtree relation coincides with the subtree relation in the associated max-weight maxmin tree. Every leaf is a descent, and every descent is a leaf. This yields a weight formula in which the weight of a permutation is obtained by summing, over non-descents, the number of leaves in the corresponding subtree and subtracting $V^o(l_{uv})$1. The same structure is then used to identify coefficients of the stabilized series $V^o(l_{uv})$2 with the partition numbers $V^o(l_{uv})$3 counted by OEIS A256193 (Huang et al., 2023).

5. Code realizations, treedepth, and minimum bag count

In coding theory, a tree decomposition of the coordinate set of a linear code is a mapping $V^o(l_{uv})$4, and a tree realization augments this with state spaces on edges and local constraint codes on vertices. For any fixed tree decomposition $V^o(l_{uv})$5, there exists a unique minimal tree realization $V^o(l_{uv})$6 that minimizes the state-space dimension at every edge. The paper further shows that this minimal realization also minimizes the local constraint code dimension at every vertex. Its state spaces satisfy

$V^o(l_{uv})$7

with

$V^o(l_{uv})$8

and the treewidth of the code is the least constraint complexity over all tree realizations. Here a minimal D-decomposition tree means either the unique minimal realization extending a fixed decomposition tree, or a tree decomposition that attains the code’s treewidth (0711.1383).

For treedepth, the corresponding object is the minimum-depth elimination tree or forest. A rooted forest $V^o(l_{uv})$9 on $\mathcal{G}=(V,E)$00 is a treedepth decomposition if every edge $\mathcal{G}=(V,E)$01 has one endpoint ancestral to the other in $\mathcal{G}=(V,E)$02, and the treedepth of $\mathcal{G}=(V,E)$03 is the minimum possible depth. The exact algorithm in the cited work recursively selects roots, deletes them, and recurses on connected components, using degree-based and path-based lower bounds, symmetry breaking, and domination rules. The implementation uses $\mathcal{G}=(V,E)$04 space. In this setting, a minimal D-decomposition tree is an optimal treedepth decomposition, equivalently a minimum-height elimination tree (Trimble, 2020).

A third minimality notion is minimum bag count under a fixed width bound. For a graph $\mathcal{G}=(V,E)$05 and fixed $\mathcal{G}=(V,E)$06, the Minimum Size Tree Decomposition problem asks for the smallest number of bags in any tree decomposition of width at most $\mathcal{G}=(V,E)$07, and the analogous path version asks the same for path decompositions. For each fixed $\mathcal{G}=(V,E)$08, both problems are NP-complete. Nevertheless, for fixed $\mathcal{G}=(V,E)$09, both can be solved exactly in time

$\mathcal{G}=(V,E)$10

and, assuming ETH, neither can be solved in time

$\mathcal{G}=(V,E)$11

for fixed $\mathcal{G}=(V,E)$12. This establishes yet another sense of minimal decomposition tree, where minimality refers to the number of bags rather than irreducibility, separator minimality, or minimum depth (Bodlaender et al., 2016).

6. Comparative interpretation and scope

Across these literatures, “minimal D-decomposition tree” denotes a family of closely related but non-identical concepts. The common pattern is that a tree-organized representation is made canonical or optimal relative to a domain-specific criterion $\mathcal{G}=(V,E)$13: minimal separators in Bayesian-network decomposition, non-redundant bags in graph triangulation, minimal tree projections in hypergraphs, minimum component count in density decomposition, minimum state or constraint complexity in code realizations, minimum depth in treedepth, or minimum bag count at fixed width (Heng et al., 6 Jul 2026, Ravid et al., 2017, Greco et al., 2012, Magee et al., 2023, 0711.1383, Trimble, 2020, Bodlaender et al., 2016).

Domain	Decomposition object	Minimality notion
Bayesian networks	Tree of d-convex clusters	Minimal d-decomposers and irreducible clusters
Graphs / hypergraphs	Proper tree decomposition or join tree	Minimal triangulation or minimal tree projection
Densities / maxmin trees	Unimodal, monotone, or minimum decomp tree	Minimum component count or minimum attachment rule
Codes / treedepth / bag-size	Tree realization, elimination tree, bounded-width decomposition	Minimum local complexity, minimum depth, or minimum number of bags

This suggests that the expression is best understood relationally: $\mathcal{G}=(V,E)$14 names the decomposition framework, and “minimal” specifies the governing partial order or optimization criterion inside that framework. In the strictest formal sense currently available, the term refers to the Bayesian-network construction based on d-convexity and minimal d-separators (Heng et al., 6 Jul 2026). In broader usage, it functions as an umbrella for structurally irredundant or cost-optimal tree-structured decompositions across several areas of combinatorics, graphical models, coding theory, and topological or density-based analysis (Ravid et al., 2017, Greco et al., 2012, Baryshnikov et al., 2018, Huang et al., 2023).