Simple Treewidth: A Refined Graph Parameter

• Simple treewidth is defined as the minimum k for which a graph admits a k-simple tree-decomposition that restricts every k-set to appear in at most two bags.
• It is tightly related to ordinary treewidth, with the parameter differing by at most one unit, thereby providing a stricter measure of graph structure.
• This refined notion facilitates sparse directed-product embeddings by controlling repeated attachments of k-cliques in simple k-trees.

Simple treewidth is a refinement of ordinary treewidth defined by restricting how often a $k$-set may recur in a width-$k$ tree-decomposition. A tree-decomposition $(T,\mathcal B)$ of a graph $G$ is $k$-simple if it has width at most $k$ and, for every set $S\subseteq V(G)$ of $k$ vertices, the number of bags containing $S$ is at most two; the simple treewidth $\operatorname{stw}(G)$ is the minimum such $k$0. Equivalently, $k$1 if and only if $k$2 is a subgraph of a simple $k$3-tree, where the same $k$4-clique is not used more than once in the recursive construction. This makes simple treewidth a one-step tightening of ordinary treewidth and, in recent work, the exact condition under which sparse directed-product embeddings exist with both factors having treewidth at most $k$5 (Hendrey et al., 15 Aug 2025).

1. Definition and equivalent formulations

For a finite graph $k$6, a tree-decomposition is a pair $k$7 in which each bag $k$8, every vertex of $k$9 appears in some bag, every edge of $(T,\mathcal B)$0 has both endpoints in some bag, and for each $(T,\mathcal B)$1 the set of bags containing $(T,\mathcal B)$2 induces a connected subtree of $(T,\mathcal B)$3. Its width is $(T,\mathcal B)$4, and ordinary treewidth $(T,\mathcal B)$5 is the minimum width over all tree-decompositions (Mazoit, 2013).

Simple treewidth adds a multiplicity restriction at the level of $(T,\mathcal B)$6-subsets. A tree-decomposition $(T,\mathcal B)$7 is $(T,\mathcal B)$8-simple if it has width at most $(T,\mathcal B)$9 and satisfies

$G$0

for every $G$1-set $G$2. The parameter $G$3 is the minimum $G$4 for which $G$5 has a $G$6-simple tree-decomposition (Hendrey et al., 15 Aug 2025).

The equivalent constructive model is given by simple $G$7-trees. A $G$8-tree is obtained from $G$9 by repeatedly adding a new vertex adjacent to all vertices of an existing $k$0-clique. A simple $k$1-tree imposes the extra rule that the same $k$2-clique is not used more than once. The equivalence

$k$3

is structurally decisive, because it turns simple treewidth into a statement about controlled recursive attachment along $k$4-cliques rather than merely about bag sizes (Hendrey et al., 15 Aug 2025).

2. Position relative to ordinary treewidth

Simple treewidth is always within one of ordinary treewidth: $k$5 It is therefore a stricter parameter, but only by one unit in the worst case (Hendrey et al., 15 Aug 2025).

At low values, simple treewidth has exact characterizations that sharply distinguish it from unconstrained treewidth.

Condition	Graph class
connected and $k$6	paths
$k$7	outerplanar graphs
$k$8	planar graphs with treewidth $k$9

These identifications are exact: a connected graph has simple treewidth $k$0 if and only if it is a path; a graph has simple treewidth at most $k$1 if and only if it is outerplanar; and a graph has simple treewidth at most $k$2 if and only if it is planar and has treewidth $k$3 (Hendrey et al., 15 Aug 2025).

These characterizations show that simple treewidth does not merely repackage ordinary treewidth. Outerplanar graphs already appear at simple treewidth $k$4, while the next step, simple treewidth $k$5, captures precisely the planar treewidth-$k$6 regime. This suggests that the extra multiplicity restriction is sensitive to how clique attachments are reused, rather than only to the largest bag size.

3. Repeated stacking and the basic obstruction

The simplest obstruction to small simple treewidth is repeated attachment to the same $k$7-clique. The paper (Hendrey et al., 15 Aug 2025) isolates the graph $k$8, obtained from $k$9 by making the $S\subseteq V(G)$0-vertex side a clique. If $S\subseteq V(G)$1 denotes that $S\subseteq V(G)$2-clique side and $S\subseteq V(G)$3 are the other three vertices, then each $S\subseteq V(G)$4 is a $S\subseteq V(G)$5-clique. Consequently, in every width-$S\subseteq V(G)$6 tree-decomposition, the set $S\subseteq V(G)$7 must lie in at least three bags, and therefore

$S\subseteq V(G)$8

This obstruction makes the gap between treewidth and simple treewidth concrete. Ordinary treewidth $S\subseteq V(G)$9 allows arbitrarily many vertices to be attached to the same $k$0-clique. Simple treewidth forbids exactly this kind of repeated stacking. In the simple $k$1-tree formulation, the same rule appears as “use each $k$2-clique at most once”; in the decomposition formulation, it appears as “no $k$3-set occurs in more than two bags” (Hendrey et al., 15 Aug 2025).

A useful interpretation is that ordinary treewidth controls bag size, whereas simple treewidth controls bag size together with the repeated use of separators or attachment cliques. This interpretation is not a separate theorem, but it matches the obstruction mechanism exhibited by $k$4.

4. Sparse directed-product embeddings

The main modern structural theorem for simple treewidth concerns embeddings into sparse directed products. For directed graphs $k$5, their directed product has vertex set $k$6, and there is an arc from $k$7 to $k$8 if either $k$9 and $S$0, or $S$1 and $S$2, or both coordinates move along arcs simultaneously; arcs of the third type are diagonal arcs. The paper works with the underlying undirected graph of this product (Hendrey et al., 15 Aug 2025).

If the factor indegrees are bounded,

$S$3

then the directed product has indegree at most $S$4, so any subgraph of the underlying undirected product is sparse in the sense that its number of edges is linearly bounded in its number of vertices (Hendrey et al., 15 Aug 2025).

Against this background, the central theorem is: $S$5 with

$S$6

Thus every graph of simple treewidth $S$7 is contained in a sparse directed product of two factors, each of treewidth at most $S$8 and maximum indegree at most $S$9 (Hendrey et al., 15 Aug 2025).

Two corollaries are especially concrete.

Graph class	Directed-product factors
outerplanar graphs	trees with maximum indegree $\operatorname{stw}(G)$0
planar graphs with treewidth $\operatorname{stw}(G)$1	graphs of treewidth $\operatorname{stw}(G)$2 and maximum indegree $\operatorname{stw}(G)$3

The first follows from $\operatorname{stw}(G)$4 for outerplanar graphs; the second from the characterization of simple treewidth $\operatorname{stw}(G)$5 (Hendrey et al., 15 Aug 2025).

The treewidth bound on the factors is best possible. For every $\operatorname{stw}(G)$6, there exists a graph $\operatorname{stw}(G)$7 with $\operatorname{stw}(G)$8 that is not contained in any directed product of two digraphs with indegrees bounded by $\operatorname{stw}(G)$9 and both factor treewidths at most $k$00 (Hendrey et al., 15 Aug 2025). In that sense, the factor bound $k$01 is optimal.

5. Proof architecture of the embedding theorem

The proof of the sparse directed-product theorem is not formulated at the level of arbitrary $k$02-simple decompositions. It first reduces to simple $k$03-trees, using the equivalence between $k$04 and containment in a simple $k$05-tree (Hendrey et al., 15 Aug 2025).

The induction maintains an embedding of the current graph $k$06 into a directed product

$k$07

together with several invariants: each factor has maximum indegree at most $k$08, each factor has treewidth at most $k$09, every $k$10-clique projects in each factor to a transitive tournament, each used big sibling of a diagonal $k$11-clique is adjacent to all vertices of that clique, and no two diagonal $k$12-cliques share a common unused big sibling (Hendrey et al., 15 Aug 2025).

A $k$13-clique is diagonal if its vertices have pairwise distinct first coordinates and pairwise distinct second coordinates. Such a clique determines a unique big diagonal edge and two corresponding big siblings. The induction then splits into two cases when a new vertex is attached along a $k$14-clique $k$15.

If $k$16 is not diagonal, then one projection of $k$17 has size at most $k$18. A new factor vertex is added in the corresponding factor, adjacent to a clique of size at most $k$19, and the new graph vertex is placed using that new coordinate together with a sink in the transitive tournament on the other projection. This preserves both the indegree bound and the factor treewidth bound (Hendrey et al., 15 Aug 2025).

If $k$20 is diagonal, the argument does not enlarge either factor. Instead it places the new vertex at one of the two big sibling coordinates of $k$21, provided that coordinate pair is unused. The reason such an unused big sibling must exist is precisely the simple-treewidth obstruction: if both big siblings were already used, then one would create a copy of $k$22, contradicting $k$23 (Hendrey et al., 15 Aug 2025).

This is the point at which simple treewidth, rather than ordinary treewidth, becomes decisive. The proof does not merely require bounded bag size; it requires the prohibition on repeated attachments to the same $k$24-clique.

6. Relation to ordinary treewidth and nearby notions

Ordinary treewidth remains the ambient reference parameter. It has the exact dual characterization

$k$25

where $k$26 is the maximum order of a bramble in $k$27 (Mazoit, 2013). Simple treewidth is not accompanied in the cited material by an analogous bramble duality, and the sparse directed-product theorem makes clear that it behaves differently from ordinary treewidth in another structural direction.

The contrast is sharpest in the directed-product setting. For ordinary treewidth, the positive theorem above fails completely: for any integers $k$28, there exists a graph $k$29 with $k$30 that is not contained in the directed product of $k$31 for any directed graphs with

$k$32

Simple treewidth is therefore not a cosmetic strengthening of treewidth; in this setting it is exactly the condition that makes bounded-indegree, lower-treewidth factorization possible (Hendrey et al., 15 Aug 2025).

A different recent direction studies decompositions with controlled vertex multiplicity rather than simple treewidth itself. Every graph of treewidth $k$33 has a tree-decomposition of width at most $k$34 in which each vertex $k$35 has spread at most $k$36, and stronger results simultaneously bound width, order, and the degree of the decomposition tree (Wood, 1 Sep 2025). The same work explicitly notes that this is not the standard parameter simple treewidth. This suggests bounded spread and simple treewidth are adjacent but distinct themes: both constrain recurrence inside decompositions, but they constrain different combinatorial objects.

The term should also be distinguished from edge-cut based analogues such as edge-cut width and slim tree-cut width. Those parameters are presented as simple or slim edge-based analogues of treewidth, but they are not variants of simple treewidth in the graph-theoretic sense discussed here (Brand et al., 2022, Ganian et al., 2022).