Path-Aligned Graph Products

• Path-aligned graph products are constructions that layer a canonical path with a bounded-treewidth graph to control structural complexity and facilitate algorithmic applications.
• They employ techniques such as BFS layering, recursive decomposition, and quotient graph construction to decompose graphs from planar, minor-closed, and other classes.
• These products yield tight bounds on layout parameters like stack and queue numbers, while transferring coloring, twin-width, and topological properties across diverse graph families.

A path-aligned graph product is a class of graph constructions in which one factor is a path (or an object organized canonically "along" a path), and the overall product is formed so that structural or combinatorial complexity is controlled by "layering" against this path. The prevailing realization of this notion is the strong graph product $P\boxtimes H$, where $P$ is a (possibly augmented) path and $H$ is a graph of bounded treewidth or similar combinatorial constraint. Such product decompositions, now central in structural graph theory, underpin results on minor-closed classes, geometric intersection graphs, and a variety of "beyond planarity" families, and provide a transfer mechanism to lift bounded treewidth properties to much broader classes (Dvořák et al., 2020).

1. Definitions and Fundamental Constructions

The strong product $G\boxtimes H$ for graphs $G,H$ has vertex set $V(G)\times V(H)$. Distinct vertices $(g,h),(g',h')$ are adjacent if and only if

• $g = g'$ and $hh'\in E(H)$; or
• $h = h'$ and $gg'\in E(G)$; or
• $gg'\in E(G)$ and $hh'\in E(H)$.

A path-aligned product structure for a graph class $\mathcal G$ means that there exists a uniform constant $k$ such that every $G\in\mathcal G$ is a subgraph $G \subseteq P \boxtimes H$ for some path $P$ and graph $H$ with $\operatorname{tw}(H)\leq k$ (Dvořák et al., 2020). More generally, some decompositions allow additional low-complexity factors (cliques or small powers of paths).

In key applications, $P$ is a canonical path capturing breadth-first layering, and $H$ is a quotient capturing all inter-layer adjacencies, so structural and coloring parameters transfer from $H$ to $G$ via the product representation.

2. Path-Aligned Product Structure Theorems

Planar and Minor-Closed Classes

Every $n$-vertex planar graph $G$ admits

$G \subseteq H \boxtimes P$

for some path $P$ and $H$ of $\operatorname{tw}(H)\leq 8$ (Morin, 2020, Dvořák et al., 2020). The construction is explicit: decompose $G$ through BFS layering, partition vertices along vertical paths, then set $H$ as the quotient contracted along these paths. An $O(n\log n)$-time algorithm computes $H,P$ and the embedding map $\varphi$ (Morin, 2020).

Extensions encompass arbitrary genus-$g$ graphs, apex-minor-free, and general proper minor-closed classes via clique-sum decompositions, each yielding analogous $P\boxtimes H$ or $P\boxtimes H\boxtimes K_c$ embeddings with explicit $\operatorname{tw}(H)$ and $c$ (Dvořák et al., 2020).

Non-Minor-Closed and Framed Graphs

The path-aligned strong product paradigm applies to $k$-planar graphs, optimal $h$-framed graphs, string graphs, map graphs, powers of graphs, and more. For $k$-planar graphs, the following holds: $G \subseteq H_k \boxtimes P_n,$ with $H_k$ of treewidth $O(k^3)$ (Dujmović et al., 2019). For $h$-framed graphs (which include 1-planar, 2-planar, and $k$-map graphs), every such $G$ embeds as

$G \subseteq P \boxtimes H \boxtimes K_c,$

$H$ planar with $\operatorname{tw}(H)\le3$, and $c=3\lfloor h/2\rfloor + \lfloor h/3\rfloor - 1$ (Bekos et al., 2022). This structural theory is fully constructive with explicit $O(n^2)$ algorithms for decomposition (Bekos et al., 2022).

3. Graph Layouts and Algorithmic Consequences

Stack and Queue Numbers

Path-aligned graph product decompositions provide tight control over layout parameters:

• For $G = P_n\boxtimes H$ with $H$ of pathwidth $p$, the stack-number $\operatorname{sn}(G) \le 5p+2$ (Pupyrev, 2020). For bipartite $H$ of treewidth $t$ and degree $\Delta$, $\operatorname{sn}(G)\le 3(t+1)\Delta+1$ (Pupyrev, 2020).
• In three dimensions, the stack-number $$\operatorname{sn}(P_n\boxtimes P_n\boxtimes P_n) = \Theta(n^{1/3})$$, resolving the first explicit bounded-degree graph family with bounded queue-number and unbounded stack-number. This separation exploits Gromov's topological overlap theorem (Eppstein et al., 2022).
• Queue-number for $G\subseteq P\boxtimes H\boxtimes K_c$ with $\operatorname{tw}(H)\le3$ is $$\operatorname{qn}(G)\le 3c\operatorname{qn}(H) + \tfrac32 c$$, yielding $\operatorname{qn}(G) = O(h)$ for $h$-framed graphs (Bekos et al., 2022).

Coloring Parameters and Twin-Width

• Non-repetitive chromatic number: $\pi(G)\le 4^{\,\operatorname{tw}(H)+1} c$ for $G\subseteq P\boxtimes H\boxtimes K_c$ (Bekos et al., 2022).
• $p$-centered coloring: $\chi_p(G)\in O(c p^3\log p)$ for $G$ in path-aligned product structure with planar $\operatorname{tw}(H)\le3$ (Bekos et al., 2022).
• Twin-width for simple planar and 1-planar graphs can be reduced to $37$ and $80$ respectively through path-aligned contraction schemes (Bekos et al., 2022).

4. Algebraic and Topological Properties: Independence Complexes

For path-aligned products $P_n\boxtimes P_m$, the independence complex $I(P_n\boxtimes P_m)$ exhibits recursive suspension–join structure, often yielding a wedge of spheres or contractible spaces. For $m=2,3,4$, detailed recursive homotopy type decompositions are established (Bravo, 9 May 2025). All induced subgraphs of such products are shown to be in the class $\mathcal{SW}$ (all induced subgraphs contractible or wedge of spheres), and the results extend to lexicographic products via polyhedral joins (Bravo, 9 May 2025).

5. Methodologies and Structural Proof Schemes

Path-aligned structure theorems commonly proceed via:

1. BFS Layering: Construct a breadth-first search tree and partition into paths or layers such that conflicts or shortcuts can be controlled.
2. Recursive Decomposition: Vertices are grouped into parts (vertical paths, superlayers, or bags) managed recursively, often leveraging cycle decomposition and bounded treewidth partitioning (Bekos et al., 2022, Morin, 2020).
3. Shortcut Systems: For non-minor-closed classes, additional edges (“shortcuts”) are modeled via controlled-length paths with bounded vertex participation; this innovation preserves bounded treewidth in extended products (Dujmović et al., 2019).
4. Quotient Graph Construction: The quotient with respect to partition encodes complex interactions, while the path index preserves linear structure. The assembly ensures that the original graph is captured as a subgraph of the product, with all adjacencies justified against the product’s three types of adjacency (Morin, 2020).

6. Extensions, Limitations, and Open Problems

Generalizations

• The product structure framework extends beyond planar graphs: graphs of bounded Euler genus, apex-minor-free, bounded-degree minor-free, bounded-parameter geometric intersection graphs, $k$-planar, map graphs, and string graphs all fall under the same paradigm via pathway-and-bounded-scaffold decompositions (Dvořák et al., 2020, Bekos et al., 2022).
• Classes of polynomial growth are precisely those admitting a finite sequence of path-aligned product embeddings, each factor of bounded pathwidth/degree, which is both necessary and sufficient (Dvořák et al., 2020).

Known Barriers and Open Questions

• Three-dimensional products demonstrate that queue-number and stack-number may diverge in path-aligned product classes, with sharp thresholds for parameter unboundedness depending on degree (Eppstein et al., 2022).
• The minimum degree $\Delta_0\in\{6,7\}$ such that a bounded-degree family with bounded queue-number and unbounded stack-number exists is unresolved (Eppstein et al., 2022).
• There are conjectures regarding the extension of efficient algorithms and coloring bounds from $G\subseteq H\boxtimes P$ classes to broader expansion regimes and more general geometric or minor-excluded classes (Dvořák et al., 2020).
• Homotopy types of independence complexes for larger product graphs, stability under induced subgraphs, and generalizations to higher-dimensional lexicographic products are current research frontiers (Bravo, 9 May 2025).

7. Illustrative Examples

Path-Aligned Product in 1-Planar Graphs

For the 1-planar “crossed quadrilateral” (4-cycle with two crossing diagonals), the skeleton is a 4-cycle, yielding a 4-framed graph. Superlayer merging groups vertices as $\{a,b,d\},\{c\}$, the quotient $H$ is a path, and the clique $K_2$ suffices. The embedding demonstrates $G\subseteq P\boxtimes H\boxtimes K_2$ (Bekos et al., 2022).

Hamiltonicity in Generalized Lexicographic Path Products

For $P_n[H_1,\dots,H_n]$, necessary and sufficient conditions for Hamiltonicity, traceability, and Hamiltonian-connectivity are given in terms of the linear forest capacity $\pi(H_i)$; for identical $H_i=H$, the criteria become linear in $n$ and $\pi(H)$ (Ekstein et al., 2020).

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Table: Structural Product Representations for Key Graph Classes

Graph Class	Product Structure	Key Bound(s)
Planar	$H\boxtimes P$	$\operatorname{tw}(H)\le 8$
Euler genus $g$	$H\boxtimes P \boxtimes K_c$	$\operatorname{tw}(H)\le 9,\,c=O(g)$
$k$-Planar	$H_k\boxtimes P$	$\operatorname{tw}(H_k) = O(k^3)$
$h$-Framed	$P\boxtimes H \boxtimes K_c$	$\operatorname{tw}(H)\le 3,\,c=O(h)$
Bounded-degree minor-free	$H\boxtimes P$	$\operatorname{tw}(H) = O(\Delta)$

Specific bounds and parameter formulas for coloring, twin-width, and layout parameters depend on the class and are given in the references (Dvořák et al., 2020, Bekos et al., 2022, Dujmović et al., 2019, Pupyrev, 2020).

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Path-aligned graph products thus constitute a central structural tool in contemporary graph theory—unifying bounded-width decompositions, enabling algorithmic and topological transfer, and exposing sharp dichotomies in graph layout parameters across dimensions and complexity regimes (Dvořák et al., 2020, Morin, 2020, Bekos et al., 2022, Dujmović et al., 2019, Bravo, 9 May 2025, Eppstein et al., 2022).