Graph-Based Open Books

• Graph-based open books are geometric structures that use a metric graph as a spine to which two-dimensional pages are attached, forming stratified domains.
• They enable dimensional reduction in nonlinear variational PDEs by linking ground state behaviors to spectral properties and symmetry-breaking transitions.
• These constructions bridge geometric topology, contact and Legendrian knot theory, and combinatorial graph embeddings, offering practical tools for both analysis and algorithm design.

A graph-based open book is a geometric or topological structure in which a base metric graph serves as the combinatorial or geometric "spine", and pages (typically modeled as 2-dimensional rectangles or strips) are attached along the edges or vertices of the graph. These constructions unify diverse domains across geometric topology, variational analysis, and combinatorial graph theory. The "open book" terminology captures the core idea: a union of pages, each sharing boundaries ("bindings"), forming a book-like stratified space whose properties are encoded or controlled by an underlying graph. In recent literature, "graph-based open books" arise as precise objects enabling the resolution of nonlinear variational PDEs via dimensional reduction, as well as in the combinatorial encoding of contact 3-manifolds and Legendrian knot theory through graph-structured Morse diagrams.

1. Geometric and Analytic Definition of Graph-Based Open Books

A graph-based open book is a two-dimensional domain, $L$, constructed from a metric graph $\Gamma$ and a finite transverse interval. Formally, for half-width $b>0$, the domain is

$\Omega_b = \Gamma \times [-b, b],$

with the structure and topology inherited from the graph and the product metric. Each "page" corresponds to a rectilinear subset $[0, L^1_k] \times [0, L^2_k]$, isometric to a rectangle, whose boundaries ("bindings") are joined along the edges or vertices in a manner mirroring the incidence structure of $\Gamma$. The Laplacian on $L$ is defined via the Sobolev space

$$H^1_D(L) = \Bigl\{\,u=(u_k):\,u_k\in H^1(P_k),\ u_k|_B \equiv u_{k'}|_B\ \forall B,\ P_k, P_{k'} \sim B \Bigr\}$$

with Kirchhoff–Neumann boundary conditions at bindings, ensuring continuity and flux conservation at the interfaces. This reduction to a graph-parametrized geometric configuration generalizes both open book decompositions in contact topology and provides a natural domain for eigenvalue problems and nonlinear PDEs (Coz et al., 29 Dec 2025).

2. Morse-Theoretic Encoding: Graph Structures in Open Book Decompositions

In contact topology, open book decompositions of closed contact 3-manifolds $(M, \xi)$ admit a purely combinatorial encoding by parameterized Morse theory. Given an open book $(B, \pi)$ with binding $B\subset M$ and fibration $\pi: M \setminus B \to S^1$, one constructs an efficient Morse–Smale pair $(f, V)$ on each page. The associated Morse diagram is a labelled trivalent graph $\Gamma$ on the disjoint union of tori (one per binding component), with edges representing the intersections of ascending manifolds (co-skeletons) with a torus at fixed Morse level $F=-\epsilon$. Edge labels correspond to index-1 critical points, and trivalent vertices encode handle-slide events.

This graph-based encoding enables Legendrian fronts (generalized front projections of Legendrian knots) to be studied combinatorially, and a full suite of generalized Reidemeister-type moves (S, B, K, H) governs isotopy classes in the complement of the binding and skeleton. The Thurston–Bennequin invariant is recovered by a combinatorial formula derived from the edge crossings, cusp count, and trace-edge multiplicities on the Morse diagram (Gay et al., 2015).

3. Variational Problems and Dimensional Reduction: NLS on Graph-Based Open Books

Let $\Omega_b = \Gamma \times [-b, b]$ be a graph-based open book. For the stationary nonlinear Schrödinger equation (NLS)

$$-\Delta u + \omega u - |u|^{p-1}u = 0 \qquad \text{on } L,$$

the analysis of ground states is governed by the geometry of $L$. The action–Nehari framework seeks minimizers of

$$S_\omega(u) = \frac12\int_L|\nabla u|^2 + \frac{\omega}{2}\int_L|u|^2 - \frac{1}{p+1}\int_L|u|^{p+1}, \quad I_\omega(u)= 0$$

on the appropriate Sobolev domain, with Kirchhoff coupling at bindings (Coz et al., 29 Dec 2025). For arbitrary finite or periodic open books, existence of minimizers is proved under spectral compactness criteria ($s_\omega < s_\omega^\infty$), yielding a rich theory for PDEs on these graph-inspired domains.

In the limiting regime $b\rightarrow 0$, the ground states exhibit a sharp transition: below a critical transverse width $b_c$, the ground states are trivial $y$-extensions of ground states on the metric graph, while above $b_c$ the solutions acquire two-dimensional structure.

4. Critical Width and the Crossover Phenomenon

The critical width $b_c$ for graph-based open books is given by

$$b_c = \left(\frac{C_p}{\lambda_1(\Gamma)}\right)^{\frac{1}{p-1}},$$

where $C_p$ is a constant depending on the nonlinearity and the transverse eigenfunction, and $\lambda_1(\Gamma)$ is the first nonzero eigenvalue of the Kirchhoff Laplacian on $\Gamma$. For $b < b_c$, all ground states satisfy $\partial_y u \equiv 0$, and the stationary NLS reduces exactly to its graph-based (1D) counterpart: $$-\phi_{xx} + (\omega + \lambda_1(b))\phi - \kappa_p |\phi|^{p-1}\phi = 0,$$ with Kirchhoff vertex conditions. At $b = b_c$, a symmetry-breaking transition occurs, marking the emergence of genuinely two-dimensional ground state profiles for $b > b_c$, as evidenced by qualitative changes in $|u(x, y)|$ (Coz et al., 29 Dec 2025).

5. Combinatorial and Algorithmic Aspects: Book Embeddings of Graphs

The combinatorial counterpart involves book embeddings of abstract graphs: a $k$-page book embedding is an assignment of vertices to a line and edges to $k$ non-crossing pages (half-planes), with the non-crossing constraint

$$u \prec w \prec v \prec x \implies \text{edges } uv, wx \text{ cannot share a page}.$$

The minimum such $k$ is the book thickness or stack-number of the graph. Both the fixed-order and general versions are NP-complete, but become fixed-parameter tractable (FPT) when parameterized by vertex cover number ($\tau(G)$) or pathwidth ($\pw(G, \prec)$).

Key algorithmic results:

• FO-BT (Fixed-Order Book Thickness) FPT in time $2^{O(\tau^3)} n$.
• FO-BT also FPT by pathwidth: $n \cdot \kappa^{O(\kappa^2)}$.
• BT (Book Thickness) FPT by kernelization, reducing to a polynomial-size kernel, followed by brute-force on the kernel (Bhore et al., 2019).

These combinatorial embeddings are foundational in the algorithmic treatment of graph representations, network visualization, and VLSI design.

6. Applications and Examples

Graph-based open books serve as natural domains for:

• Analysis of nonlinear variational PDEs (e.g., NLS ground states) where geometric control via the graph structure allows for rigorous dimensional reduction and symmetry-breaking results. For a loop graph of circumference $L_0$, explicit computation of eigenvalues and critical width is possible, and ground states' spatial profiles can be completely characterized (Coz et al., 29 Dec 2025).
• Contact topology, where Morse diagrams encode the entirety of the open book's contact structure and Legendrian knot theory becomes a combinatorial graph problem. The full dictionary from the topological open book to a labeled trivalent graph leads to effective computation of isotopy invariants (e.g., Thurston–Bennequin number) and characterization of Legendrian isotopies by generalized Reidemeister moves (Gay et al., 2015).
• Graph theory, where book embeddings capture essential combinatorial properties of graphs, with deep implications for algorithmic graph layout and parameterized complexity (Bhore et al., 2019).

7. Outlook and Extension

Graph-based open books exemplify the synthesis of combinatorial, geometric, and analytic methods. Open questions remain on the spectral properties of general open books, dimensional crossovers in nonlinear problems, and more refined kernelization and parameterization results in combinatorial book embedding problems. The identification of sharp transition phenomena (such as the critical width for dimensional reduction) and explicit combinatorial-to-analytic correspondences signal broader applications across mathematical physics, low-dimensional topology, and computational geometry.