Page Graphs in Graph Theory

• Page graphs are heterogeneous representations that treat pages as geometric constraints, semantic units, or combinatorial resources in graph analysis.
• They improve visual clarity by utilizing structured layouts like cartographic drawings and disciplined page assignments to reduce clutter and false crossings.
• Applications span book embeddings, web analytics, GUI state modeling, and document object extraction, offering actionable insights for researchers.

“Page graphs” is not a single standardized term. In the literature represented here, it refers to several page-centered graph formalisms and practices: graph representations optimized for a bounded page or screen surface; classical page- and book-embedding models in graph theory, where vertices lie on a spine and edges are assigned to pages; graphs whose nodes are literal pages, such as web pages, document pages, or GUI screens; and page-level object graphs used in document understanding. Across these uses, the common theme is that a page is treated either as a geometric constraint, a combinatorial resource, or a semantic unit of analysis (Blakley et al., 2014).

1. Page-oriented graph representation on a bounded surface

A page-oriented view of graph drawing treats the page or screen as a structured layout space rather than as a neutral canvas. In “How to Draw Graphs: Seeing and Redrafting Large Networks in Security and Biology” (Blakley et al., 2014), the central claim is that familiar node-link drawings are often the wrong medium for large or dense graphs because they are not canonical, they create accidental edge crossings that look like “false visual vertices,” they offer no consistent place for edge labels, and they quickly become visually dense. The paper gives a compact condition under which node-link drawings are especially poor: graphs satisfying

$10^6 < v^3 < e^2$

produce useless node-link “hairballs” (Blakley et al., 2014).

The proposed alternative is a cartographic representation in which vertices are horizontal latitudes and edges are vertical longitudes. A latitude represents a vertex and a longitude represents an edge. Latitudes cannot share an $x$-coordinate with or touch one another; longitudes cannot share a $y$-coordinate with or touch one another. Edge endpoints must land visibly on the latitudes corresponding to their endpoint vertices, and loops are represented when both endpoints land on the same latitude. This orthogonal regime removes ambiguous edge-edge crossings, gives natural regions for both vertex and edge labels, and makes blank space meaningful because absent edges remain visible as unoccupied positions (Blakley et al., 2014).

The construction is explicitly tied to the augmented incidence matrix. For a simple graph $G$ of order $n$, viewed as a subgraph of the complete graph $K_n$, the augmented incidence matrix $D$ has

$a(n)=\binom{n}{2}=\frac{n(n-1)}{2}$

columns, one for each possible edge of $K_n$; missing edges are represented by all-zero columns. The drawing procedure follows the matrix directly: draw one horizontal latitude per row, one vertical guide per column, and then a vertical longitude wherever a column contains two $1$s. For graphs with loops, the paper notes $x$0 columns, and for a digraph without loops, $x$1 columns (Blakley et al., 2014).

This page-based discipline also changes what can be inspected effectively. Complements become legible because a dense graph may be easier to understand through the missing edges in its complement $x$2, defined on the same vertex set as $x$3 and containing exactly the edges absent from $x$4. Likewise, graph comparison becomes a page-partitioning problem: for graphs $x$5 and $x$6, the edge space is decomposed into $x$7 for $x$8, $x$9 for $y$0, and $y$1 for $y$2, yielding

$y$3

and therefore

$y$4

The prescribed layout places $y$5 on the left, $y$6 in the center, and $y$7 on the right, making overlap and change directly inspectable on a single page (Blakley et al., 2014).

A plausible implication is that this use of “page graph” is less about geometric metaphor and more about disciplined page allocation: every vertex, every possible edge, every label region, and every comparison region has a stable place. The paper is explicit, however, that this superiority is argued visually and conceptually rather than established by user studies or formal readability metrics (Blakley et al., 2014).

2. Page number, book embeddings, and the classical graph-theoretic meaning

In graph theory, a page is a combinatorial resource in a linear layout. A book with $y$8 pages consists of a spine and $y$9 half-planes bounded by the spine. A $G$0-page drawing places all vertices on the spine and draws each edge entirely within one page; a $G$1-page embedding is such a drawing with no crossings. The page number $G$2 is the minimum $G$3 such that $G$4 admits a $G$5-page embedding, and the $G$6-page crossing number is denoted $G$7 (Klerk et al., 2012).

For complete bipartite graphs, exact finite and asymptotic results are known. “Book drawings of complete bipartite graphs” (Klerk et al., 2012) proves that for each $G$8,

$G$9

so $n$0 is the first graph in that family requiring more than $n$1 pages. The same paper gives, for $n$2, an exact formula for the $n$3-page crossing number of $n$4. Writing

$n$5

one has

$n$6

The paper also proves the asymptotic estimate

$n$7

together with the explicit bounds

$n$8

A central constructive notion is the balanced $n$9-page embedding, which lets one “blow up” a white vertex into a cluster while controlling the induced crossings; this is what makes the exact formula attainable in those cases (Klerk et al., 2012).

For directed acyclic graphs, the page-number notion is constrained by topological order. “Recognizing DAGs with Page-Number 2 is NP-complete” (Bekos et al., 2022) defines the page number of a DAG as the minimum $K_n$0 for which the DAG has a topological order and a $K_n$1-coloring of its edges such that no two edges of the same color cross, i.e. have alternating endpoints along the topological order. The paper resolves Heath and Pemmaraju’s conjecture by proving that recognizing whether a DAG has page-number $K_n$2 is NP-complete, even for $K_n$3-planar graphs and even for planar posets (Bekos et al., 2022). This completes the fixed-$K_n$4 complexity picture cited there: $K_n$5 is polynomial-time solvable, while $K_n$6 is NP-complete (Bekos et al., 2022).

A related extremal question concerns upward planar graphs. “A Sublinear Bound on the Page Number of Upward Planar Graphs” (Jungeblut et al., 2021) improves the known lower bound to $K_n$7 and proves the first asymptotic improvement over the trivial linear upper bound: $K_n$8 for every $K_n$9-vertex upward planar graph. More specifically, it proves

$D$0

for width $D$1, and

$D$2

for height $D$3, combining these via a balancing argument (Jungeblut et al., 2021).

Within bounded-degree planar families, stronger positive results are known. “Two-Page Book Embeddings of 4-Planar Graphs” (Bekos et al., 2014) proves that every planar graph of maximum degree $D$4 has page number at most $D$5, equivalently that every 4-planar graph is subhamiltonian, with a linear-time algorithm for the triconnected case and an $D$6-time algorithm in general. “Embedding 5-planar graphs in three pages” (Guan et al., 2018) extends the bounded-degree line by giving an $D$7-time algorithm that embeds every planar graph of maximum degree $D$8 in three pages (Bekos et al., 2014, Guan et al., 2018).

These results show that in the classical sense, “page graph” concerns the interaction between a single linear vertex order and page-wise forbidden patterns. The page is not a document page but a layer in a book embedding.

3. Relaxations and variants of page-based linear layouts

Several recent parameters relax or hybridize classical page number rather than replacing it outright. “The Mixed Page Number of Graphs” (Alam et al., 2021) introduces the mixed page number

$D$9

Here a linear layout uses a total vertex order, and each page is allowed to be either a stack page, forbidding crossings, or a queue page, forbidding nestings. Because pure stack and pure queue layouts are special cases,

$a(n)=\binom{n}{2}=\frac{n(n-1)}{2}$0

The paper proves that for every fixed vertex order and every graph with $a(n)=\binom{n}{2}=\frac{n(n-1)}{2}$1 edges, the mixed page number is at most

$a(n)=\binom{n}{2}=\frac{n(n-1)}{2}$2

and that this is asymptotically tight up to constants for the fixed-order setting. It also derives an edge-density bound: if $a(n)=\binom{n}{2}=\frac{n(n-1)}{2}$3 on $a(n)=\binom{n}{2}=\frac{n(n-1)}{2}$4 vertices, then $a(n)=\binom{n}{2}=\frac{n(n-1)}{2}$5 has at most

$a(n)=\binom{n}{2}=\frac{n(n-1)}{2}$6

For complete graphs,

$a(n)=\binom{n}{2}=\frac{n(n-1)}{2}$7

and $a(n)=\binom{n}{2}=\frac{n(n-1)}{2}$8 satisfies

$a(n)=\binom{n}{2}=\frac{n(n-1)}{2}$9

witnessing a strict advantage of mixed layouts (Alam et al., 2021).

Another relaxation shifts attention from global page count to local load. “Local and Union Page Numbers” (Merker et al., 2019) defines the local page number $K_n$0 as the minimum $K_n$1 such that there exists a book embedding in which each vertex is incident to edges on at most $K_n$2 pages, and the union page number $K_n$3 as the minimum number of pages in a union embedding, where each page is a vertex-disjoint union of crossing-free components. These parameters satisfy

$K_n$4

and are always within a multiplicative factor of $K_n$5: $K_n$6 At the same time, the classical page number is not bounded in terms of either relaxed parameter: for every $K_n$7 and infinitely many $K_n$8, there exist $K_n$9-vertex graphs with

$1$0

The paper also relates the local and union parameters to density, proving

$1$1

where $1$2 is the maximum average degree (Merker et al., 2019).

A different extension introduces weighted pages through data-structure semantics. “Linear Layouts of Graphs with Priority Queues” (Giacomo et al., 30 Jun 2025) defines a priority queue layout for an edge-weighted graph $1$3, where edges on the same page must be compatible with deletion from a priority queue keyed by edge weights during a left-to-right scan. The forbidden configuration on a page is: two edges $1$4, $1$5 with

$1$6

The corresponding robust parameter is

$1$7

The paper proves linear lower bounds for dense graphs,

$1$8

characterizes the graphs with $1$9 by eight forbidden minors, proves

$x$00

for graphs of pathwidth $x$01, and shows that priority queue number is unbounded already on graphs of treewidth $x$02 (Giacomo et al., 30 Jun 2025).

On the algorithmic side, crossing minimization in page drawings also admits parameterized results. “Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth” (Bannister et al., 2014) proves, via $x$03 formulations and Courcelle’s theorem, that computing the 1-page crossing number is fixed-parameter tractable with respect to the number of crossings, testing 2-page planarity is fixed-parameter tractable with respect to treewidth, and computing the 2-page crossing number is fixed-parameter tractable with respect to the sum of crossing number and treewidth (Bannister et al., 2014).

Taken together, these variants show that page-based graph theory has diversified well beyond the original minimization of a single total page count. The page remains the primitive resource, but the resource can be mixed, localized, weighted, or constrained algorithmically in different ways.

4. Page graphs as networks whose nodes are pages

In another major usage, a page graph is a graph whose vertices are literal pages. The nodes may be web pages, document pages, or GUI screens, and the edges represent hyperlinks, continuations, or action-induced transitions. In this sense, the page is the semantic atom rather than the drawing surface.

For directed web-like graphs, “Rich-club and page-club coefficients for directed graphs” (Smilkov et al., 2011) distinguishes between “rich” nodes, defined by high in-degree, and “popular” or prestigious nodes, defined by high PageRank. For a PageRank threshold $x$04, the page-club coefficient is

$x$05

where $x$06 is the number of nodes with PageRank larger than $x$07, and $x$08 is the number of directed edges among those nodes. PageRank itself is defined via the random-surfer equation

$x$09

with $x$10 in the experiments. The normalized coefficient

$x$11

tests whether high-PageRank pages link to one another more often than expected under the null model. In the Google web network with $x$12 pages and $x$13 hyperlinks, the paper reports partial page-club ordering up to an intermediate range, followed by declining normalized coefficients at higher thresholds; the top 20 PageRank nodes share no links among themselves (Smilkov et al., 2011). The paper’s interpretation is that top web pages do not form a strongly self-linking elite, even when PageRank and in-degree are highly correlated (Smilkov et al., 2011).

Page graphs also arise in interactive analytics systems. “A Web-based Interactive Visual Graph Analytics Platform” (Ahmed et al., 2015) describes a browser-based environment for graph mining and visualization in which a graph file can be drag-and-dropped into a browser and explored “within seconds.” The platform is graph-generic rather than web-specific, but the paper explicitly notes that metrics such as degree, PageRank, clustering coefficient, k-core number, triangle counts, betweenness, CDFs and CCDFs of graph properties, community detection, role discovery, filtering, ranking, and temporal exploration are all relevant to page-level hyperlink analysis. The platform computes macro-level statistics including maximum and average degree, total number of triangles, global clustering, maximum k-core number, diameter, mean distance, approximate chromatic number, number of communities, number of roles, and maximum triangle-core number, and supports linked statistical views such as scatter plot matrices, semantic zooming, top-$x$14 ranking, and dynamic filtering (Ahmed et al., 2015).

A related but structurally different use of page graphs appears in GUI agents. “PG-Agent: An Agent Powered by Page Graph” (Chen et al., 27 Aug 2025) treats GUI screens as nodes and action-induced page jumps as directed edges. The paper states that “the pages of GUI screens naturally form a page graph connected by the actions, and a sequential episode is essentially a path sampling on this graph.” From an episode with task $x$15, an action tuple $x$16 is summarized by

$x$17

page jumps are detected by

$x$18

and destination pages are summarized as

$x$19

Deduplicated nodes are stored as

$x$20

while edges are

$x$21

where $x$22 is a queue of accumulated in-page operations. Retrieval over the page graph yields guidelines

$x$23

using top-$x$24 node retrieval with $x$25 and breadth-first search depth $x$26. The retrieved graph knowledge is then injected into a multi-agent planner (Chen et al., 27 Aug 2025).

This page-as-node sense emphasizes relational semantics rather than layout. A plausible implication is that “page graph” has become a general design pattern for any environment in which page transitions matter structurally: the web, citation-like page prestige systems, and GUI state spaces all fit that pattern, though they use different edge semantics.

5. Page-object and chunk-page graphs in document understanding

In visually rich documents, a page may be both a retrieval target and a graph node embedded in a richer page-centered structure. Two recent works make this explicit.

“PubTables-v2” (Smock et al., 11 Dec 2025) introduces a large-scale dataset organized into Cropped Tables, Single Pages, and Full Documents. The single-page collection contains $x$27 samples and $x$28 tables in full-page context, with 16 detection classes: 8 base classes and 8 rotated counterparts. The 8 base classes are table, column, row, column header, spanning cell, projected row header, caption, and footer. The single-page relation schema is hierarchical: the table object is the parent and the other 7 object classes associated with it are connected as children. This yields a page-level object graph centered on each table. For full documents, the dataset contains $x$29 samples and $x$30 tables, including $x$31 multi-page tables and $x$32 single-page but multi-column split tables (Smock et al., 11 Dec 2025).

The same paper introduces the Page-Object Table Transformer (POTATR) as an image-to-graph extension of Table Transformer. POTATR predicts both page objects and explicit relations between them, rather than relying on overlap heuristics to assign rows, columns, captions, and footers to tables. It uses 250 object queries, compared with 125 in the predecessor model, and adds a relation prediction head while preserving compatibility with pretrained table-structure weights. On page-level table extraction, POTATR-v1.0-Pub achieves

$x$33

substantially above the VLM baselines reported there. On a small-scale image-to-graph comparison, POTATR attains Edge F1 $x$34, compared with $x$35 for EGTR and $x$36 for Relationformer (Smock et al., 11 Dec 2025). In this usage, the page graph is a table-centered object graph within a single page, not a hyperlink-style graph among pages.

“EviProp: Seeded Relevance Diffusion on Chunk-Page Graphs for Long Multimodal Document Retrieval” (Zhang et al., 8 Jun 2026) uses a different construction: a per-document heterogeneous Chunk–Page graph

$x$37

where $x$38 are page nodes and $x$39 are content chunk nodes. Text chunks come from OCR text lines or paragraphs; visual chunks are cropped table, chart, or figure regions with GPT-4o-generated captions. The graph has three families of weighted edges:

• Hierarchical membership edges between a chunk and its parent page,

$x$40

• Sequential page edges between adjacent pages,

$x$41

• Similarity edges, consisting of page-page visual similarity and chunk-chunk semantic similarity.

For page-page similarity, mean-pooled ColPali page vectors are $x$42-normalized,

$x$43

and connected with clipped cosine weights. For chunk-chunk similarity, the appendix clarifies thresholded positive cosine cubed, with

$x$44

Given a query, EviProp combines dense visual page priors and sparse chunk seeds in a restart vector $x$45, then runs Personalized PageRank: $x$46 until

$x$47

Final page scores interpolate the original visual page score and the diffused posterior: $x$48 On MMLongBench-Doc and LongDocURL, EviProp improves evidence-page retrieval over independent visual retrieval. For example, Recall@3 rises from $x$49 to $x$50 on MMLongBench-Doc and from $x$51 to $x$52 on LongDocURL (Zhang et al., 8 Jun 2026).

These two works show two different page-graph regimes in document understanding. PubTables-v2 uses page-level object graphs to structure extraction within a page, whereas EviProp uses heterogeneous page-centered graphs to propagate relevance across document structure. Both treat the page as an explicit graph primitive rather than as a mere image extent.

6. Hierarchies, large-scale exploration, and interpretability

A final line of work treats page graphs as large graphs that must be summarized, explored, or explained across levels of abstraction. “SuperGraph Visualization” (Rodrigues et al., 2015) introduces a hierarchy of graph partitions, called a SuperGraph,

$x$53

where $x$54 are SuperNodes, $x$55 are LeafSuperNodes, and $x$56 are SuperEdges. A LeafSuperNode is an induced subgraph on a subset $x$57, the leaf node sets are pairwise disjoint,

$x$58

and together cover all original vertices,

$x$59

The closure of a SuperNode is the set of original graph nodes it contains recursively,

$x$60

A SuperEdge $x$61 stores the set of original edges whose endpoints fall in the closures of $x$62 and $x$63, and its weight is $x$64 (Rodrigues et al., 2015).

To preserve recoverable original connectivity, the paper defines internal edges, external edges, and open nodes. A node is open with respect to a community if it participates in an external edge leaving that community. The paper then characterizes connectivity between arbitrary communities through open-node sets and the relevant ancestor-level SuperEdge. This supports drill-down and drill-out analysis in very large graphs, demonstrated on an email network and a DBLP graph with $x$65 nodes and $x$66 edges (Rodrigues et al., 2015). For page graphs, the same machinery applies directly to communities of pages, site sections, or topic clusters.

Interpretability can also be directed at learned graph models rather than at graph datasets. “PAGE: Prototype-Based Model-Level Explanations for Graph Neural Networks” (Shin et al., 2022) is unrelated to document pages or book pages; here “PAGE” is an acronym. Nevertheless, it is relevant to page-graph research insofar as page graphs are often analyzed with graph neural networks. PAGE is a post hoc model-level explainer for graph classification. For a target class $x$67, it first clusters graph-level embeddings with a Gaussian mixture model

$x$68

selects representative graphs near each centroid by Mahalanobis distance, and then discovers a prototype graph using node-level embeddings and the prototype scoring function

$x$69

The method yields class-level prototype graphs explaining what the GNN has learned, rather than instance-level attributions for one graph at a time (Shin et al., 2022).

A common misconception is that “page graph” always denotes one formal graph class. The literature here shows otherwise. In some papers, pages are layers in a linear layout; in others, pages are semantic nodes in a web, GUI, or document graph; in others still, page-level objects become graph nodes within a page. Another misconception is that page-centric graph methods are necessarily only visual. The record is broader: it includes combinatorial extremal theory, NP-completeness, fixed-parameter tractability, browser-based analytics, retrieval with Personalized PageRank, and model-level GNN explanation (Alam et al., 2021, Bekos et al., 2022, Ahmed et al., 2015, Zhang et al., 8 Jun 2026, Shin et al., 2022).

A plausible synthesis is that “page graphs” now names a family resemblance rather than a single object. The family is unified by one operation: impose page structure on graphs, or represent pages themselves as graph structure, in order to make adjacency, hierarchy, absence, comparison, navigation, or evidence propagation legible.