Extremum Graph Analysis

• Extremum graphs are graph-based representations that capture adjacency between extrema and saddles in scalar-field topology, enabling effective visualization and analysis.
• They simplify the full Morse–Smale complex by focusing on critical connectivity, supporting both static and time-varying data through methods like GPU parallelism and temporal matching.
• Different formulations, including applications in quantum trees, illustrate the versatility of extremum graphs in reducing complex topological data for efficient computation and feature tracking.

Searching arXiv for recent and foundational papers on “extremum graph”. An extremum graph is a graph-based representation of extremal structure. In scalar-field topology, it is a substructure of the Morse–Smale complex that captures adjacency relationships between extrema and neighboring saddles, and it is used as a succinct representation of the Morse decomposition of a scalar field (Ande et al., 2023). In time-varying settings, it admits a temporal extension that links extrema across successive time steps (Das et al., 2024). In a distinct usage from quantum graph theory, the term can denote the subgraph induced by the extremal points of an eigenfunction; for metric trees with standard vertex conditions, the extremum graph of the second Laplacian eigenfunction is always contained in the boundary vertices (Kennedy et al., 2018). These uses share a common purpose: reducing extremal information to a graph that preserves salient structure while remaining amenable to analysis, visualization, or computation.

1. Definitions and scope

The scalar-field literature defines the extremum graph for a smooth function $f: \mathbb{M} \rightarrow \mathbb{R}$ in terms of critical points, where $\nabla f(x) = 0$, and uses Morse index to distinguish minima, maxima, and saddles (Ande et al., 2023). For maxima, the graph is bipartite: the node set consists of maxima and $(n-1)$-saddles, and an arc $(s,m)$ exists iff $s$ lies on the boundary of the descending manifold of $m$ (Ande et al., 2023). A related formulation for a piecewise-linear scalar field $f: \mathbb{X} \to \mathbb{R}$ defines nodes as critical points and edges as monotonic integral paths connecting extrema to saddles; separate extremum graphs can be constructed for minima and maxima (Li et al., 1 Apr 2026).

For a fixed time step $t$, the time-varying extremum graph literature writes the static extremum graph as $G^t(V^t,E^t)$, with

$V^t = M^t \cup S^t$

and

$\nabla f(x) = 0$0

where $\nabla f(x) = 0$1 is the set of maxima and $\nabla f(x) = 0$2 the set of $\nabla f(x) = 0$3-saddles (Das et al., 2024). This places the extremum graph strictly below the full Morse–Smale complex in descriptive complexity, while retaining the adjacency information relevant to extrema.

The term also has a narrower meaning in the setting of quantum trees. There, an extremum graph is the subgraph induced by the set of extremal points of a function, specifically the second Laplacian eigenfunction on a metric tree (Kennedy et al., 2018). This suggests that “extremum graph” is not a single universal construction but a family of graph abstractions organized around extrema.

2. Static extremum graphs in scalar-field topology

As a topological descriptor, the extremum graph records how extrema are connected to saddles through gradient structure. The TACHYON formulation emphasizes maxima and $\nabla f(x) = 0$4-saddles, with arcs representing the existence of an integral line connecting a saddle to a maximum (Ande et al., 2023). The time-varying extremum graph work characterizes the extremum graph as a substructure of the Morse–Smale complex that captures the adjacency relationship between cells in the Morse decomposition of a scalar field (Das et al., 2024).

This representation sits between local critical-point analysis and full combinatorial topology. It is more structured than a list of critical points, because it records adjacency and gradient-flow connectivity, yet less elaborate than the entire Morse–Smale complex. The literature explicitly associates this simplification with topological feature-directed visualization of $\nabla f(x) = 0$5D/$\nabla f(x) = 0$6D scalar fields, dimensionality reduction, exploratory analysis of high-dimensional scalar fields, segmentation, and feature tracking (Ande et al., 2023).

In practical terms, the extremum graph retains geometric embedding and adjacency but with much lower complexity than the full Morse–Smale complex (Das et al., 2024). That distinction is important: it explains why the structure is repeatedly used as an intermediate topological skeleton rather than as an end in itself.

3. Construction, classification, and simplification

The shared-memory parallel computation described in TACHYON proceeds from a piecewise-linear scalar field on an $\nabla f(x) = 0$7-dimensional rectilinear grid, tessellated into linear simplices for a consistent scalar extension (Ande et al., 2023). Critical point classification is based on upper and lower links and their zeroth Betti numbers. In that formulation, a maximum satisfies $\nabla f(x) = 0$8, and an $\nabla f(x) = 0$9-saddle satisfies $(n-1)$0 (Ande et al., 2023). GPU parallelism is used for per-vertex classification, while CPU task parallelism is used for gradient path tracing from saddles to extrema, since path lengths are irregular and can cause GPU thread divergence (Ande et al., 2023).

The resulting workflow is explicitly hybrid. The GPU handles the embarrassingly parallel stage of classification; the CPU handles gradient path tracing; and, for datasets that do not fit in device memory, the method partitions data into blocks with ghost cells and pipelines GPU classification with CPU tracing (Ande et al., 2023). Partial paths that cross block boundaries are suspended and resumed when adjacent blocks are available (Ande et al., 2023).

Simplification is an integral part of extremum-graph computation rather than a purely downstream concern. TACHYON provides arc-bundling, persistence-driven cancellation of low-persistence saddle-extremum pairs, and saturated simplification to prune spatially long arcs (Ande et al., 2023). In the time-varying setting, extremum graphs are simplified using persistent pair cancellation before temporal matching (Das et al., 2024). The consistent theme is that raw extremum graphs often reflect noise, symmetry, or sampling artifacts, so topological simplification is needed to obtain salient structure.

Empirically, TACHYON reports superior performance and good scaling behavior, with comparisons indicating that it often outperforms TTK and pyms3d by orders of magnitude and supports higher-dimensional domains, including $(n-1)$1D/$(n-1)$2D/$(n-1)$3D cases (Ande et al., 2023). A plausible implication is that the practicality of extremum-graph analysis depends as much on parallel construction and simplification as on the underlying mathematical definition.

4. Time-varying extremum graphs and topology preservation

The time-varying extremum graph (TVEG) extends the static construction to dynamic scalar fields. Its global vertex and edge sets are

$(n-1)$4

where $(n-1)$5 denotes temporal arcs between maxima of consecutive time steps (Das et al., 2024). Spatial arcs come from the static extremum graph, while temporal arcs encode feature correspondence over time (Das et al., 2024).

The temporal matching is formulated as an optimization problem. For a candidate correspondence $(n-1)$6, the score is

$(n-1)$7

where the terms respectively measure differences in topological persistence, function value, Euclidean distance, and neighborhood profiles within the extremum graph (Das et al., 2024). Each maximum at time $(n-1)$8 can have up to two temporal correspondences at time $(n-1)$9, which supports split events; the algorithm then detects continuation, split, merge, generation, and deletion events, while removing ambiguous “z-shaped” configurations (Das et al., 2024).

A different but closely related line of work studies preservation of extremum graphs under lossy compression. EXaCTz preserves extremum graphs and contour trees by enforcing three classes of constraints on decompressed data: extremum graph constraints preserving critical-point classification and steepest-neighbor identities, saddle ordering constraints preserving the global ordering among saddles, and event constraints preserving which extrema are paired with which saddles during topological events (Li et al., 1 Apr 2026). Rather than reconstructing topology explicitly, EXaCTz enforces consistent min/max neighbors of all vertices together with global ordering among critical points (Li et al., 1 Apr 2026).

Its correction scheme is iterative and monotonic, with bounded step size $(s,m)$0, and its convergence is bounded by

$(s,m)$1

where $(s,m)$2 is the longest path in the reduced vulnerability graph (Li et al., 1 Apr 2026). Reported results include a single-GPU throughput of up to $(s,m)$3 GB/s, speedups of up to $(s,m)$4 and $(s,m)$5 over the cited contour-tree-preserving baseline under different comparison modes, scaling to $(s,m)$6 GPUs with $(s,m)$7 efficiency, processing datasets of up to $(s,m)$8 GB in under $(s,m)$9 seconds, and an aggregate correction throughput of up to $s$0 GB/s (Li et al., 1 Apr 2026). These results clarify that extremum graphs have become not only analysis objects but also topological invariants to be preserved in large-scale data workflows.

5. Visualization, dimensionality reduction, and multimodal fusion

The scalar-field literature repeatedly treats the extremum graph as a visualization and abstraction device. TACHYON describes it as a useful data structure for topological feature-directed visualization, segmentation, feature tracking, and dimensionality reduction for high-dimensional scalar fields (Ande et al., 2023). TVEG extends that role to dynamic data by representing feature tracks through space-time and supporting queries such as longest-living features, merges, splits, and user-selected spatial or temporal neighborhoods (Das et al., 2024).

A recent application appears in spectral computed tomography. The topology-aware volume-fusion method for PCCT constructs an extremum graph from a $s$1D histogram of two minimally correlated yet complementary volumes (Sharma et al., 20 Aug 2025). The histogram is treated as a scalar field $s$2, critical points satisfy $s$3, and the extremum graph is the maximum graph connecting maxima and saddles via ascending $s$4-separatrices (Sharma et al., 20 Aug 2025). Edge weights are assigned by

$s$5

The fusion pipeline then computes a minimum spanning tree of the extremum graph, extracts the longest simple path, fits a cubic B-spline to that path, and projects each histogram point to the spline. If $s$6 is a histogram point and $s$7 its closest point on the spline, the fused scalar is the normalized arc length

$s$8

For a spatial location $s$9, the fused volume is

$m$0

In multi-branch cases, the encoding becomes $m$1, with branch number $m$2 and normalized arc length $m$3 (Sharma et al., 20 Aug 2025).

The significance of this application is methodological rather than domain-specific. The extremum graph is not merely visualized; it is used as the organizing skeleton for dimension reduction from a $m$4D histogram to a $m$5D representative scalar field while preserving dominant joint structure (Sharma et al., 20 Aug 2025). This suggests a broader role for extremum graphs as topology-aware latent structures.

6. Extremum graphs on quantum trees

In quantum graph theory, the phrase “extremum graph” is used differently. For a metric tree $m$6, it denotes the subgraph induced by the set of extremal points of a function, specifically the second eigenfunction of the Laplacian with standard vertex conditions (Kennedy et al., 2018). The standard conditions are continuity at every vertex and Kirchhoff conditions, meaning that at every vertex the sum of outgoing derivatives along all incident edges is zero (Kennedy et al., 2018).

The central statement is Theorem 2.1: if $m$7 is a finite, compact, connected metric tree and $m$8 is an eigenfunction associated to the first positive eigenvalue $m$9, then all global minima and maxima of $f: \mathbb{X} \to \mathbb{R}$0 are located at the boundary of $f: \mathbb{X} \to \mathbb{R}$1, where the boundary is the set of vertices of degree one (Kennedy et al., 2018). Consequently, the extremum graph of the second Laplacian eigenfunction is always a subset of the boundary vertices (Kennedy et al., 2018).

The same work gives a counterexample to a natural intuition about extremal placement. Although one might expect the maximum and minimum to occur at leaves that realize the diameter of the tree, the paper constructs a tree of diameter $f: \mathbb{X} \to \mathbb{R}$2 for which the eigenfunction associated with $f: \mathbb{X} \to \mathbb{R}$3 is supported only on a doubled star component, not on the path components (Kennedy et al., 2018). Hence the extremal vertices need not realize the diameter, and the separation between maximum and minimum can be made arbitrarily small by adjusting the construction (Kennedy et al., 2018).

This alternative usage is important because it prevents a common conflation. In scalar-field topology, extremum graphs encode adjacency between extrema and saddles within a Morse-theoretic decomposition. In quantum trees, the extremum graph is induced by extremal locations of a specific eigenfunction. The two notions are related by their focus on extrema, but they arise from different mathematical objects and support different questions.

7. Conceptual status and related structures

The extremum graph occupies an intermediate conceptual position among topological descriptors. It is explicitly described as a substructure of the Morse–Smale complex (Das et al., 2024), and EXaCTz uses the preservation of extremum graphs together with additional ordering constraints to preserve contour trees and merge/split events (Li et al., 1 Apr 2026). This indicates that extremum-graph information is strong enough to control significant parts of the global topology, but not identical to the whole contour-tree or Morse–Smale-complex structure.

A recurring misconception is to treat the extremum graph as a synonym for any graph of critical points. The literature is more specific. In one formulation it is bipartite, with maxima and $f: \mathbb{X} \to \mathbb{R}$4-saddles as nodes and integral-line connectivity as arcs (Ande et al., 2023). In another it is the static or time-varying adjacency graph extracted from the Morse decomposition, augmented with temporal arcs only between maxima (Das et al., 2024). In the quantum-tree setting it is induced by extremal points of an eigenfunction and is confined, for the second Laplacian eigenfunction on a tree with standard conditions, to degree-one boundary vertices (Kennedy et al., 2018).

Taken together, these works show that the extremum graph is best understood as a structured reduction of extremal information. Its enduring value lies in that reduction: enough topology is retained to support visualization, tracking, compression, and fusion, while enough complexity is discarded to make large-scale computation and interpretation feasible (Ande et al., 2023).