Topology Representation Form (TRF)

• TRF is a representation that maps graph structures into multimodal inputs, exposing key topological cues for downstream model reasoning.
• In zero-shot graph QA, TRF leverages both visual and textual encodings to balance structure clarity and computational efficiency.
• In autonomous driving, TRF constructs a directed scene-graph that fuses lane geometry and semantics for robust topology inference.

Searching arXiv for the cited TRF-related papers and recent context. Tool unavailable in this environment; proceeding with the supplied arXiv records and citing them directly. Topology Representation Form (TRF) denotes the representational form through which topology is exposed to a downstream model or task. In the supplied literature, the term appears explicitly in zero-shot graph question answering, where a TRF is a mapping $f_{TRF}:(G,Q)\to R$ or, when conditioned on model and task, $f_{TRF}:(G,Q,M)\to R$ that renders graph structure into text, image, or multimodal input for a Large Multimodal Model (LMM) (Wei et al., 8 Aug 2025). A closely related but differently instantiated use appears in autonomous driving, where an equivalent TRF can be reconstructed for OpenLane-V2 from the TopoHR framework as a directed, typed graph $G=(V,E,S)$ whose nodes are centerline instances with geometry and semantics and whose edges encode lane-lane connectivity, including cyclic structures at intersections (Bai et al., 27 Apr 2026).

1. Concept and terminological scope

The supplied literature uses TRF in two technically distinct but structurally analogous senses. In graph QA, TRF is an input rendering problem: the question is how a graph’s nodes, edges, weights, directions, capacities, and constraints should be encoded so that an LMM can reason over them efficiently. In OpenLane-V2 lane topology reasoning, TRF is a scene-graph schema: the problem is how detected centerlines, semantics, and directed lane-lane relations should be represented so that geometry and topology can be inferred jointly.

This dual usage is important because it prevents an overly narrow definition. In one setting, TRF is a modality-dependent interface between a symbolic graph and an LMM; in the other, it is a topology-bearing representation of a driving scene. The common denominator is that TRF is not merely a storage format. It determines which structural cues are made explicit, which relations are easy to infer, and which failure modes are likely to occur. The graph-QA literature states this directly by emphasizing that different TRFs lead LMMs down distinct cognitive paths, while the TopoHR reconstruction shows that the choice of geometry, node identity, and connectivity representation affects both centerline detection and topology reasoning (Wei et al., 8 Aug 2025, Bai et al., 27 Apr 2026).

Context	Formal object	Representation target
Zero-shot graph QA	$f_{TRF}:(G,Q,M)\to R$	LMM-consumable text, image, or multimodal input
OpenLane-V2 / TopoHR	$G=(V,E,S)$ over centerlines	Directed, typed topology of a driving scene

2. TRF as an LMM-facing graph encoding

In zero-shot graph QA, the underlying graph is formalized as $G=(V,E,A)$, where $V$ is the set of nodes, $E\subseteq V\times V$ is the set of edges, and $A$ collects attributes such as edge weights $w:E\to\mathbb{R}_{\ge 0}$, edge capacities $f_{TRF}:(G,Q,M)\to R$0, direction indicators $f_{TRF}:(G,Q,M)\to R$1, node features $f_{TRF}:(G,Q,M)\to R$2, and task-specific labels or constraints. A natural-language question $f_{TRF}:(G,Q,M)\to R$3 specifies the task, such as connectivity, shortest path, maximum flow, or topological sort, and may identify queried nodes such as source and target. The TRF then renders $f_{TRF}:(G,Q,M)\to R$4 into a representation $f_{TRF}:(G,Q,M)\to R$5 that exposes the structural cues needed for the question while minimizing cognitive overhead such as token length or extraneous detail (Wei et al., 8 Aug 2025).

The zero-shot study defines a model-agnostic TRF set

$f_{TRF}:(G,Q,M)\to R$6

The visual members are Graphviz layouts provided as images to the LMM. $f_{TRF}:(G,Q,M)\to R$7 is hierarchical and tree-like, with clear directional layers and reduced crossings, and is best suited to topological relations, flowcharts, DAG overviews, and perceptual tasks such as bipartite matching; its reported weakness is that it can distort distances and hide cycles behind layers. $f_{TRF}:(G,Q,M)\to R$8 is a spring-model layout that makes global network structure salient and is well suited to connectivity, cycle detection, and global structure perception, but crowded regions can hide multi-edges or weights. $f_{TRF}:(G,Q,M)\to R$9 emphasizes cycles and ring structures and is best for cycle detection and low-degree cyclic motifs, but it makes long chains wrap and impairs shortest-path reasoning. $G=(V,E,S)$0 and $G=(V,E,S)$1 are force-directed layouts targeted at larger graphs, with $G=(V,E,S)$2 designed for scalability to thousands of nodes, but both can obscure fine-grained local semantics such as weights or capacities.

The textual members expose graph structure sequentially. $G=(V,E,S)$3 is an edge-set encoding that is dense, explicit, and compact, especially for weighted edges in shortest path and maximum flow, but neighborhoods remain implicit and can require longer scans. $G=(V,E,S)$4 is an adjacency-list encoding that is node-centric, sorted, and traversal-friendly, supporting shortest path, Hamiltonian path, and neighbor reasoning, though large degrees can inflate token count. $G=(V,E,S)$5 is an adjacency-matrix encoding that makes global connectivity patterns, symmetry, block structure, and explicit directionality visible, and is suited to bipartite matching and some flow settings, but redundant zeros make it inefficient for sparse graphs. The same study distinguishes these zero-shot TRFs from learned embeddings or GraphTokens, which compress structure through aligned encoders but are excluded from $G=(V,E,S)$6 because they require training and are not suitable for closed-source zero-shot settings (Wei et al., 8 Aug 2025).

A central claim of this literature is that there is no universally optimal TRF. Visual TRFs favor perceptual-intensive tasks such as connectivity, cycle detection, and some matching problems. Textual TRFs favor algorithmic reasoning when edge weights, capacities, or ordered decompositions must be manipulated explicitly. The documented failure mode of one-size-fits-all TRFs is either under-perception of salient topology, which produces incorrect answers, or overly long responses when the chosen representation forces inefficient reasoning (Wei et al., 8 Aug 2025).

3. Adaptive TRF selection and Graph Response Efficiency

DynamicTRF formalizes the choice among alternative TRFs as an efficiency-accuracy optimization problem. Its evaluation metric is Graph Response Efficiency (GRE),

$G=(V,E,S)$7

where $G=(V,E,S)$8 is the fraction of correct answers across $G=(V,E,S)$9 stochastic runs, $f_{TRF}:(G,Q,M)\to R$0 is the average output token count, and $f_{TRF}:(G,Q,M)\to R$1 controls the trade-off between correctness and brevity. The normalization described in the study implies that $f_{TRF}:(G,Q,M)\to R$2 when either accuracy goes to zero or token count grows without bound, and that $f_{TRF}:(G,Q,M)\to R$3 reduces GRE to $f_{TRF}:(G,Q,M)\to R$4, ignoring brevity (Wei et al., 8 Aug 2025).

The DynamicTRF pipeline begins by constructing a TRF Preference (TRFP) dataset. The reported setup generates 7K in-domain QA instances across seven algorithmic tasks using Erdős–Rényi graphs with $f_{TRF}:(G,Q,M)\to R$5 and edge probability in $f_{TRF}:(G,Q,M)\to R$6, adding weights where required. For each question $f_{TRF}:(G,Q,M)\to R$7, each $f_{TRF}:(G,Q,M)\to R$8 is executed for $f_{TRF}:(G,Q,M)\to R$9 trials, with $G=(V,E,S)$0 in TRFP creation, and the preferred set is labeled as

$G=(V,E,S)$1

Correctness is validated by task-specific algorithms including Union-Find for connectivity, DFS for cycles, Dijkstra for shortest path, Edmonds–Karp for maximum flow, Hopcroft–Karp for bipartite matching, and backtracking for Hamiltonian path. A router $G=(V,E,S)$2, implemented by default as DeBERTaV3-base, is then trained with multi-label binary cross-entropy so that it can predict a preferred TRF at inference time. The probabilistic routing formulation also allows conditioning on graph features and model identity, $G=(V,E,S)$3, after which the selected TRF is rendered and passed to the LMM (Wei et al., 8 Aug 2025).

The reported empirical pattern is that adaptive routing outperforms any fixed single-TRF strategy in GRE. On GPT-4o, DynamicTRF yields Conn GRE $G=(V,E,S)$4 versus $G=(V,E,S)$5–$G=(V,E,S)$6 for CoT and GITA, with $G=(V,E,S)$7 accuracy and only $G=(V,E,S)$8 tokens; Cyc GRE $G=(V,E,S)$9 versus $G=(V,E,A)$0 for GITA; BGM GRE $G=(V,E,A)$1 versus $G=(V,E,A)$2 for GITA, with $G=(V,E,A)$3 accuracy; HP GRE $G=(V,E,A)$4 versus $G=(V,E,A)$5 for the $G=(V,E,A)$6 baseline, with tokens near $G=(V,E,A)$7; and SP/MF GRE $G=(V,E,A)$8 versus $G=(V,E,A)$9 baselines. On Gemini-2.5 Pro, Conn GRE reaches $V$0 versus $V$1 for NLGraph, with $V$2 accuracy and about $V$3 tokens, while Cyc reaches $V$4 versus $V$5 for GITA, with $V$6 accuracy and about $V$7 tokens. The same paper reports out-of-domain gains on link prediction and node classification for both GPT-4o and Gemini-2.5 Pro. These results support the paper’s central claim that TRF preference is task-specific and partially model-specific, rather than fixed across graph reasoning workloads (Wei et al., 8 Aug 2025).

4. TRF as a lane/centerline topology schema in OpenLane-V2

In the TopoHR line of work, TRF is not explicitly named in the paper, but the supplied reconstruction gives a precise equivalent for OpenLane-V2. A centerline instance $V$8 is represented geometrically as an ordered polyline in BEV coordinates,

$V$9

where $E\subseteq V\times V$0 is the number of sampled points along the polyline. A driving scene is then represented as a directed, typed graph

$E\subseteq V\times V$1

where $E\subseteq V\times V$2 are nodes corresponding to centerline instances, $E\subseteq V\times V$3 are directed edges with relation types such as successor, merge, split, and lane-change-left/right, and $E\subseteq V\times V$4 contains semantic attributes attached to nodes or edges, including turn type, intersection participation, and approach or exit legs. This schema also defines a binary adjacency matrix $E\subseteq V\times V$5, typed adjacencies $E\subseteq V\times V$6, predicted edge probabilities $E\subseteq V\times V$7 with $E\subseteq V\times V$8, and, when needed, typed probabilities $E\subseteq V\times V$9 (Bai et al., 27 Apr 2026).

A notable feature of this TRF is that cyclic topology is allowed. The reconstruction states that cycles may occur across intersections or roundabouts, while local continuity constraints still apply to simple lane segments in homogeneous stretches. This is aligned with the motivation of TopoHR, which explicitly targets cyclic interaction between centerline detection and topology reasoning. A common simplification in lane topology reasoning is therefore not generally valid: acyclicity may hold for local segments, but the scene-level TRF must be able to represent loops, merges, and splits at intersections (Bai et al., 27 Apr 2026).

Geometric supervision is strengthened through a Directional Distance Transform (DDT) mask. For centerline instance $A$0, the reconstruction defines

$A$1

where the mask is defined over each half-lane width, exemplified as $A$2, normalized to $A$3, and discretized into $A$4 bins. With the default $A$5, the per-bin lateral resolution is $A$6 meters on each side, approximately $A$7. Coarser $A$8 and finer $A$9 discretizations were also tested, with six bins reported as best. In the reconstructed TRF view, the centerline geometry $w:E\to\mathbb{R}_{\ge 0}$0, node semantics $w:E\to\mathbb{R}_{\ge 0}$1, typed directed edges, and optional probability scores together form a scene graph aligned to the OpenLane-V2 task definition (Bai et al., 27 Apr 2026).

5. Hierarchical and cyclic TRF reasoning in TopoHR

TopoHR introduces a hierarchical centerline representation built from point queries, instance queries, and semantic representations, and uses cyclic interaction between detection and topology reasoning so that both can iteratively enhance each other. In the reconstructed mapping to TRF, point-level features encode fine-grained geometry for $w:E\to\mathbb{R}_{\ge 0}$2, instance-level features define node identity $w:E\to\mathbb{R}_{\ge 0}$3, and semantic-level features supply attributes in $w:E\to\mathbb{R}_{\ge 0}$4. The decoder proceeds by updating instance-level queries, feeding them into a hierarchical integrator that enriches point-level queries through point-to-instance (P2I) interaction, and then aggregating the enhanced point features back into instances before applying instance-to-instance (I2I) topology reasoning. This closes the loop between geometry and topology rather than treating them as sequentially decoupled stages (Bai et al., 27 Apr 2026).

The P2I mechanism is formulated with Integrator Attention. With point-level queries $w:E\to\mathbb{R}_{\ge 0}$5 and instance-level keys and values $w:E\to\mathbb{R}_{\ge 0}$6,

$w:E\to\mathbb{R}_{\ge 0}$7

The attention weights realize a soft association between points and instances before aggregation back to the instance level. The DDT mask acts as a geometric prior for spatially coherent grouping, and the paper notes that masks are initialized to all zeros before the first decoder layer, following Mask2Former, so that early interactions are unrestricted and become informative only as decoding progresses. The I2I module uses Topo-Aware Attention with instance features on both sides,

$w:E\to\mathbb{R}_{\ge 0}$8

where $w:E\to\mathbb{R}_{\ge 0}$9 is a topology-aware bias derived from initial relations obtained using TopoLogic’s endpoint–startpoint distance strategy. Edge probabilities are then produced by an MLP head, with the supplied reconstruction giving

$f_{TRF}:(G,Q,M)\to R$00

The same reconstruction gives a loss decomposition consistent with the method description: a centerline detection loss with DDT mask supervision, an optional geometry loss if polylines are regressed, a P2I aggregation loss when explicit assignments are used, and an I2I topology loss over adjacency labels. The total objective is expressed as a weighted sum

$f_{TRF}:(G,Q,M)\to R$01

At inference time, the supplied outline describes BEV feature extraction from multi-view images, initialization of hierarchical queries and zero masks, iterative decoder updates, final edge and semantic prediction, centerline formation from point features and DDT masks, edge thresholding and pruning, graph validation that preserves intersection cycles, and emission of $f_{TRF}:(G,Q,M)\to R$02 (Bai et al., 27 Apr 2026).

The reported architecture uses $f_{TRF}:(G,Q,M)\to R$03 decoder layers, $f_{TRF}:(G,Q,M)\to R$04 attention heads, and feature dimension $f_{TRF}:(G,Q,M)\to R$05. Parameter counts are $f_{TRF}:(G,Q,M)\to R$06 for TopoHR without cyclic enhancement and $f_{TRF}:(G,Q,M)\to R$07 with cyclic enhancement. Reported runtime is about $f_{TRF}:(G,Q,M)\to R$08 FPS for TopoHR, $f_{TRF}:(G,Q,M)\to R$09 FPS for TopoHR with cyclic enhancement, and $f_{TRF}:(G,Q,M)\to R$10 FPS for the TopoLogic baseline. The appendix numbers quoted in the supplied reconstruction are as follows:

Model	$f_{TRF}:(G,Q,M)\to R$11	$f_{TRF}:(G,Q,M)\to R$12
TopoLogic (6 layers, 8 heads, 256 dim)	29.9	23.9
Deeper TopoLogic (12 layers)	30.2	24.8
TopoHR (hierarchical representation, no cyclic)	34.6	30.6
TopoHR + cyclic enhancement	36.1	31.8

The same source reports that a memory-efficient baseline removing GNN yields $f_{TRF}:(G,Q,M)\to R$13 for $f_{TRF}:(G,Q,M)\to R$14, and the abstract states that, relative to previous best results on OpenLane-V2, TopoHR achieves $f_{TRF}:(G,Q,M)\to R$15 in $f_{TRF}:(G,Q,M)\to R$16 and $f_{TRF}:(G,Q,M)\to R$17 in $f_{TRF}:(G,Q,M)\to R$18 on subset_A, and $f_{TRF}:(G,Q,M)\to R$19 in $f_{TRF}:(G,Q,M)\to R$20 and $f_{TRF}:(G,Q,M)\to R$21 in $f_{TRF}:(G,Q,M)\to R$22 on subset_B (Bai et al., 27 Apr 2026).

6. Empirical patterns, misconceptions, and limitations

Across the two research lines, a recurring misconception is that topology can be represented once and used uniformly across tasks. The graph-QA study rejects this directly: no single TRF dominates all tasks, visual TRFs are favored for perceptual-intensive problems such as connectivity, cycle detection, and bipartite graph matching, while textual TRFs are favored for edge-weighted or ordered-decomposition problems such as shortest path, maximum flow, topological sort, and Hamiltonian path, with Hamiltonian path occasionally benefiting from concise visuals on small graphs (Wei et al., 8 Aug 2025). The TopoHR results reject an analogous simplification in autonomous driving: topology reasoning is not best handled as a purely sequential MLP stage after detection, because decoupling geometry from topology causes detection errors to propagate without corrective feedback and makes complex cyclic intersections harder to resolve (Bai et al., 27 Apr 2026).

The empirical evidence also points to representation-specific failure modes. In graph QA, visual TRFs without explicit weights lead to wrong numeric results in shortest path and maximum flow; textual TRFs can become verbose on large sparse graphs, especially with adjacency matrices full of zeros or very long adjacency lists; and style-fixed visuals can obscure local multiplicity or clutter hubs. In TopoHR, the supplied analysis emphasizes that six DDT bins balance sensitivity and robustness better than coarser or finer discretizations, that gains are not attributable merely to parameter count because deeper TopoLogic improves only marginally, and that cyclic enhancement is most effective for centerlines because the traffic branch follows TopoLogic and lacks internal topology to form cyclic relations (Wei et al., 8 Aug 2025, Bai et al., 27 Apr 2026).

The limitations are likewise domain-specific. DynamicTRF notes that GRE labels depend on finite $f_{TRF}:(G,Q,M)\to R$23 and tokenization, that noisy outputs and outliers can affect TRFP quality, that the TRFP dataset is synthetic and validator-driven, that the router adds preprocessing overhead, and that $f_{TRF}:(G,Q,M)\to R$24 excludes learned embeddings in order to remain model-agnostic. TopoHR reports strict adherence to official OpenLane-V2 protocols, with subset_A and subset_B trained and tested independently and no overlap, but its reported runtime shows a clear throughput trade-off relative to the TopoLogic baseline (Wei et al., 8 Aug 2025, Bai et al., 27 Apr 2026).

The forward-looking directions in the supplied literature also converge. DynamicTRF proposes richer TRF modalities such as overlays of numeric cues on visuals, compact structured tables or JSON-like schemas, learned TRF generation conditioned on question and graph, structure-aware LMM components, stronger routing signals from graph statistics and model telemetry, and more robust GRE estimation. The TopoHR reconstruction concludes with a parallel set of design principles for TRF quality in driving scenes: hierarchical queries, topology-aware biases from endpoint/startpoint distances, DDT-based geometric supervision, and post-processing that validates geometric continuity while allowing intersection cycles. This suggests that, across both graph QA and autonomous driving, TRF is most effective when it is optimized jointly for structural fidelity and downstream reasoning behavior rather than treated as a static serialization (Wei et al., 8 Aug 2025, Bai et al., 27 Apr 2026).