Consistent Path Systems in Graph Theory

• Consistent path systems are defined as families of designated paths in a graph where every vertex pair is connected by a unique path that maintains subpath closure.
• They abstract the notion of unique geodesics, bridging combinatorial graph theory with metric realizability through linear inequalities and LP methods.
• Research focuses on classifying metric realizability, studying forbidden subdivisions, and developing algorithmic approaches for efficient identification and enumeration.

Searching arXiv for recent and foundational papers on consistent path systems and closely related notions. Consistent path systems are families of designated paths that satisfy a coherence condition under restriction to subpaths or intersections. In the graph-theoretic formulation that has become central to recent work, a connected finite simple graph $G=(V,E)$ carries a consistent path system $\mathcal P$ if for every pair $u,v\in V$ there is exactly one chosen $uv$-path $P_{u,v}$, and the intersection of any two chosen paths is either empty, a single vertex, or itself a chosen path. This axiomatizes the combinatorics of unique geodesics without presupposing edge lengths, and it leads to the realization problem: when does a consistent path system arise from shortest paths under a positive weighting, and when does every consistent path system on a graph admit such a realization (Chudnovsky et al., 2023).

1. Definitions and equivalent formulations

A consistent path system on a graph $G=(V,E)$ is a collection $\mathcal P$ of paths such that

$$\forall u,v\in V,\ \exists!\ P_{u,v}\in\mathcal P \text{ joining }u\text{ and }v,$$

and for all $P,Q\in\mathcal P$,

$$P\cap Q\in \{\emptyset,\text{ a single vertex},\text{ a path in }\mathcal P\}.$$

This is the precise graph-based formulation of being intersection-closed (Chudnovsky et al., 2023).

An equivalent formulation emphasizes subpaths: if $\mathcal P$0, then the subpath of $\mathcal P$1 between $\mathcal P$2 and $\mathcal P$3 is exactly $\mathcal P$4. In this form, consistency says that designated routes are closed under taking internal segments (Cizma et al., 2020). The same idea appears in the abstract path-system setting, where a path system $\mathcal P$5 is consistent when any two paths containing the same ordered pair $\mathcal P$6 have the same continuous $\mathcal P$7 subpath. In prose, no two paths may intersect, split apart, and then intersect again (Bodwin, 2018).

The motivation is inherited from shortest-path geometry. If a path is a unique shortest path, then every continuous subpath is again a unique shortest path. Consistency isolates exactly this closure phenomenon at the combinatorial level. A plausible implication is that consistent path systems should be viewed as abstract geodesic structures: they encode one coherent route between every pair, but not yet the metric that would certify those routes as shortest.

2. Metric realization, metrizability, and strong metrizability

A consistent path system $\mathcal P$8 is called metric if there exists a positive edge-length assignment

$\mathcal P$9

such that every path in $u,v\in V$0 is a $u,v\in V$1-shortest path. A graph $u,v\in V$2 is metrizable if every consistent path system in $u,v\in V$3 is metric (Chudnovsky et al., 2023). In the stronger abstract formulation, a path system $u,v\in V$4 is strongly metrizable if there exists a directed weighted graph $u,v\in V$5 such that every $u,v\in V$6 is the unique shortest path between its endpoints (Bodwin, 2018).

For a fixed consistent path system, realizability can be expressed as a linear feasibility problem. If $u,v\in V$7 is the chosen $u,v\in V$8-path, then for every alternative $u,v\in V$9-path $uv$0,

$uv$1

Non-metrizability is certified when the resulting inequalities force some $uv$2, contradicting positivity (Chudnovsky et al., 2023). In the strong setting, consistency is necessary but not sufficient: there exist consistent abstract path systems, such as the octahedral example $uv$3, whose unique-shortest-path inequalities sum to an impossibility (Bodwin, 2018).

This distinction between consistency and realizability is quantitatively sharp. The number of consistent path systems on $uv$4 vertices is

$uv$5

whereas the number of consistent path systems realizable as the unique geodesics with respect to some metric is only

$uv$6

Thus abstract consistency is much less restrictive than unique-geodesic realizability (Cizma et al., 29 Aug 2025).

3. Structural theory on graphs

Recent structural results show that graph metrizability is extremely rigid. A major inheritance property is closure under topological minors: if a graph contains a subdivision of a non-metrizable graph, then it is itself non-metrizable (Chudnovsky et al., 2023). This makes forbidden subdivisions central to the theory.

A key structural device is the compliant edge. A branch vertex is a vertex of degree at least $uv$7. A flat path is a path of length at least $uv$8 whose internal vertices all have degree $uv$9. An edge $P_{u,v}$0 is compliant if $P_{u,v}$1 and $P_{u,v}$2 are also connected by a flat path. Such edges are metrically inessential: $P_{u,v}$3 for any compliant edge $P_{u,v}$4 (Chudnovsky et al., 2023).

After deleting compliant edges, large $P_{u,v}$5-connected metrizable graphs have a short list of possible cores. If $P_{u,v}$6 is $P_{u,v}$7-connected, metrizable, $P_{u,v}$8, and has no compliant edges, then

$P_{u,v}$9

or $G=(V,E)$0 is a subdivision of

$G=(V,E)$1

A strong corollary is that every $G=(V,E)$2-connected metrizable graph with at least $G=(V,E)$3 vertices has a vertex $G=(V,E)$4 such that $G=(V,E)$5 is outerplanar; this is the “outerplanar plus one vertex” phenomenon (Chudnovsky et al., 2023).

These results are asymptotic rather than converse characterizations. Many subdivisions of $G=(V,E)$6, $G=(V,E)$7, $G=(V,E)$8, and $G=(V,E)$9 are still non-metrizable, and the complete classification of metrizable theta graphs $\mathcal P$0 remains open (Chudnovsky et al., 2023). At the same time, there are infinite positive families: cycles are strictly metrizable, and every outerplanar graph is strictly metrizable (Cizma et al., 2020).

4. Forbidden patterns, topology, and the abstract theory

The abstract theory of path systems strengthens the graph-based picture by treating paths as node sequences independent of any ambient graph. In that setting, the complete directed characterization is: $\mathcal P$1 Here $\mathcal P$2 denotes a path-system homomorphism preserving subpath structure and branching or merging structure (Bodwin, 2018).

The forbidden objects are polyhedral path systems. They admit two complementary descriptions. Combinatorially, they are pairs of path systems that are flat at every node, with local structure decomposing into pinwheels. Topologically, they correspond to balanced two-colored cell decompositions of compact orientable $\mathcal P$3-manifolds; in the undirected case, non-orientable obstructions also appear (Bodwin, 2018). This yields a topological obstruction theory for unique-shortest-path realizability.

The significance of this characterization is that pairwise consistency is only the first obstruction. Consistency forbids the visible defect that two paths intersect, diverge, and later rejoin. Polyhedral obstructions capture higher-order global incompatibilities among many paths. This suggests that the geometry of unique shortest paths is governed not only by local subpath coherence but also by global surface-like cancellation patterns.

5. Enumeration, approximation, and algorithmic questions

Consistent path systems are abundant. The asymptotic count

$\mathcal P$4

is obtained by constructing many diameter-$\mathcal P$5 neighborly systems, while the strictly metric count remains only exponential in $\mathcal P$6 (Cizma et al., 29 Aug 2025). The same work connects these counts to bounds on the number of faces of the metric cone and to the enumeration of maximum VC-classes (Cizma et al., 29 Aug 2025).

Approximate realizability has also been formalized. A path system $\mathcal P$7 is $\mathcal P$8-metric if there exists a metric $\mathcal P$9 on the vertex set such that for every designated path $$\forall u,v\in V,\ \exists!\ P_{u,v}\in\mathcal P \text{ joining }u\text{ and }v,$$0,

$$\forall u,v\in V,\ \exists!\ P_{u,v}\in\mathcal P \text{ joining }u\text{ and }v,$$1

The infimum of such $$\forall u,v\in V,\ \exists!\ P_{u,v}\in\mathcal P \text{ joining }u\text{ and }v,$$2 is denoted $$\forall u,v\in V,\ \exists!\ P_{u,v}\in\mathcal P \text{ joining }u\text{ and }v,$$3. There are infinitely many $$\forall u,v\in V,\ \exists!\ P_{u,v}\in\mathcal P \text{ joining }u\text{ and }v,$$4-point consistent path systems with

$$\forall u,v\in V,\ \exists!\ P_{u,v}\in\mathcal P \text{ joining }u\text{ and }v,$$5

showing that consistency alone can be very far from metric geodesicity. At the same time, $$\forall u,v\in V,\ \exists!\ P_{u,v}\in\mathcal P \text{ joining }u\text{ and }v,$$6 is an algebraic number and can be computed in polynomial time (Cizma et al., 29 Jan 2026).

Algorithmically, the realization problem has several distinct forms. For graph-based metrizability, feasibility of the shortest-path inequality system gives an LP-style criterion (Chudnovsky et al., 2023). For strong metrizability of abstract path systems, there is an LP-based recognition method with $$\forall u,v\in V,\ \exists!\ P_{u,v}\in\mathcal P \text{ joining }u\text{ and }v,$$7 variables and $$\forall u,v\in V,\ \exists!\ P_{u,v}\in\mathcal P \text{ joining }u\text{ and }v,$$8 constraints, together with a dual obstruction theory based on multiflow rigidity (Bodwin, 2018). A plausible implication is that computation is substantially better understood than structural classification in the converse direction.

6. Related notions and terminological extensions

The phrase “consistent path system” is not uniform across adjacent literatures. In computational geometry, two labelled point sets admit compatible geometric paths when the same permutation of labels yields a noncrossing straight-line path in both sets. Deciding existence in general is NP-complete, but polynomial-time algorithms are known in three restricted regimes: $$\forall u,v\in V,\ \exists!\ P_{u,v}\in\mathcal P \text{ joining }u\text{ and }v,$$9 time for points in convex position, $P,Q\in\mathcal P$0 time for two simple polygons with the paths constrained to stay inside the polygons, and $P,Q\in\mathcal P$1 time for compatible monotone paths on general point sets (Arseneva et al., 2020). This is a consistency notion across embeddings rather than across subpaths.

In oriented-graph decomposition, “consistent” denotes something else entirely: a digraph $P,Q\in\mathcal P$2 is consistent when its path number equals the degree-imbalance lower bound

$P,Q\in\mathcal P$3

There the problem is not shortest-path realizability but decomposition of directed edges into as few paths as possible. For fixed odd $P,Q\in\mathcal P$4, a uniformly random $P,Q\in\mathcal P$5-vertex $P,Q\in\mathcal P$6-regular graph is strongly consistent with probability tending to $P,Q\in\mathcal P$7 as $P,Q\in\mathcal P$8, and every odd-$P,Q\in\mathcal P$9 regular graph with girth at least $$P\cap Q\in \{\emptyset,\text{ a single vertex},\text{ a path in }\mathcal P\}.$$0 is strongly consistent (Patel et al., 2024).

These extensions do not preserve the graph-geodesic definition, but they show that “consistency” is repeatedly used to express a global coherence condition on families of paths. In the graph-metric literature, that coherence is subpath closure and metric realizability; in geometric compatibility it is a common noncrossing embedding; in oriented decompositions it is attainment of an optimal degree-based lower bound. This suggests that the modern theory of consistent path systems is best understood as a cluster of closely related coherence notions, with the graph-theoretic geodesic formulation as the most developed structural core (Chudnovsky et al., 2023).