Routed Graph: Models & Algorithms

• Routed graph is a graph equipped with routing semantics—admissible moves, path constraints, and performance criteria that distinguish it from standard graph models.
• It encompasses diverse models such as permutation routing, deadlock-free schemes, electrical oblivious routing, and local geometric routing, each with unique structural and computational characteristics.
• Studies show that routing complexity depends more on graph geometry, embedded constraints, and bottleneck substructures than on raw vertex count, influencing practical network design.

Taken together, the literature suggests that a routed graph is best understood not as a single canonical graph class, but as a graph equipped with routing semantics: admissible moves, path constraints, state variables, and performance or reachability criteria. In this sense, routed graph theory encompasses permutation routing via matchings on connected undirected graphs, deadlock-free message routing on directed graphs, oblivious flow routing on weighted graphs, local routing on embedded graphs, and scheduled all-to-all communication on regular digraphs (Banerjee et al., 2016, Mendlovic et al., 6 Mar 2025, 0909.2859).

1. Formal models and routing semantics

Across the cited works, routing is defined by the pair consisting of a graph and an admissible communication primitive. In the routing-via-matching model, the network is an undirected connected graph $G$ with vertices labeled $[1,\dots,n]$; a permutation $\pi$ specifies pebble destinations; and a routing step is a matching whose edges swap pebbles in parallel. The routing time is $rt(G,\pi)$, and the routing number is $rt(G)=\max_{\pi} rt(G,\pi)$ (Banerjee et al., 2016). In the Dally–Seitz deadlock model, the network is a strongly connected directed graph $G=(V,E)$, routing is deterministic and depends only on source and destination, and deadlock freedom is equivalent to acyclicity of the dependency directed graph $\mathcal D=(E,D)$ (Mendlovic et al., 6 Mar 2025). In oblivious electrical routing, the graph is undirected, positively weighted, and routing is a linear operator $R:\mathbb{R}^V\to\mathbb{R}^E$, instantiated as

$R(d)=WB\,L^\dagger d,$

with competitive ratio $\eta_R$ defined against congestion (0909.2859). In local geometric routing, the next hop is chosen using only local information, often with $[1,\dots,n]$0 memory and a competitive routing ratio (Ashvinkumar et al., 2019). In all-to-all routing on regular digraphs, a routing scheme assigns a directed walk to every ordered source–destination pair, and the main global measures are dilation, congestion, and makespan (Faber et al., 2023).

Model	Graph setting	Central quantity
Routing via matching	Connected undirected graph	$[1,\dots,n]$1, $[1,\dots,n]$2
Deadlock-free routing	Strongly connected directed graph	Acyclic dependency graph
Electrical oblivious routing	Undirected weighted graph	$[1,\dots,n]$3
Local routing	Embedded graph	Routing ratio / delivery guarantee
All-to-all digraph routing	Connected regular digraph	Dilation, congestion, makespan

A persistent point of clarification is that routing on graphs is not restricted to shortest-path computation. The literature includes swap-based permutation routing, cut- and dependency-based admissibility, face-walking and homology-based local navigation, and linear-algebraic reachability systems. This suggests that “routed graph” is a model-dependent notion whose precise meaning is fixed by the routing primitive rather than by graph structure alone.

2. Structural conditions for routability

Several works turn routing existence into a structural graph theorem. The sharpest example is the 2025 characterization of deadlock-free routing for arbitrary directed networks: deadlock-free routing can be constructed for a network $[1,\dots,n]$4 if and only if $[1,\dots,n]$5 contains two edge-disjoint directed trees rooted at the same vertex $[1,\dots,n]$6, one directed into $[1,\dots,n]$7 and one directed away from $[1,\dots,n]$8. Equivalently, the graph must contain an in-arborescence and an out-arborescence rooted at the same $[1,\dots,n]$9, with no common edges (Mendlovic et al., 6 Mar 2025). The same paper observes that deciding whether such a pair exists is NP-complete, so deadlock-free routability is not merely a question of route design but of underlying graph structure.

A related structural viewpoint appears in the Broadcast Routing Problem. Given an undirected graph $\pi$0, a root $\pi$1, and a nonnegative weight function $\pi$2, one seeks an $\pi$3-acyclic orientation and a maximum packing of $\pi$4-arborescences. The quantities

$\pi$5

and

$\pi$6

formalize the optimization objective. For outerplanar graphs, BRP and its rooted variant are solvable in polynomial time (Chaourar, 2018). Here, routability is constrained simultaneously by acyclicity, root choice, and arborescence-packing capacity.

Symmetry provides a third structural route to routability. In regular directed graphs, a spanning factorization is a 1-factorization together with a set of words $\pi$7, $\pi$8, such that for every vertex $\pi$9, the endpoints $rt(G,\pi)$0 are all distinct. A schedule then assigns times to factor occurrences so that no factor receives the same time twice and times strictly increase along each word. This yields conflict-free transfer of a single-source routing template to all vertices (Dougherty et al., 2014). A plausible implication is that high symmetry allows routing to be designed at the level of a generating template rather than by explicit per-pair path synthesis.

3. Routing number and permutation routing

Permutation routing via matchings is one of the most developed routed-graph models. The fundamental problem is to route every pebble to its destination under a permutation $rt(G,\pi)$1 using minimum matching-steps. The complexity landscape is sharply split: determining whether $rt(G,\pi)$2 can be done in $rt(G,\pi)$3 deterministic time for any arbitrary connected graph $rt(G,\pi)$4, whereas determining whether $rt(G,\pi)$5 for any $rt(G,\pi)$6 is NP-complete (Banerjee et al., 2016). The two-step case is characterized by an auxiliary cycle graph $rt(G,\pi)$7, with

$rt(G,\pi)$8

Within this model, substantial effort has gone into graph-specific routing numbers. For the $rt(G,\pi)$9-dimensional pyramid $rt(G)=\max_{\pi} rt(G,\pi)$0, with level sizes $rt(G)=\max_{\pi} rt(G,\pi)$1 and total number of vertices

$rt(G)=\max_{\pi} rt(G,\pi)$2

the main theorem is

$rt(G)=\max_{\pi} rt(G,\pi)$3

The proof is constructive and uses five rounds, alternating routing within levels and routing along disjoint vertical paths in a multi-grid subgraph $rt(G)=\max_{\pi} rt(G,\pi)$4 (Banerjee et al., 2016). The dominant cost comes from routing inside the bottom $rt(G)=\max_{\pi} rt(G,\pi)$5-dimensional mesh, while the even rounds contribute only $rt(G)=\max_{\pi} rt(G,\pi)$6.

A second major generalization concerns convex pieces of the infinite square lattice. If $rt(G)=\max_{\pi} rt(G,\pi)$7 is the induced subgraph enclosed by a convex polygon and is connected, then

$rt(G)=\max_{\pi} rt(G,\pi)$8

and, since the routing number is at least the diameter, the paper notes that

$rt(G)=\max_{\pi} rt(G,\pi)$9

This extends the rectangular benchmark $G=(V,E)$0 to all connected convex grid pieces (Alpert et al., 2021). The proof proceeds through ramp-like polygons, monotonic configurations, burger bun polygons, and a final shear-and-skin reduction. The resulting picture is that convexity restores linear routing behavior, whereas arbitrary nonconvex lattice pieces can behave as badly as trees.

These results underscore a broader theme: the routing number is controlled less by raw vertex count than by geometry, layer structure, or large mesh dimensions. This suggests that routed graph complexity often tracks the dominant bottleneck substructure rather than the ambient graph alone.

4. Geometry, embedding, and local routing

In geometric and embedded settings, routing is often constrained to be local. For planar geometric graphs, a routing algorithm is 1-local if each decision uses only the current vertex, its immediate neighborhood, and a constant-size header or memory. One notable construction starts from the Delaunay triangulation, prunes it to a bounded-degree graph $G=(V,E)$1, and proves

$G=(V,E)$2

The resulting routing simulation on $G=(V,E)$3 is 1-local, uses $G=(V,E)$4 memory, and has routing ratio at most

$G=(V,E)$5

A further pruning step yields a light graph $G=(V,E)$6 with the same degree bound, bounded total weight relative to $G=(V,E)$7, and final routing ratio

$G=(V,E)$8

(Ashvinkumar et al., 2019).

For graphs embedded on surfaces of genus $G=(V,E)$9, ordinary planar Face Routing fails because homologically non-trivial closed walks may trap the right-hand rule. Generalized Face Routing (GFR) remedies this by using a homology basis

$\mathcal D=(E,D)$0

and a virtual multigraph whose vertices are non-trivial border walks. The main theorem is that GFR is a local routing algorithm transporting memory of $\mathcal D=(E,D)$1, and guaranteeing delivery in time $\mathcal D=(E,D)$2 (Fraser, 2012). Here, local routability depends on topological information external to the graph adjacency relation itself.

Geographic routing has also been extended beyond lattices. In fixed-growth graph families $\mathcal D=(E,D)$3 of dimensionality $\mathcal D=(E,D)$4, balls satisfy

$\mathcal D=(E,D)$5

and long-range contacts are chosen with probability proportional to $\mathcal D=(E,D)$6. With highway constant $\mathcal D=(E,D)$7, the paper proves tight bounds for greedy routing and diameter. In particular, for $\mathcal D=(E,D)$8, greedy routing between far-apart nodes succeeds in $\mathcal D=(E,D)$9 hops w.h.p. in $R:\mathbb{R}^V\to\mathbb{R}^E$0, while for $R:\mathbb{R}^V\to\mathbb{R}^E$1 the guarantee is

$R:\mathbb{R}^V\to\mathbb{R}^E$2

hops (Gila et al., 5 Feb 2025). The empirical application to all 50 U.S. states plus DC indicates that the optimal clustering exponent is better modeled by $R:\mathbb{R}^V\to\mathbb{R}^E$3 than by network size.

A recurring misconception is that locality necessarily precludes strong guarantees. The cited results show otherwise: constant-competitive local routing exists in bounded-degree planar spanners, guaranteed-delivery routing exists on arbitrary-genus surfaces, and polylogarithmic greedy routing exists on fixed-growth graphs under the appropriate long-range distribution.

5. Congestion, obliviousness, and dynamic routing

Another major routed-graph theme is the separation between routing quality and traffic awareness. Electric routing is an oblivious routing scheme on undirected weighted graphs, defined by the pseudoinverse of the Laplacian: $R:\mathbb{R}^V\to\mathbb{R}^E$4 Its congestion competitive ratio satisfies

$R:\mathbb{R}^V\to\mathbb{R}^E$5

and, for unweighted bounded-degree graphs, the paper derives

$R:\mathbb{R}^V\to\mathbb{R}^E$6

in terms of vertex expansion $R:\mathbb{R}^V\to\mathbb{R}^E$7. The same operator-theoretic control also bounds latency of the electric walk (0909.2859). The central insight is that congestion and latency can be tied to the $R:\mathbb{R}^V\to\mathbb{R}^E$8 norm of the inverse Laplacian rather than to shortest-path structure.

A contrasting framework treats congestion structurally rather than edgewise. In “global congestion avoidance,” the congested core $R:\mathbb{R}^V\to\mathbb{R}^E$9 is the subgraph induced by congested edges, and a congested subnetwork is a dense subgraph of $R(d)=WB\,L^\dagger d,$0. A Congestion-Avoiding Path avoids every densest subgraph of $R(d)=WB\,L^\dagger d,$1 under a chosen density measure $R(d)=WB\,L^\dagger d,$2. More generally, the density index

$R(d)=WB\,L^\dagger d,$3

leads to the PATH WITH PRESCRIBED DENSITY INDEX problem, which is polynomial-time solvable for efficient density measures (Mojaveri et al., 2021). This replaces path-length optimization by dense-substructure avoidance.

Time-varying routed graphs introduce a further distinction between the full contact graph and a sparse geometric backbone. In Delay Tolerant Networking, the graph is modeled as $R(d)=WB\,L^\dagger d,$4 with time-dependent edge weights, and the TDSP objective minimizes arrival time subject to the FIFO condition. For satellites moving along elliptic curves, the number of topological critical events that change a Delaunay-based subgraph $R(d)=WB\,L^\dagger d,$5 is polynomial, stated as $R(d)=WB\,L^\dagger d,$6 up to Davenport–Schinzel-sequence-based factors (Piekenbrock, 2024). Experimental results on STARLINK TLE-derived networks report an order-of-magnitude reduction in cumulative edges and topological critical events for the $R(d)=WB\,L^\dagger d,$7-stable Delaunay graph compared with the full visibility graph. This suggests that geometric structure can compress dynamic routing state.

Adaptive, learning-based routing provides yet another interpretation of routed graphs. In packet routing on a directed graph $R(d)=WB\,L^\dagger d,$8, DGATR models each router as an agent in a MA-POMDP, uses a graph attention network over local features $R(d)=WB\,L^\dagger d,$9, and defines reward

$\eta_R$0

The greedy action is

$\eta_R$1

Simulation results show that DGATR outperforms Q-routing, Hybrid, and DQN-routing in average packet E2E delay and affordable load, with cooperated learning often giving the best stability/delay tradeoff (Mai et al., 2021). A plausible implication is that, once queueing and load are endogenous, the routed graph becomes partly a learned state-space object rather than only a combinatorial one.

6. Symmetry, schedules, and algebraic reachability

In regular digraphs, routing is often studied as a scheduling problem over a fixed path system. Every connected regular digraph has an all-to-all routing scheme and associated schedule with no waiting. If the graph has degree $\eta_R$2 and diameter $\eta_R$3, the makespan is bounded by

$\eta_R$4

For dense $\eta_R$5-regular digraphs of diameter $\eta_R$6, the ratio of the optimal all-to-all makespan to $\eta_R$7 approaches $\eta_R$8 as $\eta_R$9, where

$[1,\dots,n]$00

For Kautz graphs, the line-graph structure yields better walk covers, and the farthest-distance-first scheduler is proved optimal under the dominance conditions developed in the paper (Faber et al., 2023). This continues the earlier spanning-factorization program, where short factorizations and minimum schedules were constructed for Cayley graphs, cycle prefix digraphs, and the graphs $[1,\dots,n]$01 (Dougherty et al., 2014).

Rotor-routing introduces an algebraic extension of routed-graph reachability. In the linear extension on Generalized Rotor Mechanism multigraphs, the state is $[1,\dots,n]$02, and the routing map is

$[1,\dots,n]$03

Standard rotor-routing is recovered as the cyclic GRM special case. Free routing satisfies

$[1,\dots,n]$04

while legal reachability requires both a legal routing vector and a guiding support condition. The paper establishes that legal reachability in GRM multigraphs is NP-complete in general, but polynomial-time for cyclic GRM multigraphs (Auger et al., 2024). This places routed graphs alongside vector addition systems and the abelian sandpile model as additive state-transition systems with legality constraints.

Viewed across these works, routed graphs admit two complementary abstractions. One treats routing as a scheduled use of graph edges under symmetry, factorization, and congestion bounds. The other treats routing as reachability in a constrained dynamical system on graph-indexed states. The coexistence of these abstractions is one reason the subject spans combinatorics, distributed algorithms, topology, geometry, optimization, and algebra.