Temporal Graph Exploration (TEXP)

• TEXP is defined as the problem of finding the shortest temporal walk that visits every vertex in a dynamic graph, where each time step uses only the edges present in the corresponding snapshot.
• It examines both offline and online explorations under adversarial and stochastic models, revealing that even always-connected snapshots can lead to quadratic worst-case exploration times.
• Recent studies leverage structural properties, frequency constraints, and parameterized complexity approaches to achieve subquadratic and near-linear exploration bounds in practical settings.

The Temporal Graph Exploration Problem (TEXP) asks for a shortest temporal walk that visits all vertices of a temporal graph, where a temporal graph is a sequence of graphs on a common vertex set and the $i$-th move may use only an edge present in the $i$-th snapshot. In the notation used in recent work, for a temporal graph $\mathcal{G}$, the exploration time is

$\texp(\mathcal{G}) = \min\{t : \exists \text{ a temporal walk starting at some vertex that visits all vertices within time } t\},$

with $\texp(\mathcal{G})=\infty$ if exploration is impossible (Baguley et al., 5 Aug 2025). TEXP is one of the central optimization problems in temporal graph theory because it exposes the interaction between temporal reachability, snapshot structure, and algorithmic access to the future.

1. Formal model and core notions

A temporal graph is typically written as a sequence of snapshots on a fixed vertex set,

$\mathcal{G} = (G_1,G_2,\dots),$

or, in finite-lifetime settings,

$\mathcal{G}=\langle G_1,G_2,\dots,G_L\rangle,$

where each $G_t=(V,E_t)$ and the underlying graph is the static union of all edges that ever appear (Baguley et al., 5 Aug 2025). A time-edge is a pair $(e,t)$ with $e\in E_t$. A temporal path is a sequence of time-edges with strictly increasing times whose underlying edges form a static path; a temporal walk is defined analogously with repeated vertices allowed. In the strict model, the agent traverses at most one edge per time step; in the non-strict model, an arbitrary number of edges may be traversed within one snapshot, which can equivalently be expressed by moving inside connected components with non-decreasing snapshot indices (Erlebach et al., 2022).

The always-connected regime is the foundational deterministic setting: every snapshot $i$0 is connected. In that regime, a standard lemma states that in any sliding window of $i$1 consecutive snapshots, every pair of vertices is temporally connected; equivalently, moving between arbitrary vertices never takes more than $i$2 time steps in such a window (Baguley et al., 5 Aug 2025). This property underlies many upper bounds, because it converts per-snapshot connectivity into a uniform temporal reachability guarantee.

TEXP is also studied in online and offline forms. In offline, or clairvoyant, exploration the full snapshot sequence is known in advance; in online exploration the algorithm only knows the past and current snapshot and must move without knowledge of future snapshots (Baguley et al., 5 Aug 2025). Much of the sharpest recent progress concerns the offline setting, while several lower bounds show that online exploration can be strictly harder.

2. Worst-case deterministic landscape

In the adversarial always-connected model, TEXP has a sharp quadratic baseline. Every always-connected temporal graph on $i$3 vertices admits an exploration in $i$4 time, and there are always-connected temporal graphs requiring $i$5 time; thus the worst-case exploration time is $i$6 up to constants (Erlebach et al., 2015). This remains true even though every individual snapshot is connected. A common misconception is that per-snapshot connectivity already implies static-like exploration behavior; the quadratic lower bound shows that temporal ordering alone can force repeated waiting and revisitation.

The optimization problem is also hard in the approximation sense. For every constant $i$7, it is NP-hard to approximate TEXP within a factor $i$8 (Erlebach et al., 2015). The same work also gives bounded-degree lower bounds of order $i$9 and a planar degree-4 construction, with each snapshot a path, requiring $\mathcal{G}$0 exploration time. These results identify two separate sources of difficulty: global temporal adversariality, which yields quadratic worst-case behavior, and sparse structured dynamics, which can still force superlinear exploration.

This baseline has become the point of comparison for essentially all later work. Subsequent papers either restrict the temporal process, restrict the underlying graph, or alter the objective, with the aim of escaping the quadratic barrier.

3. Structured classes with improved exploration bounds

A large body of recent work isolates structural properties that reduce exploration complexity. The following bounds are representative.

Class or model	Exploration guarantee	Source
Always-connected, average temporal maximum degree $\mathcal{G}$1	$\mathcal{G}$2	(Bastide et al., 27 Nov 2025)
Always-connected $\mathcal{G}$3-edge-deficient temporal graphs	$\mathcal{G}$4; for constant $\mathcal{G}$5, $\mathcal{G}$6	(Lahtin et al., 15 May 2026)
$\mathcal{G}$7-frequent / $\mathcal{G}$8-regular temporal graphs	$\mathcal{G}$9 and $\texp(\mathcal{G}) = \min\{t : \exists \text{ a temporal walk starting at some vertex that visits all vertices within time } t\},$0; more generally, at most $\texp(\mathcal{G}) = \min\{t : \exists \text{ a temporal walk starting at some vertex that visits all vertices within time } t\},$1 where $\texp(\mathcal{G}) = \min\{t : \exists \text{ a temporal walk starting at some vertex that visits all vertices within time } t\},$2 is the MWST weight in the frequency-weighted underlying graph	(Adamson, 20 May 2025)
Word-representable temporal graphs	$\texp(\mathcal{G}) = \min\{t : \exists \text{ a temporal walk starting at some vertex that visits all vertices within time } t\},$3 if connected in every timestep; $\texp(\mathcal{G}) = \min\{t : \exists \text{ a temporal walk starting at some vertex that visits all vertices within time } t\},$4 under the stated word-length condition	(Adamson, 11 Feb 2025)
Always $\texp(\mathcal{G}) = \min\{t : \exists \text{ a temporal walk starting at some vertex that visits all vertices within time } t\},$5-connected temporal graphs with $\texp(\mathcal{G}) = \min\{t : \exists \text{ a temporal walk starting at some vertex that visits all vertices within time } t\},$6 agents	$\texp(\mathcal{G}) = \min\{t : \exists \text{ a temporal walk starting at some vertex that visits all vertices within time } t\},$7; treewidth-$\texp(\mathcal{G}) = \min\{t : \exists \text{ a temporal walk starting at some vertex that visits all vertices within time } t\},$8 single-agent corollary $\texp(\mathcal{G}) = \min\{t : \exists \text{ a temporal walk starting at some vertex that visits all vertices within time } t\},$9	(Adamson et al., 23 Feb 2026)

The parameter $\texp(\mathcal{G})=\infty$0, the average temporal maximum degree, unifies several earlier subquadratic bounds. If each snapshot has maximum degree at most $\texp(\mathcal{G})=\infty$1, then $\texp(\mathcal{G})=\infty$2, yielding $\texp(\mathcal{G})=\infty$3; if the underlying graph has bounded average degree, the same theorem gives the first subquadratic upper bound in that regime (Bastide et al., 27 Nov 2025). The same framework improves earlier bounds for planar and bounded-treewidth underlying graphs.

The near-static regime of $\texp(\mathcal{G})=\infty$4-edge-deficient temporal graphs takes a different route. Here each snapshot differs from a fixed spanning tree in at most $\texp(\mathcal{G})=\infty$5 missing edges, so a stable backbone is almost always present. The resulting exploration time $\texp(\mathcal{G})=\infty$6 removes the extraneous $\texp(\mathcal{G})=\infty$7 factor from previous bounds, and for constant $\texp(\mathcal{G})=\infty$8 it yields order-optimal $\texp(\mathcal{G})=\infty$9 exploration time (Lahtin et al., 15 May 2026). This identifies a setting in which temporal exploration essentially retains the linear-time character of static graph traversal.

Frequency and regularity assumptions replace structural restrictions by temporal recurrence constraints. If every edge is active at least once in every $\mathcal{G} = (G_1,G_2,\dots),$0 consecutive time steps, then exploration is possible in $\mathcal{G} = (G_1,G_2,\dots),$1 steps; if edges obey periodic activation with regularity at most $\mathcal{G} = (G_1,G_2,\dots),$2, the analogous bound is $\mathcal{G} = (G_1,G_2,\dots),$3 (Adamson, 20 May 2025). The underlying technique is to weight each static edge by its worst waiting time and then explore a minimum-weight spanning tree of the frequency-weighted underlying graph.

Word-representable temporal graphs impose a combinatorial structure via a representing word. In the always-connected case, every edge reappears within $\mathcal{G} = (G_1,G_2,\dots),$4 steps, where $\mathcal{G} = (G_1,G_2,\dots),$5 is the minimum degree of the underlying graph, leading to exploration in at most $\mathcal{G} = (G_1,G_2,\dots),$6 time. In the general connected case, under the stated word-length condition, every edge appears in every window of $\mathcal{G} = (G_1,G_2,\dots),$7 steps, where $\mathcal{G} = (G_1,G_2,\dots),$8 is the underlying diameter, yielding exploration in at most $\mathcal{G} = (G_1,G_2,\dots),$9 time; a matching $\mathcal{G}=\langle G_1,G_2,\dots,G_L\rangle,$0 lower bound is also given (Adamson, 11 Feb 2025).

A further generalization replaces per-snapshot connectivity by always $\mathcal{G}=\langle G_1,G_2,\dots,G_L\rangle,$1-connectedness: each snapshot may have up to $\mathcal{G}=\langle G_1,G_2,\dots,G_L\rangle,$2 connected components, each containing a vertex of $\mathcal{G}=\langle G_1,G_2,\dots,G_L\rangle,$3. With $\mathcal{G}=\langle G_1,G_2,\dots,G_L\rangle,$4 agents, this yields $\mathcal{G}=\langle G_1,G_2,\dots,G_L\rangle,$5 exploration, and when combined with $\mathcal{G}=\langle G_1,G_2,\dots,G_L\rangle,$6-divisions it produces improved single-agent bounds for bounded-treewidth and certain interval graphs (Adamson et al., 23 Feb 2026). This suggests that separator structure and controlled component interfaces can substitute for full snapshot connectivity.

4. Random spanning-tree models and probabilistic exploration

A major recent shift is the study of TEXP in stochastic temporal graphs. The Random Spanning Tree (RST) model consists of a set $\mathcal{G}=\langle G_1,G_2,\dots,G_L\rangle,$7 of trees on $\mathcal{G}=\langle G_1,G_2,\dots,G_L\rangle,$8 together with a probability distribution $\mathcal{G}=\langle G_1,G_2,\dots,G_L\rangle,$9, and a random temporal graph is obtained by taking i.i.d. samples $G_t=(V,E_t)$0 from $G_t=(V,E_t)$1 (Baguley et al., 5 Aug 2025). Each snapshot is therefore a spanning tree, so the model is sparse but always-connected.

The main theorem is a sharp separation from the adversarial case: any RST model can, with high probability, be explored offline in $G_t=(V,E_t)$2 time, and this is tight up to constant factors (Baguley et al., 5 Aug 2025). The lower bound is realized by star-based RSTs. For a uniform distribution over $G_t=(V,E_t)$3 distinct $G_t=(V,E_t)$4-vertex stars, the exploration time is concentrated around

$G_t=(V,E_t)$5

up to an additive $G_t=(V,E_t)$6 term. Thus randomness lowers the worst-case order from $G_t=(V,E_t)$7 to $G_t=(V,E_t)$8 in this model.

The key structural phenomenon is temporal closeness. For sufficiently large $G_t=(V,E_t)$9, every vertex has at least $(e,t)$0 vertices that are $(e,t)$1-close, meaning reachable within $(e,t)$2 time steps with probability at least $(e,t)$3 (Baguley et al., 5 Aug 2025). The proof uses a temporal analogue of depth-first search, coupled runs on independent samples, and concentration arguments. This closeness relation defines an auxiliary meta-graph $(e,t)$4 whose connected components all have size at least $(e,t)$5; there are therefore at most $(e,t)$6 such components. An Euler-tour traversal inside components, together with deterministic $(e,t)$7-time travel between components from always-connectedness, yields the $(e,t)$8 exploration bound.

A stronger result holds when all sampled trees are spanning trees of a fixed connected graph $(e,t)$9 with $e\in E_t$0 edges. In that case, exploration is possible offline in time at most $e\in E_t$1 with probability at least $e\in E_t$2 (Baguley et al., 5 Aug 2025). The construction weights each edge $e\in E_t$3 by $e\in E_t$4, where $e\in E_t$5 is its appearance probability, uses a minimum-weight spanning tree, and refines it into a backbone forest and fast components joined by meta-edges. The same paper proves that such linear-in-$e\in E_t$6 behavior is impossible for online algorithms in general: there are sparse instances where every randomized online explorer needs $e\in E_t$7 time with high probability.

5. Parameterized complexity and kernelization

Parameterized work has focused especially on strict and non-strict exploration variants. In the strict model, a temporal walk uses edges at strictly increasing times; in the non-strict model, one may traverse arbitrarily many edges within a snapshot component (Erlebach et al., 2022). For the strict variant, $e\in E_t$8-fixed TEXP can be solved in $e\in E_t$9 time, and $i$00-arbitrary TEXP admits a Monte Carlo algorithm with running time $i$01 and a deterministic counterpart with running time $i$02 (Erlebach et al., 2022). In the non-strict model, NS-TEXP is fixed-parameter tractable in the lifetime $i$03, solvable in $i$04, and if every snapshot has at most two connected components then NS-TEXP, $i$05-fixed NS-TEXP, $i$06-arbitrary NS-TEXP, and Set NS-TEXP are all solvable in $i$07 time (Erlebach et al., 2022).

The negative side is equally pronounced. Set TEXP and Set NS-TEXP are W[2]-hard when parameterized by the lifetime $i$08 (Erlebach et al., 2022). For non-strict exploration, NS-TEXP is NP-complete for $i$09, where $i$10 is the maximum number of connected components in any snapshot, and remains NP-complete even when the underlying graph is a tree of depth two (Arrighi et al., 2023). That result rules out hopes that simple static structure alone makes temporal exploration easy.

Kernelization results sharpen this picture. Under the standard assumption that $i$11, neither NS-TEXP nor $i$12-arb NS-TEXP admits polynomial kernels for the standard parameters $i$13, $i$14, $i$15, or $i$16, nor for the combined parameters $i$17, $i$18, and $i$19 (Arrighi et al., 2023). The positive exception is structural: for

$i$20

which measures the excess of total edge appearances over a spanning-tree baseline, Weighted $i$21-arb NS-TEXP admits a kernel of size $i$22, with $i$23, $i$24, and $i$25 (Arrighi et al., 2023). The proof identifies a small core containing all repeated edges and compresses the attached tree components by local gadget replacements plus Frank–Tardos weight compression. This suggests that temporal sparsity, rather than standard graph parameters, is the more effective kernelization axis for exploration.

6. Related objectives and specialized variants

Several closely related formulations refine or specialize TEXP. The Shortest Temporal Exploration Problem (STEXP) keeps the always-connected model and the same strict movement rule, but minimizes the number of traversed edges subject to completing within a given lifetime $i$26 (Balev et al., 9 Apr 2025). Every constantly connected temporal graph with $i$27 vertices can be explored with $i$28 edges within $i$29 time steps. If every snapshot has diameter at most $i$30, there exists an exploration using $i$31 edges within $i$32 time. For temporal cycles with $i$33, the worst-case edge complexity is tightly characterized: if $i$34, the bound is $i$35, whereas for $i$36 there are instances requiring exactly $i$37 edges (Balev et al., 9 Apr 2025).

Another specialized line studies temporal stars with a return-to-base requirement. In this model the walk starts at the center, visits leaves, and returns to the center. The decision problem $i$38 and the maximization problem $i$39 depend on the maximum number $i$40 of time labels per edge. $i$41 and $i$42 are solvable in $i$43, while for every $i$44, $i$45 is NP-complete and $i$46 is APX-hard; a polynomial-time 2-approximation exists for $i$47 for every $i$48 (Akrida et al., 2018). Although this is not standard TEXP, it shows that temporal hardness can arise even on the simplest underlying topology once time labels are sufficiently expressive.

A more historical antecedent is the exploration of periodically varying graphs, where edges appear according to periodic carrier routes rather than arbitrary snapshot sequences. In anonymous systems, exploration is impossible without an upper bound on the period; with node IDs, exploration is impossible without either a period bound or the total number of sites $i$49. These limitations are tight: worst-case optimal algorithms are given for anonymous systems with a period bound and for labelled systems with node IDs (0909.4369). This route-based model predates the current snapshot-centric formulation but already exhibits the same core phenomenon as TEXP: temporal recurrence and switching opportunities, rather than static connectivity alone, determine what exploration can achieve.

Taken together, these results suggest that the main algorithmic divide in temporal exploration is not simply between connected and disconnected snapshots. The stronger distinction is between adversarial temporal orderings, which preserve the quadratic worst-case barrier, and models with exploitable temporal regularity, stochastic independence, bounded recurrence, or compressible structure, which admit subquadratic or even near-linear exploration regimes.