Improved exploration of temporal graphs

TL;DR

The paper studies the temporal exploration problem on always-connected temporal graphs and introduces the average temporal maximum degree D as a unifying parameter. It proves a general bound: an exploration exists with length $O(n^{3/2}\sqrt{D\log n})$, from which subquadratic bounds follow in several natural settings. The approach hinges on two structural lemmas—one guaranteeing temporally connected walks inside large vertex subsets and another producing small dominating subsets—coupled with a time-interval partitioning strategy to iteratively cover all vertices. These results unify and improve many existing bounds, providing algorithmic constructions and expanding applicability to planar, treewidth-bounded, and minor-free underlying graphs.

Abstract

A temporal graph $G$ is a sequence $(G_t)_{t \in I}$ of graphs on the same vertex set of size $n$. The \emph{temporal exploration problem} asks for the length of the shortest sequence of vertices that starts at a given vertex, visits every vertex, and at each time step $t$ either stays at the current vertex or moves to an adjacent vertex in $G_t$. Bounds on the length of a shortest temporal exploration have been investigated extensively. Perhaps the most fundamental case is when each graph $G_t$ is connected and has bounded maximum degree. In this setting, Erlebach, Kammer, Luo, Sajenko, and Spooner [ICALP 2019] showed that there exists an exploration of $G$ in $\mathcal{O}(n^{7/4})$ time steps. We significantly improve this bound by showing that $\mathcal{O}(n^{3/2} \sqrt{\log n})$ time steps suffice. In fact, we deduce this result from a much more general statement. Let the \emph{average temporal maximum degree} $D$ of $G$ be the average of $\max_{t \in I} d_{G_t}(v)$ over all vertices $v \in V(G)$, where $d_{G_t}(v)$ denotes the degree of $v$ in $G_t$. If each graph $G_t$ is connected, we show that there exists an exploration of $G$ in $\mathcal{O}(n^{3/2} \sqrt{D \log n})$ time steps. In particular, this gives the first subquadratic upper bound when the underlying graph has bounded average degree. As a special case, this also improves the previous best bounds when the underlying graph is planar or has bounded treewidth and provides a unified approach for all of these settings. Our bound is subquadratic already when $D=o(n/\log n)$.

Figures (2)

• Figure 1: The figure depicts two consecutive snapshots from the construction that shows that \ref{['lemma:no_big_stable_set']} is tight. The orange vertices represent the set $X$.
• Figure 2: The figure depicts the situation described in the proof of \ref{['lemma:dominating_set']}. The fact that $G_t$ is connected implies the existence of a vertex $w_{u,t}$ outside of but adjacent to $F(u,t-1) \cap B(u,t+1)$. By construction, $w_{u,t}$ will be in $F(u,t)\cap B(u,t)$. Crucially, $d_{G_t}(w_{u,t}) \leqslant d_{\max}(w_{u,t})$. Therefore, $w_{u,t}$ is recorded by at most $d_{\max}(w_{u,t})$ vertices at time step $t$. We stress that even though some sets are represented to be disjoint for clarity, this is not always the case.

Theorems & Definitions (11)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Lemma 2.1
• proof : Proof of Lemma \ref{['lemma:no_big_stable_set']}
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4: ErlebachHoffmannKammer21