Resolving Sets in Temporal Graphs

TL;DR

The paper studies temporal resolving sets in temporal graphs, generalizing the metric dimension via the temporal distance $dist_t(u,v)$ to capture earliest feasible journeys with strictly increasing time steps. It establishes NP-hardness for several restricted labeling regimes (e.g., temporal complete graphs with 1-labeling and two steps, temporal subdivided stars with 2-labeling, and temporal trees with consecutive steps) and provides constructive polynomial-time algorithms for specific subclasses, notably temporal paths and temporal stars under 1-labelings. It also analyzes $p$-periodic $1$-labelings, deriving exact or tight bounds for key graph classes and showing fixed-parameter tractability with respect to the number of leaves (and XP results for subdivided stars). Collectively, these results map the complexity landscape of locating information in dynamic networks and suggest directions for further work on temporal distances and parameterized approaches.

Abstract

A \emph{resolving set} $R$ in a graph $G$ is a set of vertices such that every vertex of $G$ is uniquely identified by its distances to the vertices of $R$. Introduced in the 1970s, this concept has been since then extensively studied from both combinatorial and algorithmic points of view. We propose a generalization of the concept of resolving sets to temporal graphs, \emph{i.e.}, graphs with edge sets that change over discrete time-steps. In this setting, the \emph{temporal distance from $u$ to $v$} is the earliest possible time-step at which a journey with strictly increasing time-steps on edges leaving $u$ reaches $v$, \emph{i.e.}, the first time-step at which $v$ could receive a message broadcast from $u$. A \emph{temporal resolving set} of a temporal graph $\mathcal{G}$ is a subset $R$ of its vertices such that every vertex of $\mathcal{G}$ is uniquely identified by its temporal distances from vertices of $R$. We study the problem of finding a minimum-size temporal resolving set, and show that it is NP-complete even on very restricted graph classes and with strong constraints on the time-steps: temporal complete graphs where every edge appears in either time-step~1 or~2, temporal trees where every edge appears in at most two consecutive time-steps, and even temporal subdivided stars where every edge appears in at most two (not necessarily consecutive) time-steps. On the other hand, we give polynomial-time algorithms for temporal paths and temporal stars where every edge appears in exactly one time-step, and give a combinatorial analysis and algorithms for several temporal graph classes where the edges appear in periodic time-steps.

Figures (2)

• Figure 1: The construction of the proof of \ref{['thm-NPhardnessOnSubStars']}. Only the branches 1 and $s$ are detailed together with the control branch. We have $\{a_i,b_i,c_i\}=\{x_i,y_i,z_i\}$ where $a_i<b_i<c_i$.
• Figure 2: The construction of the proof of \ref{['thm-NPhardnessOnTrees']}. Only the branch 1 is detailed, we have $x_1=1$, $y_1=n$ and $z_1=3$. Dashed lines represent longer paths.

Theorems & Definitions (34)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Remark 4
• Lemma 5
• proof
• Theorem 6