Distance to Transitivity: New Parameters for Taming Reachability in Temporal Graphs

TL;DR

This work addresses reachability in temporal graphs by introducing two transitivity-distance parameters, $\delta_{\mathrm{vd}}$ and $\delta_{\mathrm{am}}$, defined on the reachability graph $G_R$ to quantify proximity to transitive reachability. It provides fixed-parameter tractable algorithms for Open-TCC under these parameters, including a $3^{\delta_{\mathrm{vd}}}$-time algorithm and a $4^{\delta_{\mathrm{am}}}$-time algorithm with polynomial kernels given a modification set, while also establishing kernel-lower-bound results and hardness for Closed-TCC. The results hold without restrictions on the input graph, snapshots, or lifetime, highlighting non-transitivity as a key driver of tractability barriers in temporal reachability problems. The paper also discusses limits of these approaches for Closed-TCC and outlines future directions in transitivity-aware parameterizations and modifications of the temporal graph itself. Overall, it isolates non-transitivity as a core source of complexity and provides a framework for targeted FPT algorithms and kernel techniques in temporal graph analysis.

Abstract

A temporal graph is a graph whose edges only appear at certain points in time. Reachability in these graphs is defined in terms of paths that traverse the edges in chronological order (temporal paths). This form of reachability is neither symmetric nor transitive, the latter having important consequences on the computational complexity of even basic questions, such as computing temporal connected components. In this paper, we introduce several parameters that capture how far a temporal graph $\mathcal{G}$ is from being transitive, namely, \emph{vertex-deletion distance to transitivity} and \emph{arc-modification distance to transitivity}, both being applied to the reachability graph of $\mathcal{G}$. We illustrate the impact of these parameters on the temporal connected component problem, obtaining several tractability results in terms of fixed-parameter tractability and polynomial kernels. Significantly, these results are obtained without restrictions of the underlying graph, the snapshots, or the lifetime of the input graph. As such, our results isolate the impact of non-transitivity and confirm the key role that it plays in the hardness of temporal graph problems.

Figures (1)

• Figure 3: Left: the original instance of Clique from \ref{['kernel if set given']} constructed from the reachability graph of the considered temporal graph. Right: the obtained compressed instance of Clique after exhaustive application of all reduction rules. In both parts, the blue vertices are the vertices from the inherent transitivity modulator $B$ and the cycles at the bottom indicate the white clusters. Note that in both graphs, each blue vertex has neighbors in at most one white cluster (see \ref{['claim:red_deg_one']}). Intuitively, RR 1 ensures that small clusters are removed, RR 1 and RR 2 ensure that there are no isolated white clusters, and RR 3 reduces the size of each white cluster to at most $|B| + 1$.

Theorems & Definitions (15)

• Lemma 2: BF03
• Lemma 3
• Corollary 4
• Proposition 5
• Lemma 6
• Theorem 7
• Theorem 8
• Theorem 9
• Corollary 11
• Theorem 12