Realization of Temporally Connected Graphs Based on Degree Sequences
Arnaud Casteigts, Michelle Döring, Nils Morawietz
TL;DR
This work resolves the TC-realizability question for degree sequences by giving complete, constructive characterizations for graphical and multigraphical sequences. It shows that TC-realizable sequences are exactly those that admit either a C_4-pivotable realization or a pair of spanning trees sharing at most one edge, with multigraphical sequences following an analogous dichotomy. The authors provide linear-time recognition and, when feasible, an O(n+m) construction that outputs a TC-realizing (multi)graph along with a TC-labeling, using a carefully designed data structure and gadget-based inductive constructions. The results substantially advance temporal graph realizability by reducing it to degree-sequence realizability and enabling efficient generation of TC-realizations, with potential extensions to directed settings and higher connectivity notions.
Abstract
Given an undirected graph $G$, the problem of deciding whether $G$ admits a simple and proper time-labeling that makes it temporally connected is known to be NP-hard (Göbel et al., 1991). In this article, we relax this problem and ask whether a given degree sequence can be realized as a temporally connected graph. Our main results are a complete characterization of the feasible cases, and a recognition algorithm that runs in $O(n)$ time for graphical degree sequences (realized as simple temporal graphs) and in $O(n+m)$ time for multigraphical degree sequences (realized as non-simple temporal graphs, where the number of time labels on an edge corresponds to the multiplicity of the edge in the multigraph). In fact, these algorithms can be made constructive at essentially no cost. Namely, we give a constructive $O(n+m)$ time algorithm that outputs, for a given (multi)graphical degree sequence $\mathbf{d}$, a temporally connected graph whose underlying (multi)graph is a realization of $\mathbf{d}$, if one exists.