Knots are commonly represented and manipulated via diagrams, which are decorated planar graphs. When such a knot diagram has low treewidth, parameterized graph algorithms can be leveraged to ensure the fast computation of many invariants and properties of the knot. It was recently proved that there exist knots which do not admit any diagram of low treewidth, and the proof relied on intricate low-dimensional topology techniques. In this work, we initiate a thorough investigation of tree decompositions of knot diagrams (or more generally, diagrams of spatial graphs) using ideas from structural graph theory. We define an obstruction on spatial embeddings that forbids low tree width diagrams, and we prove that it is optimal with respect to a related width invariant. We then show the existence of this obstruction for knots of high representativity, which include for example torus knots, providing a new and self-contained proof that those do not admit diagrams of low treewidth. This last step is inspired by a result of Pardon on knot distortion.
While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a class of knots and links called Hopf arborescent links, which are obtained as the boundaries of some iterated plumbings of Hopf bands. We show that for such links, computing the genus defects, which measure how much the four-dimensional genera differ from the classical genus, is decidable. Our proof is non-constructive, and is obtained by proving that Seifert surfaces of Hopf arborescent links under a relation of minors defined by containment of their Seifert surfaces form a well-quasi-order.
Intelligent Transportation Systems increasingly rely on Vehicle-to-Everything (V2X) communication for real-time traffic management, yet existing models assume continuous network connectivity, an assumption that fails catastrophically during disasters. This paper addresses a critical challenge in vehicular communication reliability: the simultaneous degradation of road network topology and communication infrastructure (dark zones) caused by disaster-induced failures of Roadside Units (RSUs). We present a Dynamic Population Evacuation (DPE) framework that integrates Vehicular Ad Hoc Networks (VANETs) with adaptive traffic management under infrastructure disruption. The framework couples offline strategic planning (the Shelter Allocation Problem and Dynamic Traffic Assignment via Stochastic User Equilibrium) with an online rolling-horizon management system that detects and responds to dark-zone evolution in real time. Unlike conventional VANET applications that prioritize low latency, we demonstrate that evacuation scenarios require system-optimal solutions dependent on network-wide data aggregation, making centralized Cloud architectures superior to decentralized Fog computing despite higher end-to-end delays. We incorporate driver behavioral compliance as a stochastic variable to quantify how non-compliance with rerouting instructions degrades system performance. Validated through co-simulation on the Camp Fire wildfire in Paradise, California, our framework achieves 27% clearance time improvement over naive allocation, 19% over traditional planning, and 12% over Fog-based architectures, demonstrating the criticality of dark zone-aware V2X design for resilient ITS.
Let G = (g1, ..., g & ell;) be a vector of positive integers and set n = & sum;& ell; i=1 gi. Writing Ch(a, b) for the Christoffel word with parameters (a, b), we study the following synchronization problem: choose one conjugate of each word Ch(gi, n-gi) so that, at every position 0, ..., n-1, exactly one of the chosen words contains the letter 1. This gives a gap-free form of superimposition, motivated by Fraenkel's conjecture. We encode the choice of conjugates by a shift vector V = (v1, ..., v & ell;). On the cyclic Cayley graph of Z/nZ, this yields an orbital matrix O(G, V), with entries oi,j = (vi +jgi) mod n, and a Christoffel-conjugate matrix C(G, V), which records the columns where a wraparound occurs. The vector V is a synchronizing seed precisely when each column of C(G, V) contains exactly one nonzero entry. The main algebraic tool is a vertical invariant: the column sums of O(G, V) are constant if and only if this one-wraparound-per-column condition holds. Using this criterion, we give explicit synchronizing seeds for every pair of Christoffel words of common length, for triples in which two generators coincide, for triples with three equal generators, and for the all-distinct family proportional to (1, 2, 4). Finally, we give a geometric interpretation: each synchronized row is the Freeman chain code of a standard 4-connected R & eacute;veill & egrave;s segment, whose parameter & micro;i is the unique integer in {1-n, ..., 0} satisfying & micro;i equivalent to -vi (mod n). (c) 2026 Published by Elsevier B.V.
This paper studies crossing numbers of d-dimensional simplicial complexes drawn in ℝ^2d or 𝕊^2d. We clarify and generalize a number of known results and conjectures from the graph case to higher dimensions: the crossing lemma (including the exceptional case of two-dimensional complexes drawn in dimension 4), a generalized Harary-Hill conjecture (and evidence for it), existence of a midrange crossing constant (in a weak sense), and upper bound constructions based on stochastic geometry (related to Sylvester's problem) and the cyclic polytope. Some of our results are new even in the case of graphs: we give a simple proof of the Harary-Hill conjecture for antipodal drawings, and sharpen the threshold above which the rectilinear and spherical crossing numbers of the Erdős-Rényi random graph behave like those of the complete graph.
