The 5G core network is the backbone of modern mobile communication, providing high-speed, low-latency, and diverse services for users and industries. Artificial Intelligence (AI) plays an important role in this network by optimizing per-formance, supporting dynamic resource scaling, and improving security through anomaly detection and threat mitigation. Testing AI in 5G environments is difficult because of the complexity of the network and the many possible attack vectors. In this paper, we present a modular and reproducible testbed for evaluating AI-based security mechanisms in the 5G core. The testbed emulates key 5G components and traffic patterns, enabling systematic experiments under realistic conditions. It also provides reliable measurements of Key Performance Indicators (KPIs) to evaluate the effectiveness, robustness, and operational impact of AI solutions, including their ability to detect and mitigate threats. Our work provides a structured framework for testing AI solutions and supports the development of secure, resilient, and AI -enhanced 5G networks.

Abstract. Deciding whether a diagram of a knot can be untangled with a given number of moves (as a part of the input) is known to be NP-complete. In this paper we determine the parameterized complexity of this problem with respect to a natural parameter called defect. Roughly speaking, it measures the efficiency of the moves used in the shortest untangling sequence of Reidemeister moves. We show that in a shortest untangling sequence the [Formula: see text] moves, that is, the moves removing two adjacent crossings, can be essentially performed greedily. Using that, we show that this problem belongs to W[P] when parameterized by the defect. We also show that this problem is W[P]-hard by a reduction from Minimum axiom set.

We study the problem of reconfiguring one minimum $s$-$t$-separator $A$ into another minimum $s$-$t$-separator $B$ in some $n$-vertex graph $G$ containing two non-adjacent vertices $s$ and $t$. We consider several variants of the problem as we focus on both the token sliding and token jumping models. Our first contribution is a polynomial-time algorithm that computes (if one exists) a minimum-length sequence of slides transforming $A$ into $B$. We additionally establish that the existence of a sequence of jumps (which need not be of minimum length) can be decided in polynomial time (by an algorithm that also outputs a witnessing sequence when one exists). In contrast, and somewhat surprisingly, we show that deciding if a sequence of at most $\ell$ jumps can transform $A$ into $B$ is an $\textsf{NP}$-complete problem. To complement this negative result, we investigate the parameterized complexity of what we believe to be the two most natural parameterized counterparts of the latter problem; in particular, we study the problem of computing a minimum-length sequence of jumps when parameterized by the size $k$ of the minimum \stseps and when parameterized by the number of jumps $\ell$. For the first parameterization, we show that the problem is fixed-parameter tractable, but does not admit a polynomial kernel unless $\textsf{NP} \subseteq \textsf{coNP/poly}$. We complete the picture by designing a kernel with $\mathcal{O}(\ell^2)$ vertices and edges for the length $\ell$ of the sequence as a parameter.

Abstract. For an integer [Formula: see text] and a graph [Formula: see text], let [Formula: see text] be the graph that has vertex set all proper [Formula: see text]-colorings of [Formula: see text], and an edge between two vertices [Formula: see text] and [Formula: see text] whenever the coloring [Formula: see text] can be obtained from [Formula: see text] by a single Kempe change. A theorem of Meyniel from 1978 states that [Formula: see text] is connected with diameter [Formula: see text] for every planar graph [Formula: see text]. We significantly strengthen this result by showing that there is a positive constant [Formula: see text] such that [Formula: see text] has diameter [Formula: see text] for every planar graph [Formula: see text].

Deciding whether a diagram of a knot can be untangled with a given number of moves (as a part of the input) is known to be NP-complete. In this paper we determine the parameterized complexity of this problem with respect to a natural parameter called defect. Roughly speaking, it measures the efficiency of the moves used in the shortest untangling sequence of Reidemeister moves. We show that the II- moves in a shortest untangling sequence can be essentially performed greedily. Using that, we show that this problem belongs to W[P] when parameterized by the defect. We also show that this problem is W[P]-hard by a reduction from Minimum axiom set.