Dimension and cut vertices: an application of Ramsey theory

Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every $d\geq 1$, if $P$ is a poset and the dimension of a subposet $B$ of $P$ is at most $d$ whenever the cover graph of $B$ is a block of the cover graph of $P$, then the dimension of $P$ is at most $d+2$. We also construct examples which show that this inequality is best possible. We consider the proof of the upper bound to be fairly elegant and relatively compact. However, we know of no simple proof for the lower bound, and our argument requires a powerful tool known as the Product Ramsey Theorem. As a consequence, our constructions involve posets of enormous size.Comment: Final published version with updated reference

On the dimension of posets with cover graphs of treewidth 22

In 1977, Trotter and Moore proved that a poset has dimension at most $3$ whenever its cover graph is a forest, or equivalently, has treewidth at most $1$. On the other hand, a well-known construction of Kelly shows that there are posets of arbitrarily large dimension whose cover graphs have treewidth $3$. In this paper we focus on the boundary case of treewidth $2$. It was recently shown that the dimension is bounded if the cover graph is outerplanar (Felsner, Trotter, and Wiechert) or if it has pathwidth $2$ (Bir\'o, Keller, and Young). This can be interpreted as evidence that the dimension should be bounded more generally when the cover graph has treewidth $2$. We show that it is indeed the case: Every such poset has dimension at most $1276$.Comment: v4: minor changes made following helpful comments by the referee

Dimension of posets with planar cover graphs excluding two long incomparable chains

It has been known for more than 40 years that there are posets with planar cover graphs and arbitrarily large dimension. Recently, Streib and Trotter proved that such posets must have large height. In fact, all known constructions of such posets have two large disjoint chains with all points in one chain incomparable with all points in the other. Gutowski and Krawczyk conjectured that this feature is necessary. More formally, they conjectured that for every $k\geq 1$, there is a constant $d$ such that if $P$ is a poset with a planar cover graph and $P$ excludes $\mathbf{k}+\mathbf{k}$, then $\dim(P)\leq d$. We settle their conjecture in the affirmative. We also discuss possibilities of generalizing the result by relaxing the condition that the cover graph is planar.Comment: New section on connections with graph minors, small correction

Tree-width and dimension

Over the last 30 years, researchers have investigated connections between dimension for posets and planarity for graphs. Here we extend this line of research to the structural graph theory parameter tree-width by proving that the dimension of a finite poset is bounded in terms of its height and the tree-width of its cover graph.Comment: Updates on solutions of problems and on bibliograph

Collaborative Honeypot Defense in UAV Networks: A Learning-Based Game Approach

The proliferation of unmanned aerial vehicles (UAVs) opens up new opportunities for on-demand service provisioning anywhere and anytime, but also exposes UAVs to a variety of cyber threats. Low/medium interaction honeypots offer a promising lightweight defense for actively protecting mobile Internet of things, particularly UAV networks. While previous research has primarily focused on honeypot system design and attack pattern recognition, the incentive issue for motivating UAV's participation (e.g., sharing trapped attack data in honeypots) to collaboratively resist distributed and sophisticated attacks remains unexplored. This paper proposes a novel game-theoretical collaborative defense approach to address optimal, fair, and feasible incentive design, in the presence of network dynamics and UAVs' multi-dimensional private information (e.g., valid defense data (VDD) volume, communication delay, and UAV cost). Specifically, we first develop a honeypot game between UAVs and the network operator under both partial and complete information asymmetry scenarios. The optimal VDD-reward contract design problem with partial information asymmetry is then solved using a contract-theoretic approach that ensures budget feasibility, truthfulness, fairness, and computational efficiency. In addition, under complete information asymmetry, we devise a distributed reinforcement learning algorithm to dynamically design optimal contracts for distinct types of UAVs in the time-varying UAV network. Extensive simulations demonstrate that the proposed scheme can motivate UAV's cooperation in VDD sharing and improve defensive effectiveness, compared with conventional schemes.Comment: Accepted Aug. 28, 2023 by IEEE Transactions on Information Forensics & Security. arXiv admin note: text overlap with arXiv:2209.1381