We consider the problem of finding a Hamiltonian path or cycle with precedence constraints in the form of a partial order on the vertex set. We study the complexity for graph width parameters for which the ordinary problems Hamiltonian Path and Hamiltonian Cycle are in FPT. In particular, we focus on parameters that describe how many vertices and edges have to be deleted to become a member of a certain graph class. We show that the problems are W[1]-hard for such restricted cases as vertex distance to path and vertex distance to clique. We complement these results by showing that the problems can be solved in XP time for vertex distance to outerplanar and vertex distance to block. Furthermore, we present some FPT algorithms, e.g., for edge distance to block. Additionally, we prove para-NP-hardness when considered with the edge clique cover number.

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing several applications: we prove tight upper bounds on the edge density of various beyond-planar graph classes, including so-called $k$-planar graphs with $k=1,2$, fan-crossing / fan-planar graphs, $k$-bend RAC-graphs with $k=0,1,2$, quasiplanar graphs, and $k^+$-real face graphs. In some cases ($1$-bend and $2$-bend RAC-graphs and fan-crossing / fan-planar graphs), we thereby obtain the first tight upper bounds on the edge density of the respective graph classes. In other cases, we give new streamlined and significantly shorter proofs for bounds that were already known in the literature. Thanks to the Density Formula, all of our proofs are mostly elementary counting and mostly circumvent the typical intricate case analysis found in earlier proofs. Further, in some cases (simple and non-homotopic quasiplanar graphs), our alternative proofs using the Density Formula lead to the first tight lower bound examples.

Abstract Let $$S \subseteq \mathbb {R}^2$$ S ⊆ R 2 be a set of nsites in the plane, so that every site $$s \in S$$ s ∈ S has an associated radius $$r_s > 0$$ r s > 0 . Let $$\mathcal {D}(S)$$ D ( S ) be the disk intersection graph defined by S, i.e., the graph with vertex set S and an edge between two distinct sites $$s, t \in S$$ s , t ∈ S if and only if the disks with centers s, t and radii $$r_s$$ r s , $$r_t$$ r t intersect. Our goal is to design data structures that maintain the connectivity structure of $$\mathcal {D}(S)$$ D ( S ) as sites are inserted and/or deleted in S. First, we consider unit disk graphs, i.e., we fix $$r_s = 1$$ r s = 1 , for all sites $$s \in S$$ s ∈ S . For this case, we describe a data structure that has $$O(\log ^2 n)$$ O ( log 2 n ) amortized update time and $$O(\log n/\log \log n)$$ O ( log n / log log n ) query time. Second, we look at disk graphs with bounded radius ratio $$\Psi $$ Ψ , i.e., for all $$s \in S$$ s ∈ S , we have $$1 \le r_s \le \Psi $$ 1 ≤ r s ≤ Ψ , for a parameter $$\Psi $$ Ψ that is known in advance. Here, we not only investigate the fully dynamic case, but also the incremental and the decremental scenario, where only insertions or only deletions of sites are allowed. In the fully dynamic case, we achieve amortized expected update time $$O(\Psi \log ^{4} n)$$ O ( Ψ log 4 n ) and query time $$O(\log n/\log \log n)$$ O ( log n / log log n ) . This improves the currently best update time by a factor of $$\Psi $$ Ψ . In the incremental case, we achieve logarithmic dependency on $$\Psi $$