In the longest plane spanning tree problem, we are given a finite planar point set \(\mathcal{P}\) , and our task is to find a plane (i.e., noncrossing) spanning tree for \(\mathcal{P}\) with maximum total Euclidean edge length. Despite more than two decades of research, it remains open whether this problem is NP-hard. Thus, previous results have focused on polynomial-time algorithms that produce plane trees whose total edge length approximates \(\mathrm{OPT}\) , the maximum possible length. The approximate trees in these algorithms all have small unweighted diameter, typically two to four. It is natural to ask whether this is a common feature of longest plane spanning trees, or an artifact of the specific approximation algorithms. We provide three results to elucidate the interplay between the approximation guarantee and the unweighted diameter of the approximate trees. First, we describe a polynomial-time algorithm to construct a plane tree with diameter at most four and total edge length at least \(0.546\cdot\mathrm{OPT}\) . This constitutes a substantial improvement over the state of the art. Second, we show that a longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any candidate diameter \(d\geq 3\) , we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most \( d \) (compared to a longest plane tree without constraints).

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.

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 $$\Psi $$

Let $S \subset \mathbb{R}^2$ be a set of $n$ sites, where each $s \in S$ has an associated radius $r_s > 0$. The disk graph $D(S)$ is the undirected graph with vertex set $S$ and an undirected edge between two sites $s, t \in S$ if and only if $|st| \leq r_s + r_t$, i.e., if the disks with centers $s$ and $t$ and respective radii $r_s$ and $r_t$ intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph $T(S)$ is the directed graph with vertex set $S$ and a directed edge from a site $s$ to a site $t$ if and only if $|st| \leq r_s$, i.e., if $t$ lies in the disk with center $s$ and radius $r_s$. We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in $D(S)$ and in $T(S)$. These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in $D(S)$ and in $T(S)$ can be found in $O(n \log n)$ expected time, and that the (weighted) girth of $D(S)$ can be found in $O(n \log n)$ expected time. For this, we develop new tools for batched range searching that may be of independent interest.