We initiate the study of the parameterized complexity of the Collective Graph Exploration (CGE) problem. In CGE, the input consists of an undirected connected graph G and a collection of k robots, initially placed at the same vertex r of G, and each one of them has an energy budget of B. The objective is to decide whether G can be explored by the k robots in B time steps, i.e., there exist k closed walks in G, one corresponding to each robot, such that every edge is covered by at least one walk, every walk starts and ends at the vertex r, and the maximum length of any walk is at most B. Unfortunately, this problem is NP-hard even on trees [Fraigniaud et al., 2006]. Further, we prove that the problem remains W[1]-hard parameterized by k even for trees of treedepth 3. Due to the para-NP-hardness of the problem parameterized by treedepth, and motivated by real-world scenarios, we study the parameterized complexity of the problem parameterized by the vertex cover number (vc) of the graph, and prove that the problem is fixed-parameter tractable (FPT) parameterized by vc. Additionally, we study the optimization version of CGE, where we want to optimize B, and design an approximation algorithm with an additive approximation factor of O(vc).

Over the past decade, we witness an increasing amount of interest in the design of exact exponential-time and parameterized algorithms for problems in Graph Drawing. Unfortunately, we still lack knowledge of general methods to develop such algorithms. An even more serious issue is that, here, "standard" parameters very often yield intractability. In particular, for the most common structural parameter, namely, treewidth, we frequently observe NP-hardness already when the input graphs are restricted to have constant (often, being just $1$ or $2$) treewidth. Our work deals with both drawbacks simultaneously. We introduce a novel form of tree decomposition that, roughly speaking, does not decompose (only) a graph, but an entire drawing. As such, its bags and separators are of geometric (rather than only combinatorial) nature. While the corresponding parameter -- like treewidth -- can be arbitrarily smaller than the height (and width) of the drawing, we show that -- unlike treewidth -- it gives rise to efficient algorithms. Specifically, we get slice-wise polynomial (XP) time algorithms parameterized by our parameter. We present a general scheme for the design of such algorithms, and apply it to several central problems in Graph Drawing, including the recognition of grid graphs, minimization of crossings and bends, and compaction. Other than for the class of problems we discussed in the paper, we believe that our decomposition and scheme are of independent interest and can be further extended or generalized to suit even a wider class of problems. Additionally, we discuss classes of drawings where our parameter is bounded by $O(\sqrt{n})$ (where $n$ is the number of vertices of the graph), yielding subexponential-time algorithms. Lastly, we prove which relations exist between drawn treewidth and other width measures, including treewidth, pathwidth, (dual) carving-width and embedded-width.

Grid graphs, and, more generally, k×r grid graphs, form one of the most basic classes of geometric graphs. Over the past few decades, a large body of works studied the (in)tractability of various computational problems on grid graphs, which often yield substantially faster algorithms than general graphs. Unfortunately, the recognition of a grid graph (given a graph G, decide whether it can be embedded into a grid graph) is particularly hard - it was shown to be NP-hard even on trees of pathwidth 3 already in 1987. Yet, in this paper, we provide several positive results in this regard in the framework of parameterized complexity (additionally, we present new and complementary hardness results). Specifically, our contribution is threefold. First, we show that the problem is fixed-parameter tractable (FPT) parameterized by k+mcc where mcc is the maximum size of a connected component of G. This also implies that the problem is FPT parameterized by td+k where td is the treedepth of G, as td ≤ mcc (to be compared with the hardness for pathwidth 2 where k = 3). (We note that when k and r are unrestricted, the problem is trivially FPT parameterized by td.) Further, we derive as a corollary that strip packing is FPT with respect to the height of the strip plus the maximum of the dimensions of the packed rectangles, which was previously only known to be in XP. Second, we present a new parameterization, denoted a_G, relating graph distance to geometric distance, which may be of independent interest. We show that the problem is para-NP-hard parameterized by a_G, but FPT parameterized by a_G on trees, as well as FPT parameterized by k+a_G. Third, we show that the recognition of k× r grid graphs is NP-hard on graphs of pathwidth 2 where k = 3. Moreover, when k and r are unrestricted, we show that the problem is NP-hard on trees of pathwidth 2, but trivially solvable in polynomial time on graphs of pathwidth 1.

We study a motion-planning problem inspired by the game Snake that models scenarios like the transportation of linked wagons towed by a locomotor to the movement of a group of agents that travel in an ``ant-like'' fashion. Given a ``snake-like'' robot with initial and final positions in an environment modeled by a graph, our goal is to decide whether the robot can reach the final position from the initial position without intersecting itself. Already on grid graphs, this problem is PSPACE-complete [Biasi and Ophelders, 2018]. Nevertheless, we prove that even on general graphs, it is solvable in time k^{O(k)}|I|^{O(1)} where k is the size of the robot, and |I| is the input size. Towards this, we give a novel application of color-coding to sparsify the configuration graph of the problem. We also show that the problem is unlikely to have a polynomial kernel even on grid graphs, but it admits a treewidth-reduction procedure. To the best of our knowledge, the study of the parameterized complexity of motion problems has been~largely~neglected, thus our work is pioneering in this regard.