Robert Scheffler

Jesse Beisegel, Ekkehard Köhler, Robert Scheffler et al.

The Maximum (Minimum) Leaf Spanning Tree problem asks for a spanning tree with the largest (smallest) number of leaves. As spanning trees are often computed using graph search algorithms, it is natural to restrict this problem to the set of search trees of some particular graph search, e.g., find the Breadth-First Search (BFS) tree with the largest number of leaves. We study this problem for Generic Search (GS), BFS and Lexicographic Breadth-First Search (LBFS) using search trees that connect each vertex to its first neighbor in the search order (first-in trees) just like the classic BFS tree. In particular, we analyze the complexity of these problems, both in the classical and in the parameterized sense. Among other results, we show that the minimum and maximum leaf problems are in FPT for the first-in trees of GS, BFS and LBFS when parameterized by the number of leaves in the tree. However, when these problems are parameterized by the number of internal vertices of the tree, they are W[1]-hard for the first-in trees of GS, BFS and LBFS.

Robert Scheffler

We consider two different problem families that deal with domination in graphs. On the one hand, we focus on dominating sequences. In such a sequence, every vertex dominates some vertex of the graph that was not dominated by any earlier vertex in the sequence. The problem of finding the longest dominating sequence is known as $\mathsf{Grundy~Domination}$. Depending on whether the closed or the open neighborhoods are used for domination, there are three other versions of this problem: $\mathsf{Grundy~Total~Domination}$, $\mathsf{L\text{-}Grundy~Domination}$, and $\mathsf{Z\text{-}Grundy~Domination}$. We show that all four problem variants are $\mathsf{W[1]}$-complete when parameterized by the solution size. On the other hand, we consider the family of zero forcing problems which form the parametric duals of the Grundy domination problems. In these problems, one looks for the smallest set of vertices initially colored blue such that certain color change rules are able to color all other vertices blue. Bhyravarapu et al. [IWOCA 2025] showed that the dual of $\mathsf{Z\text{-}Grundy~Domination}$, known as $\mathsf{Zero~Forcing~Set}$, is in $\mathsf{FPT}$ when parameterized by the treewidth or the solution size. We extend their treewidth result to the other three variants of zero forcing and their respective Grundy domination problems. Our algorithm also implies an $\mathsf{FPT}$ algorithm for $\mathsf{Grundy~Domination}$ when parameterized by the number of vertices that are not in the dominating sequence. In contrast, we show that $\mathsf{L\text{-}Grundy~Domination}$ is $\mathsf{W[1]}$-hard for that parameter.