Breadth-First Search Trees with Many or Few Leaves
This addresses computational complexity issues in graph theory for researchers and practitioners, but it is incremental as it extends known problems to specific search tree contexts.
The paper tackles the problem of finding spanning trees with the maximum or minimum number of leaves, specifically for Breadth-First Search (BFS) and related graph search algorithms, showing that these problems are fixed-parameter tractable when parameterized by leaf count but W[1]-hard when parameterized by internal vertex count.
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.