Breadth-First Search Trees with Many or Few Leaves

TL;DR

This paper investigates the complexity of finding BFS trees with many or few leaves, analyzing classical and parameterized complexity for various search algorithms, and shows fixed-parameter tractability results.

Contribution

It provides a detailed complexity analysis of leaf optimization problems for BFS and related search trees, including fixed-parameter tractability results.

Findings

Maximum and minimum leaf problems are FPT when parameterized by the number of leaves.

These problems are W[1]-hard when parameterized by the number of internal vertices.

The study covers classical and parameterized complexity for GS, BFS, and LBFS search trees.

Abstract

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.