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Breadth-First Search Trees with Many or Few Leaves

Jesse Beisegel, Ekkehard Köhler, Robert Scheffler, Martin Strehler

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.

Breadth-First Search Trees with Many or Few Leaves

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.

Paper Structure

This paper contains 4 sections, 9 theorems, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our results.
  3. Preliminaries
  4. Number of Leaves as Parameter

Key Result

Lemma 1

Let $G$ be a graph with a two pair $\{u, w\}$, i.e., a pair of vertices such that every induced $u$-$w$-path in $G$ has length 2. Then $G$ is weakly chordal if and only if the graph that is constructed by adding the edge $uw$ to $G$ is weakly chordal. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: In the left graph the minimum number of leaves in a spanning tree is 1, whereas any BFS-tree has at least $n/2$ leaves. The right graph is a star of $k$ ladders with $2k$ vertices each. This graph has a spanning tree with $k^2$ leaves while every BFS-tree has at most $3k$ leaves.

Theorems & Definitions (10)

  • Lemma 1: Spinrad and Sritharan spinrad1995algorithms
  • Lemma 2: Beisegel et al. beisegel2021recognition
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Corollary 8
  • Corollary 9
  • Lemma 10
  • Definition 11