Graph Search Trees and Their Leaves

TL;DR

This paper investigates the complexity of determining whether a vertex can be a leaf in graph search trees, revealing easy cases for certain searches and hardness results for others, with additional results on chordal graphs.

Contribution

It combines the problems of vertex leaf placement and search tree characterization, providing new complexity results and structural insights for chordal graphs.

Findings

Leaf placement is easy for some searches if the leaf is first in the order.

The problem is hard for DFS and BFS when the leaf cannot be first.

Structural results are provided for search tree leaves in chordal graphs.

Abstract

Graph searches and their respective search trees are widely used in algorithmic graph theory. The problem whether a given spanning tree can be a graph search tree has been considered for different searches, graph classes and search tree paradigms. Similarly, the question whether a particular vertex can be visited last by some search has been studied extensively in recent years. We combine these two problems by considering the question whether a vertex can be a leaf of a graph search tree. We show that for particular search trees, including DFS trees, this problem is easy if we allow the leaf to be the first vertex of the search ordering. We contrast this result by showing that the problem becomes hard for many searches, including DFS and BFS, if we forbid the leaf to be the first vertex. Additionally, we present several structural and algorithmic results for search tree leaves of chordal graphs.