Pegging Numbers For Various Tree Graphs

TL;DR

This paper investigates the pegging and optimal-pegging numbers in various tree graphs, providing tight bounds and exact values for specific classes, revealing how graph modifications affect these parameters.

Contribution

It introduces new bounds and exact calculations for pegging numbers in complete binary, infinitary, caterpillar, and lobster trees, highlighting the impact of graph structure changes.

Findings

Tight bounds for complete binary trees' pegging numbers

Exact optimal-pegging numbers for complete infinitary trees

Removal of a leaf can increase a tree's optimal-pegging number

Abstract

In the game of pegging, each vertex of a graph is considered a hole into which a peg can be placed. A pegging move is performed by jumping one peg over another peg, and then removing the peg that has been jumped over from the graph. We define the pegging number as the smallest number of pegs needed to reach all the vertices in a graph no matter what the distribution. Similarly, the optimal-pegging number of a graph is defined as the smallest distribution of pegs for which all the vertices in the graph can be reached. We obtain tight bounds on the pegging numbers and optimal-pegging numbers of complete binary trees and compute the optimal-pegging numbers of complete infinitary trees. As a result of these computations, we deduce that there is a tree whose optimal-pegging number is strictly increased by removing a leaf. We also compute the optimal-pegging number of caterpillar graphs and the tightest upper bound on the optimal-pegging numbers of lobster graphs.