Pegging Numbers For Various Tree Graphs

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.

💡 Research Summary

The paper studies a graph‑theoretic version of the classic peg solitaire game, introducing two quantitative parameters for any graph G: the pegging number P(G), the smallest integer k such that every distribution of k pegs can eventually occupy every vertex, and the optimal‑pegging number p(G), the size of the smallest distribution that can achieve this goal. After reviewing basic definitions (distribution, move, reachability) the authors develop a weight‑based analytic tool: for a fixed target vertex t, each peg at distance d contributes a weight ω^d, where ω = (√5 − 1)/2 (the conjugate of the golden ratio). Lemmas establish that weights never increase under legal moves and that reachability is monotone with respect to inclusion of distributions. The crucial consequence (Theorem 2.5) is that a vertex can be reached only if the total weight of the initial distribution with respect to that vertex is at least 1. This “weight‑≥ 1” condition provides a systematic way to obtain lower bounds on P(G) and p(G) by constructing distributions whose weight falls below 1 for some target set L.

The authors first prove two general facts for trees: (1) any reachable vertex can be reached using only moves that strictly decrease the distance to that vertex, and (2) the pegging number of a subtree never exceeds that of the whole tree. These observations are later used repeatedly.

The core of the paper focuses on several families of trees.

Complete binary trees T_h.
Using the weight formula, the total weight of the whole tree with respect to the root is computed as ((2ω)^{h+1}−1)/(2ω−1). By carefully selecting a set of subtrees at various depths (specifically those at distance k+2 from a leaf t, for 8 ≤ k ≤ h−1) and removing the vertices of a small subtree of height 7, the authors construct a distribution D that leaves the leaf t unreachable because w_t(D) < 1. The size of D equals |V(T_h)| − |V(T_7)| + 83, which yields the lower bound
 P(T_h) ≥ |V(T_h)| − 172 for sufficiently large h.
Since |V(T_h)| = 2^{h+1} − 1, the bound shows that asymptotically almost every vertex must initially contain a peg; indeed, Corollary 4.3 proves that P(T_h)/|V(T_h)| → 1 as h → ∞.

Complete infinitary trees.
A complete infinite‑ary tree (each node has infinitely many children) is examined for its optimal‑pegging number. The authors show that the minimal number of pegs needed to reach all vertices of height h is exactly the (h+2)‑th Fibonacci number minus 1, i.e. p(T_∞,h) = F_{h+2} − 1. The proof follows from the observation that to cover a level you must have enough pegs to cover the two preceding levels, mirroring the Fibonacci recurrence. An immediate corollary is the existence of a tree whose optimal‑pegging number strictly increases when a leaf is removed—a counter‑intuitive phenomenon not seen in many other graph parameters.

Caterpillar and lobster trees.
For caterpillars (a central path with leaves attached directly to it) the optimal‑pegging number is determined exactly: p(C_n) = ⌈n/2⌉, where n is the number of vertices on the central spine. For lobsters (caterpillars with additional “leg” subpaths) the authors do not obtain an exact formula but prove an upper bound p(L_n) ≤ ⌈2n/3⌉, showing that the optimal‑pegging number grows linearly with the size of the spine but remains below the total vertex count.

The paper concludes with a discussion of open problems, such as extending the weight method to other graph families (Cartesian products, random trees) and investigating whether tighter bounds can be achieved for lobster‑type graphs.

Overall, the work introduces a powerful weight‑based framework for analyzing peg‑solitaire‑type games on trees, delivers tight asymptotic and exact results for several important tree families, and uncovers surprising structural effects (e.g., leaf removal increasing p(G)). These contributions deepen the connection between combinatorial game theory and graph invariants, and they lay groundwork for future explorations of pegging phenomena in broader graph classes.