Abstract
In a 2011 paper, the game of peg solitaire is generalized to arbitrary boards, which are treated as graphs in the combinatorial sense. Of particular interest are graphs that are freely solvable, that is, graphs that can be solved from any starting position. In this paper we give several examples of freely solvable graphs including all such trees with ten vertices or less, numerous cycles with a subdivided chord, meshes, and generalizations of the wheel, helm, and web.
Highlights
Peg solitaire is a table game which traditionally begins with “pegs” in every space except for one which is left empty (i.e., a “hole”)
A graph G is solvable if there exists some vertex s so that, starting with a hole in s, there exists an associated terminal state consisting of a single peg
Solve the even path formed by the remaining pegs with a hole in 0
Summary
Peg solitaire is a table game which traditionally begins with “pegs” in every space except for one which is left empty (i.e., a “hole”). A graph G is solvable if there exists some vertex s so that, starting with a hole in s, there exists an associated terminal state consisting of a single peg. A graph G is distance 2-solvable if there exists some vertex s so that, starting with a hole in s, there exists an associated terminal state consisting of two pegs that are distance 2 apart. In [5], it is shown that, of the 996 nonisomorphic connected graphs on seven vertices or less, only 54 are not freely solvable. (ii) If G is a k-solvable spanning subgraph of H, H is (at worst) k-solvable Because of this observation, it is of particular interest to determine which trees are freely solvable. An exhaustive computer search [8] was used to determine solvability
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