Making graphs solvable in peg solitaire

Jan-Hendrik de Wiljes (Institute of Mathematics, Freie Universität Berlin, Germany)
Martin Kreh (Institute of Mathematics and Applied Computer Science, University of Hildesheim, Germany)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 10, No 2 (2022): Electronic Journal of Graph Theory and Applications

In 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. Since then peg solitaire has been considered on quite a few classes of graphs. Beeler and Gray introduced the natural idea of adding edges to make an unsolvable graph solvable. Recently, the graph invariant ms(G), which is the minimal number of additional edges needed to make G solvable, has been introduced and investigated on banana trees by the authors. In this article, we determine ms(G) for several families of unsolvable graphs. Furthermore, we provide some general results for this number of Hamiltonian graphs and graphs obtained via binary graph operations.