Graph pegging numbers

Helleloid, Geir; Khalid, Madeeha; Moulton, David Petrie; Wood, Philip Matchett

Graph pegging numbers

Authors

Helleloid, Geir

Khalid, Madeeha

Moulton, David Petrie

Wood, Philip Matchett

Description

In graph pegging, we view each vertex of a graph as a hole into which a peg can be placed, with checker-like ``pegging moves'' allowed. Motivated by well-studied questions in graph pebbling, we introduce two pegging quantities. The pegging number (respectively, the optimal pegging number) of a graph is the minimum number of pegs such that for every (respectively, some) distribution of that many pegs on the graph, any vertex can be reached by a sequence of pegging moves. We prove several basic properties of pegging and analyze the pegging number and optimal pegging number of several classes of graphs, including paths, cycles, products with complete graphs, hypercubes, and graphs of small diameter.
26 pages, 3 figures