Graph Rubbling: an Extension of Graph Pebbling
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Graph Rubbling: An Extension of Graph Pebbling Christopher Andrew Belford Place a whole number of pebbles on the vertices of a simple, connected graph G; this is called a pebble distribution. A rubbling move consists of removing a total of two pebbles from some neighbor(s) of a vertex v of G and placing a single pebble on v. A vertex v of G is called reachable from an initial pebble distribution p if there is a sequence of rubbling moves which, starting from p, places a pebble on v. The rubbling number of a graph G, denoted ρ(G), is the least k such that for any distribution p of k pebbles, any given vertex of G is reachable. The optimal rubbling number of a graph G, denoted ρopt(G), is the least k such that there exists a distribution p of k pebbles for which any given vertex of G is reachable. Graph rubbling, ρ(G) and ρopt(G) are generalizations of graph pebbling, the pebbling number of a graph π(G), and the optimal pebbling number of a graph πopt(G). We modify the graph pebbling tools known as the transition digraph and the balance condition for use with graph rubbling. Original proofs are given for the No-Cycle Lemma and the Squishing Lemma, in the context of graph rubbling. Also, rolling moves are introduced as a way to modify a pebble distribution for use in computing ρopt(G). Further, ρ(G) and ρopt(G) are computed for many families of graphs, including Kn, Wn, Km1,m2,...,ml, Pn and Cn. Additionally, ρ(G) is computed for Q n and the Petersen graph.
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تاریخ انتشار 2012