Abstract:

If some pebbles are distributed on the vertices of a graph, a pebbling step takes two pebbles from one vertex and replaces one at an adjacent vertex. A distribution D of pebbles is solvable if, starting from D, a pebble can be moved to any specified vertex by a sequence of pebbling steps. The pebbling number p(G) of a connected graph is the smallest number of pebbles such that every distribution with p(G) pebbles is solvable. If we restrict our attention to distributions which are minimally solvable or maximally unsolvable, we obtain three additional pebbling parameters on G, the r-, g-, and u-critical pebbling numbers of a connected graph. We investigate properties of the r-critical pebbling number and its relationship to the pebbling number.

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Abstract:

The fixing number of a graph G is the smallest cardinality of a set of vertices S such that only the trivial automorphism of G fixes every vertex in S. The fixing set of a group Γ is the set of all fixing numbers of finite graphs with automorphism group Γ. Several authors have studied the distinguishing number of a graph, the smallest number of labels needed to label G so that the automorphism group of the labeled graph is trivial. The fixing number can be thought of as a variation of the distinguishing number in which every label may be used only once, and not every vertex need be labeled. We characterize the fixing sets of finite abelian groups, and investigate the fixing sets of symmetric groups.

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