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In a given graph G, a set of vertices S with an assignment of colors is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a z(G)-coloring of the vertices of G. The concept of a defining set has been studied, to some extent, for block designs and also under another name, a critical set, for latin(More)
Let G be a graph and τ : V (G) → N be an assignment of thresholds to the vertices of G. A subset of vertices D is said to be dynamic monopoly (or simply dynamo) if the vertices of G can be partitioned into subsets D 0 , D 1 ,. .. , D k such that D 0 = D and for any i = 1,. .. , k − 1 each vertex v in D i+1 has at least t(v) neighbors in D 0 ∪.. . ∪D i.(More)
Let G be a graph and τ : V (G) → N ∪ {0} be an assignment of thresholds to the vertices of G. A subset of vertices D is said to be a dynamic monopoly corresponding to (G, τ) if the vertices of G can be partitioned into subsets v in D i+1 has at least τ (v) neighbors in D 0 ∪. .. ∪ D i. Dynamic monopolies are in fact modeling the irreversible spread of(More)
A defining set (of vertex coloring) of a graph G is a set of vertices S with an assignment of colors to its elements which has a unique completion to a proper coloring of G. We define a minimal defining set to be a defining set which does not properly contain another defining set. If G is a uniquely vertex colorable graph, clearly its minimum defining sets(More)