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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)

In this paper we obtain some upper bounds for b-chromatic number of K 1,t-free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are in terms of either clique number or chromatic number of graphs or biclique number for bipartite graphs. We show that all the bounds are tight.

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)