(-k)-critical trees and k-minimal trees
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In a graph G=(V,E), a module is a vertex subset M of V such that every vertex outside M is adjacent to all or none of M. For example, ∅, {x} (x∈ V ) and V are modules of G, called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable. A vertex x of a prime graph G is critical if G - x is decomposable. Moreover, a prime graph with k non-critical vertices is called (-k)-critical graph. A prime graph G is k-minimal if there is some k-vertex set X of vertices such that there is no proper induced subgraph of G containing X is prime. From this perspective, I. Boudabbous proposes to find the (-k)-critical graphs and k-minimal graphs for some integer k even in a particular case of graphs. This research paper attempts to answer I. Boudabbous's question. First, it describes the (-k)-critical tree. As a corollary, we determine the number of nonisomorphic (-k)-critical tree with n vertices where k∈{1,2,⌊n/2⌋}. Second, it provide a complete characterization of the k-minimal tree. As a corollary, we determine the number of nonisomorphic k-minimal tree with n vertices where k≤ 3.
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