(-k)-critical trees and k-minimal trees.

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