Primality, criticality and minimality problems in trees

Abstract

In a graph [Formula: see text], a module is a vertex subset [Formula: see text] of [Formula: see text] such that every vertex outside [Formula: see text] is adjacent to all or none of [Formula: see text]. For example, [Formula: see text], [Formula: see text][Formula: see text] and [Formula: see text] are modules of [Formula: see text], called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable. A vertex [Formula: see text] of a prime graph [Formula: see text] is critical if [Formula: see text] is decomposable. Moreover, a prime graph with [Formula: see text] noncritical vertices is called [Formula: see text]-critical graph. A prime graph [Formula: see text] is [Formula: see text]-minimal if there is some [Formula: see text]-vertex set [Formula: see text] of vertices such that there is no proper induced subgraph of [Formula: see text] containing [Formula: see text] is prime. From this perspective, Boudabbous proposes to find the [Formula: see text]-critical graphs and [Formula: see text]-minimal graphs for some integer [Formula: see text] even in a particular case of graphs. This research paper attempts to answer Boudabbous’s question. First, we describe the [Formula: see text]-critical tree. As a corollary, we determine the number of nonisomorphic [Formula: see text]-critical tree with [Formula: see text] vertices where [Formula: see text]. Second, we provide a complete characterization of the [Formula: see text]-minimal tree. As a corollary, we determine the number of nonisomorphic [Formula: see text]-minimal tree with [Formula: see text] vertices where [Formula: see text].