On Minimal Prime Graphs and Posets

Pouzet, Maurice; Zaguia, Imed

doi:10.1007/s11083-009-9131-y

Cited by 13 publications

(15 citation statements)

References 12 publications

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“…Properties of the prime substructures of a given prime structure were determined by Schmerl and Trotter Schmerl and Trotter (1993) in their fundamental paper. Indeed, several papers within the same sphere of reference have then appeared ; ; Bouchaala et al (2013); Boudabbous (2016); Boussaïri and Chaïchaâ (2007); Ehrenfeucht et al (1997); Ille (1997); Elayech et al (2015); Pouzet and Zaguia (2009); Sayar (2011)). For instance, the path defined on N n = {1, ..., n}, denoted by P n , is prime for n ≥ 4.…”

Section: Introduction and Presentation Of The Resultsmentioning

confidence: 99%

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

Marweni

2021

Preprint

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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.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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Section: Introduction and Presentation Of The Resultsmentioning

confidence: 99%

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

Marweni

2021

Preprint

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“…We construct also 2 ℵ 0 which are prime (Theorem 25). These examples are minimal in the sense of [23], but not in the sense of [26].…”

Section: Introduction and Presentation Of The Resultsmentioning

confidence: 99%

“…In [26], the authors provided among other things, the list of minimal prime graphs with no infinite cliques or no infinite independent sets. Definition 6.…”

Section: Definition 5 [26]mentioning

confidence: 99%

Graphs containing finite induced paths of unbounded length

Pouzet,

Zaguia

2020

Preprint

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The age A(G) of a graph G (undirected and without loops) is the collection of finite induced subgraphs of G, considered up to isomorphy and ordered by embeddability. It is well-quasi-ordered (wqo) for this order if it contains no infinite antichain. A graph is path-minimal if it contains finite induced paths of unbounded length and every induced subgraph G ′ with this property embeds G. We construct 2 ℵ 0 path-minimal graphs whose ages are pairwise incomparable with set inclusion and which are wqo. Our construction is based on uniformly recurrent sequences and lexicographical sums of labelled graphs

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“…It follows that the main difficulty in many problems lies in the study of prime graphs. A book and several papers on the prime graphs and their prime subgraphs have then appeared (1,2,3,4,5,6,7,10,12,14,15,17,18). On the other hand, in many studies on the classes of graphs defined by some forbidden, the prime graphs play a major role (4,5,9)…”

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confidence: 99%

Some Structural Resuits on Prime Graphs

Alrowily

2019

JAM

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Given a graph G = (V,E), a subset M of V is a module [17] (or an interval [10] or an autonomous [11] or a clan [8] or a homogeneous set [7] ) of G provided that x ∼ M for each vertex x outside M. So V,φ and {x}, where x ∈ V , are modules of G, called trivial modules. The graph G is indecomposable [16] if all the modules of G are trivial. Otherwise we say that G is decomposable . The prime graph G is an indecomposable graph with at least four vertices. Let G and H be two graphs. Let If G has no induced subgraph isomorphic to H, then we say that G is H-free. In this paper, we will prove the next theorem

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On Minimal Prime Graphs and Posets

Cited by 13 publications

References 12 publications

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

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

Graphs containing finite induced paths of unbounded length

Some Structural Resuits on Prime Graphs

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