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Journal Articles Discrete Mathematics, Algorithms and Applications Year : 2022

## Birecognition of prime graphs, and minimal prime graphs

Houmem Belkhechine
• Function : Author
Cherifa Ben Salha
• Function : Author
Pierre Ille

#### Abstract

Given a graph [Formula: see text], a subset [Formula: see text] of [Formula: see text] is a module of [Formula: see text] if for each [Formula: see text], [Formula: see text] is adjacent to all the elements of [Formula: see text] or to none of them. For instance, [Formula: see text], [Formula: see text] and [Formula: see text] ([Formula: see text]) are the trivial modules of [Formula: see text]. A graph [Formula: see text] is prime if [Formula: see text] and all its modules are trivial. Given a prime graph [Formula: see text], consider [Formula: see text] such that [Formula: see text] is prime. Given a graph [Formula: see text] such that [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] are [Formula: see text]-similar if for each [Formula: see text], [Formula: see text] and [Formula: see text] are both prime or not. The graph [Formula: see text] is said to be [Formula: see text]-birecognizable if every graph, [Formula: see text]-similar to [Formula: see text], is prime. We study the graphs [Formula: see text] that are not [Formula: see text]-birecognizable, where [Formula: see text] such that [Formula: see text] is prime, by using the following notion of a minimal prime graph. Given a prime graph [Formula: see text], consider [Formula: see text] such that [Formula: see text] is prime. Given [Formula: see text], [Formula: see text] is [Formula: see text]-minimal if for each [Formula: see text] such that [Formula: see text], [Formula: see text] is not prime.

#### Domains

Mathematics [math] Combinatorics [math.CO]

### Dates and versions

hal-03849042 , version 1 (11-11-2022)

### Identifiers

• HAL Id : hal-03849042 , version 1
• DOI :

### Cite

Houmem Belkhechine, Cherifa Ben Salha, Pierre Ille. Birecognition of prime graphs, and minimal prime graphs. Discrete Mathematics, Algorithms and Applications, 2022, ⟨10.1142/S1793830922500380⟩. ⟨hal-03849042⟩

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