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Journal Articles Discrete Mathematics, Algorithms and Applications Year : 2021

## Birecognition of prime graphs, and minimal prime graphs

Houmem Belkhechine
• Function : Author
• PersonId : 944630
Cherifa Ben Salha
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• PersonId : 1182734
Pierre Ille

#### Abstract

Given a graph G, a subset M of V (G) is a module of G if for each v ∈ V (G) ∖ M , v is adjacent to all the elements of M or to none of them. For instance, V (G), ∅ and {v} (v ∈ V (G)) are the trivial modules of G. A graph G is prime if V (G) ≥ 4 and all its modules are trivial. Given a prime graph G, consider X ⊊ V (G) such that G[X] is prime. Given a graph H such that V (G) = V (H) and G[X] = H[X], G and H are G[X]-similar if for each W ⊊ V (G) ∖ X, G[X ∪ W ] and H[X ∪ W ] are both prime or not. The graph G is said to be G[X]-birecognizable if every graph, G[X]-similar to G, is prime. We study the graphs G that are not G[X]-birecognizable, where X ⊊ V (G) such that G[X] is prime, by using the following notion of a minimal prime graph. Given a prime graph G, consider X ⊊ V (G) such that G[X] is prime. Given v, w ∈ V (G) ∖ X, G is G[X ∪ {v, w}]-minimal if for each W ⊊ V (G) such that X ∪ {v, w} ⊆ W , G[W ] is not prime.

#### Domains

Mathematics [math] Combinatorics [math.CO]

### Dates and versions

hal-03848887 , version 1 (11-11-2022)

### Identifiers

• HAL Id : hal-03848887 , version 1
• DOI :

### Cite

Houmem Belkhechine, Cherifa Ben Salha, Pierre Ille. Birecognition of prime graphs, and minimal prime graphs. Discrete Mathematics, Algorithms and Applications, 2021, ⟨10.1142/S1793830922500380⟩. ⟨hal-03848887⟩

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