Kernels and partial line digraphs

Balbuena, C.; Guevara, Mucuy-kak

doi:10.1016/j.aml.2010.06.001

Cited by 3 publications

(4 citation statements)

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“…Some known results about the existence of kernels and (k, l)-kernels in line digraphs can be seen in [16,21]. The following theorem is proved in [2].…”

Section: (K L)-kernelsmentioning

confidence: 99%

“…Theorem 2.1 [2] Let k, l be two natural numbers such that 1 ≤ l < k, and let D be a digraph with minimum in-degree at least 1 and girth at least l + 1. Then D has a (k, l)-kernel if and only if any partial line digraph LD has a (k, l)-kernel.…”

Section: (K L)-kernelsmentioning

confidence: 99%

“…In the proof of Theorem 2.1 of [2] it was proved that f is well defined. And from Lemma 2.1 it follows that f is injective.…”

Section: (K L)-kernelsmentioning

confidence: 99%

“…Let h : K * → K be defined by h( K) = H( K) for all K ∈ K * . In the proof of Theorem 2.1 of [2] it was proved that h is well defined if l < k and the girth of D is at least l + 1. Moreover, we can check that h = f −1 because h(f (K)) = H(ω − (K) ∩ A ′ ) = K. Therefore the theorem holds.…”

Section: (K L)-kernelsmentioning

confidence: 99%

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About (k,l)-kernels, semikernels and Grundy functions in partial line digraphs

Balbuena¹,

Galeana‐Sánchez²,

Guevara³

2019

Discuss. Math. Graph Theory

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Let D = (V, A) be a digraph and consider an arc subset A ′ ⊆ A and an exhaustive mapping φ : A → A ′ such that (i) the set of heads of A ′ is H(A ′ ) = V ;(ii) the map fixes the elements of A ′ , that is, φ|A ′ = Id, and for every vertex j ∈ V ,Then, the partial line digraph of D, denoted by L (A ′ ,φ) D (for short LD if the pair (A ′ , φ) is clear from the context), is the digraph with vertex set V (LD) = A ′ and set of arcs A(LD) = {(ij, φ(j, k)) : (j, k) ∈ A}. In this paper we prove the following results: Let k, l be two natural numbers such that 1 ≤ l ≤ k, and D a digraph with minimum in-degree at least 1. Then the number of (k, l)-kernels of D is less than or equal to the number of (k, l)-kernels of LD. Moreover, if l < k and the girth of D is at least l + 1, then these two numbers are equal.The number of semikernels of D is equal to the number of semikernels of LD. Also we introduce the concept of (k, l)-Grundy function as a generalization of the concept of Grundy function and we prove that the number of (k, l)-Grundy functions of D is equal to the number of (k, l)-Grundy functions of any partial line digraph LD.

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“…Some known results about the existence of kernels and (k, l)-kernels in line digraphs can be seen in [16,21]. The following theorem is proved in [2].…”

Section: (K L)-kernelsmentioning

confidence: 99%

Section: (K L)-kernelsmentioning

confidence: 99%

“…In the proof of Theorem 2.1 of [2] it was proved that f is well defined. And from Lemma 2.1 it follows that f is injective.…”

Section: (K L)-kernelsmentioning

confidence: 99%

Section: (K L)-kernelsmentioning

confidence: 99%

See 2 more Smart Citations