Semikernels, Quasi Kernels, and Grundy Functions in the Line Digraph

Galeana‐Sánchez, Hortensia; Ramírez, Laura Pastrana; Rincón-Mejía, Hugo Alberto

doi:10.1137/0404008

Cited by 7 publications

(7 citation statements)

References 6 publications

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“…Jacob and Meyniel [17] furthermore showed in 1996 that every digraph has either a kernel or three quasi-kernels. For more results on quasi-kernels, see [5,13,16].…”

Section: Introductionmentioning

confidence: 99%

Kernels by properly colored paths in arc-colored digraphs

Bai

Fujita

Zhang

2018

Discrete Mathematics

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A kernel by properly colored paths of an arc-colored digraph D is a set S of vertices of D such that (i) no two vertices of S are connected by a properly colored directed path in D, and (ii) every vertex outside S can reach S by a properly colored directed path in D. In this paper, we conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for unicyclic digraphs, semi-complete digraphs and bipartite tournaments, respectively. Moreover, weaker conditions for the latter two classes of digraphs are given.

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“…Jacob and Meyniel [17] furthermore showed in 1996 that every digraph has either a kernel or three quasi-kernels. For more results on quasi-kernels, see [5,13,16].…”

Section: Introductionmentioning

confidence: 99%

Kernels by properly colored paths in arc-colored digraphs

Bai

Fujita

Zhang

2018

Discrete Mathematics

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“…In Theorem 2.1 of [15] it was proved that the number of semikernels of a digraph with minimum in-degree at least one is less than or equal to the number of semikernels of the line digraph. Next we improve and generalize this result by stating the equality for every partial line digraph.…”

Section: Semikernelsmentioning

confidence: 99%

“…(2) g(x) = t implies that each y ∈ N + k−1 (x) satisfies g(y) = t. In [15] it was proved that the number of Grundy functions of D is equal to number of Grundy functions of its line digraph. Next, we extend this result to (k, l)-Grundy functions and to partial line digraphs.…”

Section: Grundy Functionmentioning

confidence: 99%

“…A (k, l)-kernel of a digraph D is a subset of vertices K which is both k-independent (d D (u, v) ≥ k for all u, v ∈ K) and l-absorbing (d D (x, K) ≤ l for all x ∈ V \ K). Observe that any kernel is a (2, 1)-kernel and a quasikernel, introduced in [15], is a (2, 2)-kernel. The concept of (k, l)-kernel is a nice, wide generalization of the concept of kernel; (k, l)-kernels have been deeply studied by several authors, see for example [12,24,25,26].…”

Section: Introductionmentioning

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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“…An extension of Theorem 1.1 for semikernels, quasikernels and Grundy functions (concepts closely related to those of kernel) was considered in [6], where it was proved that: If D is a digraph such that δ − D (x) ≥ 1 for each x ∈ V (D), then the number of semikernels (quasikernels) of a digraph D is less than or equal to the number of semikernels (quasikernels) of its line digraph; and the number of Grundy functions of D is equal to the number of Grundy functions of its line digraph. Another extension of Theorem 1.1 for (k, ℓ)-kernels (a concept which generalizes that of kernel) was proved in [8].…”

Section: Introductionmentioning

confidence: 99%

Kernels in monochromatic path digraphs

Galeana‐Sánchez¹,

Ramírez²,

Mejía³

2005

Discuss. Math. Graph Theory

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We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. Let D be an m-coloured digraph. A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u, v ∈ N there is no monochromatic directed path between them and (ii) for each vertex x ∈ (V (D) − N) there is a vertex y ∈ N such that there is an xy-monochromatic directed path. In this paper is defined the monochromatic path digraph of D, MP (D), and the inner m-colouration of MP (D). Also it is proved that if D is an m-coloured digraph without monochromatic directed cycles, then the number of kernels by monochromatic paths in D is equal to the 408

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Semikernels, Quasi Kernels, and Grundy Functions in the Line Digraph

Cited by 7 publications

References 6 publications

Kernels by properly colored paths in arc-colored digraphs

Kernels by properly colored paths in arc-colored digraphs

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

Kernels in monochromatic path digraphs

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