Displaying similar documents to “Some sufficient conditions on odd directed cycles of bounded length for the existence of a kernel”

KP-digraphs and CKI-digraphs satisfying the k-Meyniel's condition

H. Galeana-Sánchez, V. Neumann-Lara (1996)

Discussiones Mathematicae Graph Theory

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A digraph D is said to satisfy the k-Meyniel's condition if each odd directed cycle of D has at least k diagonals. The study of the k-Meyniel's condition has been a source of many interesting problems, questions and results in the development of Kernel Theory. In this paper we present a method to construct a large variety of kernel-perfect (resp. critical kernel-imperfect) digraphs which satisfy the k-Meyniel's condition.

On (k,l)-kernel perfectness of special classes of digraphs

Magdalena Kucharska (2005)

Discussiones Mathematicae Graph Theory

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In the first part of this paper we give necessary and sufficient conditions for some special classes of digraphs to have a (k,l)-kernel. One of them is the duplication of a set of vertices in a digraph. This duplication come into being as the generalization of the duplication of a vertex in a graph (see [4]). Another one is the D-join of a digraph D and a sequence α of nonempty pairwise disjoint digraphs. In the second part we prove theorems, which give necessary and sufficient conditions...

On graphs all of whose {C₃,T₃}-free arc colorations are kernel-perfect

Hortensia Galeana-Sánchez, José de Jesús García-Ruvalcaba (2001)

Discussiones Mathematicae Graph Theory

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A digraph D is called a kernel-perfect digraph or KP-digraph when every induced subdigraph of D has a kernel. We call the digraph D an m-coloured digraph if the arcs of D are coloured with m distinct colours. A path P is monochromatic in D if all of its arcs are coloured alike in D. The closure of D, denoted by ζ(D), is the m-coloured digraph defined as follows: V( ζ(D)) = V(D), and A( ζ(D)) = ∪_{i} {(u,v) with colour i: there exists a monochromatic...

Kernels by Monochromatic Paths and Color-Perfect Digraphs

Hortensia Galeana-Śanchez, Rocío Sánchez-López (2016)

Discussiones Mathematicae Graph Theory

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For a digraph D, V (D) and A(D) will denote the sets of vertices and arcs of D respectively. In an arc-colored digraph, a subset K of V(D) is said to be kernel by monochromatic paths (mp-kernel) if (1) for any two different vertices x, y in N there is no monochromatic directed path between them (N is mp-independent) and (2) for each vertex u in V (D) N there exists v ∈ N such that there is a monochromatic directed path from u to v in D (N is mp-absorbent). If every arc in D has a different...

A sufficient condition for the existence of k-kernels in digraphs

H. Galeana-Sánchez, H.A. Rincón-Mejía (1998)

Discussiones Mathematicae Graph Theory

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In this paper, we prove the following sufficient condition for the existence of k-kernels in digraphs: Let D be a digraph whose asymmetrical part is strongly conneted and such that every directed triangle has at least two symmetrical arcs. If every directed cycle γ of D with l(γ) ≢ 0 (mod k), k ≥ 2 satisfies at least one of the following properties: (a) γ has two symmetrical arcs, (b) γ has four short chords. Then D has a k-kernel. This result generalizes some previous...

On (k,l)-kernels in D-join of digraphs

Waldemar Szumny, Andrzej Włoch, Iwona Włoch (2007)

Discussiones Mathematicae Graph Theory

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In [5] the necessary and sufficient conditions for the existence of (k,l)-kernels in a D-join of digraphs were given if the digraph D is without circuits of length less than k. In this paper we generalize these results for an arbitrary digraph D. Moreover, we give the total number of (k,l)-kernels, k-independent sets and l-dominating sets in a D-join of digraphs.

On the Complexity of the 3-Kernel Problem in Some Classes of Digraphs

Pavol Hell, César Hernández-Cruz (2014)

Discussiones Mathematicae Graph Theory

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Let D be a digraph with the vertex set V (D) and the arc set A(D). A subset N of V (D) is k-independent if for every pair of vertices u, v ∈ N, we have d(u, v), d(v, u) ≥ k; it is l-absorbent if for every u ∈ V (D) − N there exists v ∈ N such that d(u, v) ≤ l. A k-kernel of D is a k-independent and (k − 1)-absorbent subset of V (D). A 2-kernel is called a kernel. It is known that the problem of determining whether a digraph has a kernel (“the kernel problem”) is NP-complete, even in...

On the Existence of (k,l)-Kernels in Infinite Digraphs: A Survey

H. Galeana-Sánchez, C. Hernández-Cruz (2014)

Discussiones Mathematicae Graph Theory

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Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent (if u, v ∈ N, u 6= v, then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l) set of vertices. A k-kernel is a (k, k −1)-kernel. This work is a survey of results proving sufficient conditions for the existence of (k, l)-kernels in infinite digraphs. Despite all the previous work in this direction...

k-Kernels and some operations in digraphs

Hortensia Galeana-Sanchez, Laura Pastrana (2009)

Discussiones Mathematicae Graph Theory

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Let D be a digraph. V(D) denotes the set of vertices of D; a set N ⊆ V(D) is said to be a k-kernel of D if it satisfies the following two conditions: for every pair of different vertices u,v ∈ N it holds that every directed path between them has length at least k and for every vertex x ∈ V(D)-N there is a vertex y ∈ N such that there is an xy-directed path of length at most k-1. In this paper, we consider some operations on digraphs and prove the existence of k-kernels in digraphs formed...

New classes of critical kernel-imperfect digraphs

Hortensia Galeana-Sánchez, V. Neumann-Lara (1998)

Discussiones Mathematicae Graph Theory

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A kernel of a digraph D is a subset N ⊆ V(D) which is both independent and absorbing. When every induced subdigraph of D has a kernel, the digraph D is said to be kernel-perfect. We say that D is a critical kernel-imperfect digraph if D does not have a kernel but every proper induced subdigraph of D does have at least one. Although many classes of critical kernel-imperfect-digraphs have been constructed, all of them are digraphs such that the block-cutpoint tree of its asymmetrical part...