Displaying similar documents to “On the Existence of (k,l)-Kernels in Infinite Digraphs: A Survey”

k-kernels in generalizations of transitive digraphs

Hortensia Galeana-Sánchez, César Hernández-Cruz (2011)

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 set of vertices (if u,v ∈ N, u ≠ 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). A k-kernel is a (k,k-1)-kernel. Quasi-transitive, right-pretransitive and left-pretransitive digraphs are generalizations of transitive digraphs. In this paper the following results are proved:...

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...

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...

Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs

Hortensia Galeana-Sánchez, R. Rojas-Monroy, B. Zavala (2010)

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. We call the digraph D an m-coloured digraph if each arc of D is coloured by an element of {1,2,...,m} where m ≥ 1. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if there is no monochromatic path between two vertices of N and if for every vertex v not in N there is a monochromatic path from v...

4-Transitive Digraphs I: The Structure of Strong 4-Transitive Digraphs

César Hernández-Cruz (2013)

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 digraph D is transitive if for every three distinct vertices u, v,w ∈ V (D), (u, v), (v,w) ∈ A(D) implies that (u,w) ∈ A(D). This concept can be generalized as follows: A digraph is k-transitive if for every u, v ∈ V (D), the existence of a uv-directed path of length k in D implies that (u, v) ∈ A(D). A very useful structural characterization of transitive digraphs has been known for a...

(K − 1)-Kernels In Strong K-Transitive Digraphs

Ruixia Wang (2015)

Discussiones Mathematicae Graph Theory

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Let D = (V (D),A(D)) be a digraph and k ≥ 2 be an integer. A subset N of V (D) is k-independent if for every pair of vertices u, v ∈ N, we have d(u, v) ≥ k; it is l-absorbent if for every u ∈ V (D) − N, there exists v ∈ N such that d(u, v) ≤ l. A (k, l)-kernel of D is a k-independent and l-absorbent subset of V (D). A k-kernel is a (k, k − 1)-kernel. A digraph D is k-transitive if for any path x0x1 ・ ・ ・ xk of length k, x0 dominates xk. Hernández-Cruz [3-transitive digraphs, Discuss....

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...

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.

Underlying Graphs of 3-Quasi-Transitive Digraphs and 3-Transitive Digraphs

Ruixia Wang, Shiying Wang (2013)

Discussiones Mathematicae Graph Theory

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A digraph is 3-quasi-transitive (resp. 3-transitive), if for any path x0x1 x2x3 of length 3, x0 and x3 are adjacent (resp. x0 dominates x3). C´esar Hern´andez-Cruz conjectured that if D is a 3-quasi-transitive digraph, then the underlying graph of D, UG(D), admits a 3-transitive orientation. In this paper, we shall prove that the conjecture is true.

A note on kernels and solutions in digraphs

Matúš Harminc, Roman Soták (1999)

Discussiones Mathematicae Graph Theory

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For given nonnegative integers k,s an upper bound on the minimum number of vertices of a strongly connected digraph with exactly k kernels and s solutions is presented.

3-transitive digraphs

César Hernández-Cruz (2012)

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 digraph D is 3-transitive if the existence of the directed path (u,v,w,x) of length 3 in D implies the existence of the arc (u,x) ∈ A(D). In this article strong 3-transitive digraphs are characterized and the structure of non-strong 3-transitive digraphs is described. The results are used, e.g., to characterize 3-transitive digraphs that are transitive and to characterize 3-transitive digraphs...

Some Results on 4-Transitive Digraphs

Patricio Ricardo García-Vázquez, César Hernández-Cruz (2017)

Discussiones Mathematicae Graph Theory

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Let D be a digraph with set of vertices V and set of arcs A. We say that D is k-transitive if for every pair of vertices u, v ∈ V, the existence of a uv-path of length k in D implies that (u, v) ∈ A. A 2-transitive digraph is a transitive digraph in the usual sense. A subset N of V 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 N there exists v ∈ N such that d(u, v) ≤ l. A k-kernel of D is a k-independent and...

Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs

Hortensia Galeana-Sánchez, R. Rojas-Monroy, B. Zavala (2009)

Discussiones Mathematicae Graph Theory

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We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices of N there is no monochromatic path between them and for every vertex v ∉ N there is a monochromatic path from v to N. We denote by A⁺(u) the set of arcs of D that have u as the initial vertex. We prove that if D is an m-coloured...

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...