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Displaying similar documents to “Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs”

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...

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.

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...

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-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:...

Kernels in monochromatic path digraphs

Hortensia Galeana-Sánchez, Laura Pastrana Ramírez, Hugo Alberto Rincón Mejía (2005)

Discussiones Mathematicae 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 ∈...

Kernels in edge coloured line digraph

H. Galeana-Sánchez, L. Pastrana Ramírez (1998)

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 (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A set N ⊆ V(D) is said to be a kernel by monochromatic paths if it satisfies the two following conditions (i) for every pair of different vertices u, v ∈ N there is no monochromatic directed path between them and (ii) for every vertex x ∈ V(D)-N there is a vertex y ∈ N such that there is an...

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...

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...

Monochromatic paths and monochromatic sets of arcs in bipartite tournaments

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 and all of them are used. 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 there is no monochromatic path between them and for every vertex v in V(D)∖N there is a monochromatic path from v to some vertex in N. We denote by A⁺(u) the set of arcs of D that have u as...

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...

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...

Monochromatic paths and quasi-monochromatic cycles in edge-coloured bipartite tournaments

Hortensia Galeana-Sanchez, Rocío Rojas-Monroy (2008)

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 (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A directed cycle is called quasi-monochromatic if with at most one exception all of its arcs are coloured alike. 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...

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...