Reachability problems in edge-colored digraphs

Arpin, Peter; Linek, Václav

doi:10.1016/j.disc.2006.09.042

Cited by 26 publications

(43 citation statements)

References 8 publications

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“…In [1] Arpin and Linek called a set of vertices S ⊆ V (D)H-absorbent by walks if for every x ∈ V (D)-S there is an H-walk from x to some point of S, and a subset I of V (D) was called H-independent by walks if there is no H-walk between any two distinct vertices of I. A subset N of V (D) is said to be an H-kernel by walks if N is an H-absorbent set by walks and it is an H-independent set by walks.…”

Section: Theorem 12 If D Is a Digraph Without Cycles Of Odd Lengthmentioning

confidence: 96%

“…Linek [1], Galeana-Sánchez and Delgado-Escalante [6]. In [1] Arpin and Linek called a set of vertices S ⊆ V (D)H-absorbent by walks if for every x ∈ V (D)-S there is an H-walk from x to some point of S, and a subset I of V (D) was called H-independent by walks if there is no H-walk between any two distinct vertices of I.…”

Section: Theorem 12 If D Is a Digraph Without Cycles Of Odd Lengthmentioning

confidence: 98%

“…Since there exists a relation between the definition of H-kernel and the definition of kernel, in [1] Arpin and Linek defined the reachability digraph of an H-colored digraph D as follows. Definition 1.6.…”

Section: ) Is a Subdigraph Of H) Then D Has An H-kernelmentioning

confidence: 99%

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H-kernels and H-obstructions in H-colored digraphs

Galeana‐Sánchez

Sánchez-López

2015

Discrete Mathematics

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a b s t r a c tLet D be a digraph. V (D) and A(D) will denote the vertex and arc sets of D respectively. A kernel K of a digraph D is an independent set of vertices of D such that for every vertex w in V (D)-K there exists an arc from w to a vertex in K . Let H be a digraph possibly with loops and D a digraph without loops whose arcs are colored with the vertices of H (D is said to be an H-colored digraph). A directed path W in D is said to be an H-path if and only if the consecutive colors encountered on W form a directed walk in H. A generalization of the concept of kernel is the concept of H-kernel, where an H-kernel N of an H-colored digraph D is a set of vertices of D such that for every pair of different vertices in N there is no H-path between them, and for every vertex u in V (D)-N there exists an H-path in D from u to N.A classical result in kernel theory establishes that if D is a digraph without cycles of odd length, then D has a kernel; this result is known as Richardson's theorem and in this paper we will show an extension of this theorem which is given by the main result.Let D be an H-colored digraph. For an arc (z 1 , z 2 ) of D we will denote its color by c(z 1 , z 2 ). We introduce the concept of obstruction in an H-colored digraph as follows. Let W = (v 0 , v 1 , . . . , v n−1 , v 0 ) be a closed directed walk in D. We will say that there is an obstructionis not an arc of A(H) (indices modulo n). The main result establishes that if D is an H-colored digraph such that the number of obstructions in every closed directed trail of D is even, then D has an H-kernel. Previous interesting results are generalized, as for example Sands, Sauer and Woodrow's theorem.

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Section: Theorem 12 If D Is a Digraph Without Cycles Of Odd Lengthmentioning

confidence: 96%

Section: Theorem 12 If D Is a Digraph Without Cycles Of Odd Lengthmentioning

confidence: 98%

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H-kernels and H-obstructions in H-colored digraphs

Galeana‐Sánchez

Sánchez-López

2015

Discrete Mathematics

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“…Let

M_{1}

and

M_{2}

be the

2 \times 2

matrices

M_{1} = true(\begin{matrix} center 1 & center 1 \\ center * & center 1 \end{matrix} true), M_{2} = true(\begin{matrix} center 1 & center 0 \\ center 0 & center 1 \end{matrix} true) .

Based on the work of Arpin and Linek , Galeana‐Sánchez and Strausz proved that a digraph is a panchromatic pattern if and only if it is a looped

M_{1}

‐partitionable or

M_{2}

‐partitionable digraph . Again, it is easy to observe that having an

M_{1}

‐partition or an

M_{2}

‐partition is a hereditary property.…”

Section: Minimal Obstructions For Panchromaticitymentioning

confidence: 99%

“…Which are the digraphs

H

such that every

H

‐arc‐colored digraph has a kernel by

H

‐walks? Arpin and Linek stated this question in , and found some digraphs having this property, as well as some digraphs not having this property. A panchromatic pattern is a digraph

H

such that every

H

‐arc‐colored digraph has a kernel by

H

‐walks.…”

Section: Introductionmentioning

confidence: 99%

A dichotomy for the kernel by H‐walks problem in digraphs

Galeana‐Sánchez

Hernández-Cruz

2018

Journal of Graph Theory

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Let H = ( V H , A H ) be a digraph which may contain loops, and let D = ( V D , A D ) be a loopless digraph with a coloring of its arcs c : A D → V H. An H‐walk of D is a walk ( v 0 , … , v n ) of D such that ( c ( v i − 1 , v i ) , c ( v i , v i + 1 ) ) is an arc of H, for every 1 ≤ i ≤ n − 1. For u , v ∈ V D, we say that u reaches v by H‐walks if there exists an H‐walk from u to v in D. A subset S ⊆ V D is a kernel by H‐walks of D if every vertex in V D \ S reaches by H‐walks some vertex in S, and no vertex in S can reach another vertex in S by H‐walks. A panchromatic pattern is a digraph H such that every H‐arc‐colored digraph D has a kernel by H‐walks. In this study, we prove that every digraph H is either a panchromatic pattern, or the problem of determining whether an arc‐colored digraph D has a kernel by H‐walks is N P‐complete.

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H-Kernels in Infinite Digraphs

Galeana‐Sánchez

Sánchez-López

2012

Graphs and Combinatorics

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Reachability problems in edge-colored digraphs

Cited by 26 publications

References 8 publications

H-kernels and H-obstructions in H-colored digraphs

H-kernels and H-obstructions in H-colored digraphs

A dichotomy for the kernel by H‐walks problem in digraphs

H-Kernels in Infinite Digraphs

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