Homomorphisms and oriented colorings of equivalence classes of oriented graphs

Klostermeyer, William F.; MacGillivray, Gary

doi:10.1016/s0012-365x(03)00086-4

Cited by 50 publications

(51 citation statements)

References 12 publications

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“…It is known that planar graphs with girth at least 4 (resp. 5,6,7,14) have oriented chromatic number at most 47 (resp. 19, 11, 7, 5) (see [4][5][6]).…”

Section: Discussion and Further Workmentioning

confidence: 99%

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On the oriented chromatic index of oriented graphs

Ochem

Pinlou

Sopena

2007

Journal of Graph Theory

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Abstract:A homomorphism from an oriented graph G to an oriented graph H is a mapping ϕ from the set of vertices of G to the set of vertices of H such thatis an arc in H whenever − → uv is an arc in G. The oriented chromatic index of an oriented graph G is the minimum number of vertices in an oriented graph H such that there exists a homomorphism from the line digraph LD(G) of G to H (the line digraph LD(G) of G is given by V(LD(G)) = A(G) and − → ab ∈ A(LD(G)) whenever a = − → uv and b = − → vw). We give upper bounds for the oriented chromatic index of graphs with bounded acyclic chromatic number, of planar graphs and of graphs with bounded degree. We also consider lower and upper bounds of oriented chromatic number in terms of oriented chromatic index. We finally prove that the problem of deciding whether an oriented graph has oriented chromatic index at most k is polynomial time solvable if k ≤ 3 and is NPcomplete if k ≥ 4.

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“…It is known that planar graphs with girth at least 4 (resp. 5,6,7,14) have oriented chromatic number at most 47 (resp. 19, 11, 7, 5) (see [4][5][6]).…”

Section: Discussion and Further Workmentioning

confidence: 99%

“…Klostermeyer and MacGillivray [14] have shown that given an oriented graph G, deciding whether χ o (G) ≤ k is polynomial time if k ≤ 3 and is NP-complete if k ≥ 4. Culus and Demange [8] extended the above result to the case of bipartite oriented graphs and circuit-free oriented graphs.…”

Section: Np-completenessmentioning

confidence: 99%

On the oriented chromatic index of oriented graphs

Ochem

Pinlou

Sopena

2007

Journal of Graph Theory

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“…[7] ocn 4 is NP-complete on acyclic oriented graphs with ∆(G) = max(p + 2, 4). [8] ocn 4 is NP-complete.…”

Section: Tablementioning

confidence: 99%

“…Klostermeyer and MacGillivray [8] proved in 2004 using [1], that ocn 4 is NP-complete. They established a P versus NP-complete dichotomy with respect to the number of colors k: ocn k is polynomial if k ≤ 3 and NP-complete if k > 3.…”

mentioning

confidence: 99%

Oriented coloring in planar, bipartite, bounded degree 3 acyclic oriented graphs

Coelho¹,

Faria

Gravier

et al. 2016

Discrete Applied Mathematics

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a b s t r a c tIt has been a challenging problem to determine the smallest graph class where a problem is proved to be hard. In the literature, this has been pointed out to be very important in order to establish the real nature of a combinatorial problem.An oriented k-coloring of an oriented graph ⃗ G = (V , ⃗ E) is a partition of V into k subsets such that there are no two adjacent vertices belonging to the same subset, and all the arcs between a pair of subsets have the same orientation. The decision problem k − oriented chromatic number (ocn k ) consists of an oriented graph ⃗ G and an integer k > 0, plus the question if there exists an oriented k-coloring of ⃗ G. By its strong appeal, many papers have presented NP-completeness proofs for ocn k . It was not known the complexity status of ocn k when the input graph ⃗ G satisfies that the underlying graph G has maximum degree 3.In this paper we prove that ocn 4 is NP-complete for an acyclic oriented graph ⃗ G such that G is at same time: connected, planar, bipartite, and with maximum degree 3.Our result defines a P versus NP-complete dichotomy with respect to the maximum degree ∆(G): ocn k is polynomial if ∆(G) < 3 and NP-complete if ∆(G) ≥ 3, since it is known that ocn 3 is in P, and that ocn k is in P when the underlying graph has ∆(G) ≤ 2.

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“…It has been shown [KM04] that checking χ o (G) ≤ 3 is easy, but determining whether χ o (G) ≤ 4 is already NP-complete. Subsequently, [CD06] have shown that the problem χ o (G) ≤ 4 remains NP-complete even on acyclic digraphs.…”

Section: Oriented Colouring (Ocn)mentioning

confidence: 99%

On Digraph Width Measures in Parameterized Algorithmics

Ganian

Hliněný

Kneis

et al. 2009

Parameterized and Exact Computation

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Abstract. In contrast to undirected width measures (such as treewidth or clique-width), which have provided many important algorithmic applications, analogous measures for digraphs such as DAGwidth or Kelly-width do not seem so successful. Several recent papers, e.g. those of Kreutzer-Ordyniak, Dankelmann-Gutin-Kim, or LampisKaouri-Mitsou, have given some evidence for this. We support this direction by showing that many quite different problems remain hard even on graph classes that are restricted very beyond simply having small DAG-width. To this end, we introduce new measures K-width and DAGdepth. On the positive side, we also note that taking Kanté's directed generalization of rank-width as a parameter makes many problems fixed parameter tractable.

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Homomorphisms and oriented colorings of equivalence classes of oriented graphs

Cited by 50 publications

References 12 publications

On the oriented chromatic index of oriented graphs

On the oriented chromatic index of oriented graphs

Oriented coloring in planar, bipartite, bounded degree 3 acyclic oriented graphs

On Digraph Width Measures in Parameterized Algorithmics

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