Proper orientation of cacti

Araújo, Júlio; Havet, Frédéric; Sales, Cláudia Linhares; Silva, Ana

doi:10.1016/j.tcs.2016.05.016

Cited by 23 publications

(32 citation statements)

References 6 publications

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“…In what follows, we give an alternative and shorter proof of this theorem which results in a polynomial-time algorithm. We start by recalling a result of [3] about proper orientation of paths.…”

Section: Treesmentioning

confidence: 99%

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Proper Orientations of Planar Bipartite Graphs

Knox

Matsumoto

Maza

et al. 2017

Graphs and Combinatorics

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An orientation of a graph G is proper if any two adjacent vertices have different indegrees. The proper orientation number − → χ (G) of a graph G is the minimum of the maximum indegree, taken over all proper orientations of G. In this paper, we show that a connected bipartite graph may be properly oriented even if we are only allowed to control the orientation of a specific set of edges, namely, the edges of a spanning tree and all the edges incident to one of its leaves. As a consequence of this result, we prove that 3-connected planar bipartite graphs have proper orientation number at most 6. Additionally, we give a short proof that − → χ (G) ≤ 4, when G is a tree and this proof leads to a polynomial-time algorithm to proper orient trees within this bound.

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Section: Treesmentioning

confidence: 99%

“…Lately, Araujo et al [3] proved that the problem of determining the proper orientation number of a graph remains NP-hard for subclasses of planar graphs that are also bipartite and of bounded degree. In the same paper, they proved that the proper orientation number of any tree is at most 4.…”

Section: Introductionmentioning

confidence: 99%

Proper Orientations of Planar Bipartite Graphs

Knox

Matsumoto

Maza

et al. 2017

Graphs and Combinatorics

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“…The proper orientation number of G, denoted by − → χ (G), is the minimum integer k such that G admits a proper k-orientation. Proper orientation number is defined by Ahadi and Dehghan [1], and see the related research [2,3,6]. Knox et al [6] showed the following.…”

Section: Introductionmentioning

confidence: 99%

Proper 3-orientations of bipartite planar graphs with minimum degree at least 3

Noguchi

2020

Discrete Applied Mathematics

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“…In the same paper, they proved that the proper orientation number of any tree is at most 4; Knox et al [16] provided a shorter proof of the same result. In another paper, Araújo et al [5] proved that the proper orientation number of cacti is at most 7, and that this bound is tight.…”

Section: Introductionmentioning

confidence: 99%

Weighted proper orientations of trees and graphs of bounded treewidth

Araújo

Sales

et al. 2019

Theoretical Computer Science

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Given a simple graph G, a weight function w : E(G) → N \ {0}, and an orientationwhenever u and v are adjacent. We introduce the parameter weighted proper orientation number of G, denoted by − → χ (G, w), which is the minimum, over all weighted proper orientations D of G, of µ − (D). When all the weights are equal to 1, this parameter is equal to the proper orientation number of G, which has been object of recent studies and whose determination is NP-hard in general, but polynomial-time solvable on trees.Here, we prove that the equivalent decision problem of the weighted proper orientation number (i.e., − → χ (G, w) ≤ k?) is (weakly) NP-complete on trees but can be solved by a pseudo-polynomial time algorithm whose running time depends on k. Furthermore, we present a dynamic programming algorithm to determine whether a general graph G on n vertices and treewidth at most tw satisfies − → χ (G, w) ≤ k, running in time O(2 tw 2 · k 3tw · tw · n), and we complement this result by showing that the problem is W[1]-hard on general graphs parameterized by the treewidth of G, even if the weights are polynomial in n.

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Proper orientation of cacti

Cited by 23 publications

References 6 publications

Proper Orientations of Planar Bipartite Graphs

Proper Orientations of Planar Bipartite Graphs

Proper 3-orientations of bipartite planar graphs with minimum degree at least 3

Weighted proper orientations of trees and graphs of bounded treewidth

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