Further hardness results on rainbow and strong rainbow connectivity

Lauri, Juho

doi:10.1016/j.dam.2015.07.041

Cited by 16 publications

(17 citation statements)

References 18 publications

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“…Cacti have previously been considered in the context of rainbow coloring. In particular, Uchizawa et al (2013) show that, given a fixed edge coloring c of a cactus G, determining whether c strongly rainbow connects G can be done in polynomial time (although this problem is NP-complete on general graphs, including interval outerplanar and k-regular graphs (Lauri, 2016)). A particularly useful property of odd cacti which is not shared by other cacti, is that they are geodetic-i.e., every pair of vertices in the graph is connected by a unique shortest path (Stemple and Watkins, 1968).…”

Section: Introductionmentioning

confidence: 99%

A Polynomial Time Algorithm for Computing the Strong Rainbow Connection Numbers of Odd Cacti

Smith¹,

Mildebrath²,

Hicks³

2019

Preprint

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We consider the problem of computing the strong rainbow connection number src(G) for cactus graphs G in which all cycles have odd length. We present a formula to calculate src(G) for such odd cacti which can be evaluated in linear time, as well as an algorithm for computing the corresponding optimal strong rainbow edge coloring, with polynomial worst case run time complexity. Although computing src(G) is NP-hard in general, previous work has demonstrated that it may be computed in polynomial time for certain classes of graphs, including cycles, trees and block clique graphs. This work extends the class of graphs for which src(G) may be computed in polynomial time.

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Section: Introductionmentioning

confidence: 99%

A Polynomial Time Algorithm for Computing the Strong Rainbow Connection Numbers of Odd Cacti

Smith¹,

Mildebrath²,

Hicks³

2019

Preprint

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“…The complexity of computing the rc(G) has been studied in [7]. For 2-connected graphs it has been proved that rc(G) ≤ ⌈|V (G)|/2⌉, see [8].…”

Section: Introductionmentioning

confidence: 99%

Rainbow connectivity of Moore cages of girth 6

Balbuena

Fresán-Figueroa

González-Moreno

et al. 2018

Discrete Applied Mathematics

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Let G be an edge-colored graph. A path P of G is said to be rainbow if no two edges of P have the same color. An edge-coloring of G is a rainbow t-coloring if for any two distinct vertices u and v of G there are at least t internally vertex-disjoint rainbow (u, v)-paths. The rainbow t-connectivity rc t (G) of a graph G is the minimum integer j such that there exists a rainbow t-coloring using j colors. A (k; g)-cage is a k-regular graph of girth g and minimum number of vertices denoted n(k; g). In this paper we focus on g = 6. It is known that n(k; 6) ≥ 2(k 2 − k + 1) and when n(k; 6) = 2(k 2 − k + 1) the (k; 6)-cage is called a Moore cage. In this paper we prove that the rainbow k-connectivity of a Moore (k; 6)-cage G satisfies that k ≤ rc k (G) ≤ k 2 − k + 1. It is also proved that the rainbow 3-connectivity of the Heawood graph is 6 or 7.

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“…Hardness results for related problems [22,29] do not imply that finding an optimal coloring of a boundedtreewidth graph is hard, and it seems that new insights are needed to determine the complexity of these problems on graphs of bounded treewidth.…”

Section: Concluding Notesmentioning

confidence: 99%

On the Complexity of Rainbow Coloring Problems

Eiben

Ganian

Lauri

2016

Lecture Notes in Computer Science

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Abstract. An edge-colored graph G is said to be rainbow connected if between each pair of vertices there exists a path which uses each color at most once. The rainbow connection number, denoted by rc(G), is the minimum number of colors needed to make G rainbow connected. Along with its variants, which consider vertex colorings and/or so-called strong colorings, the rainbow connection number has been studied from both the algorithmic and graph-theoretic points of view. In this paper we present a range of new results on the computational complexity of computing the four major variants of the rainbow connection number. In particular, we prove that the Strong Rainbow Vertex Coloring problem is NP-complete even on graphs of diameter 3. We show that when the number of colors is fixed, then all of the considered problems can be solved in linear time on graphs of bounded treewidth. Moreover, we provide a linear-time algorithm which decides whether it is possible to obtain a rainbow coloring by saving a fixed number of colors from a trivial upper bound. Finally, we give a linear-time algorithm for computing the exact rainbow connection numbers for three variants of the problem on graphs of bounded vertex cover number.

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Further hardness results on rainbow and strong rainbow connectivity

Cited by 16 publications

References 18 publications

A Polynomial Time Algorithm for Computing the Strong Rainbow Connection Numbers of Odd Cacti

A Polynomial Time Algorithm for Computing the Strong Rainbow Connection Numbers of Odd Cacti

Rainbow connectivity of Moore cages of girth 6

On the Complexity of Rainbow Coloring Problems

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