On the Rainbow Connectivity of Graphs: Complexity and FPT Algorithms

Uchizawa, Kei; Aoki, Takanori; Ito, Takehiro; Suzuki, Ai; Zhou, Xiao

doi:10.1007/s00453-012-9689-4

Cited by 26 publications

(31 citation statements)

References 10 publications

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“…We mention that no results for graphs of higher treewidth are known, even for outerplanar or cactus graphs. However, for the slightly different problem of deciding whether an already given coloring forms a (strong) rainbow coloring of a given graph, a polynomial-time algorithm for cactus graphs and an NPhardness result for outerplanar graphs are known [23]. With this in mind, we focus on cactus graphs and make the first progress towards understanding the complexity of rainbow coloring problems, in particular of computing vsrc(G), on graphs of treewidth 2 with the following result.…”

Section: Our Resultsmentioning

confidence: 99%

Algorithms and Bounds for Very Strong Rainbow Coloring

Chandran

Das

Issac

et al. 2018

Lecture Notes in Computer Science

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A well-studied coloring problem is to assign colors to the edges of a graph G so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (src(G)) of the graph. When proving upper bounds on src(G), it is natural to prove that a coloring exists where, for every shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call very strong rainbow connection number (vsrc(G)). In this paper, we give upper bounds on vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on src(G) for these classes, showing that the study of vsrc(G) enables meaningful progress on bounding src(G). Then we study the complexity of the problem to compute vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for src(G). We also observe that deciding whether vsrc(G) = k is fixed-parameter tractable in k and the treewidth of G. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether vsrc(G) ≤ 3 nor to approximate vsrc(G) within a factor n 1−ε , unless P=NP.

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Section: Our Resultsmentioning

confidence: 99%

Algorithms and Bounds for Very Strong Rainbow Coloring

Chandran

Das

Issac

et al. 2018

Lecture Notes in Computer Science

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“…It was shown by Chakraborty et al [4] that Rainbow Connectivity is NP-complete. Later on, the complexity of both edge variants was studied by Uchizawa et al [13]. For instance, the authors showed both problems remain NP-complete for outerplanar graphs, and that Rainbow Connectivity is NPcomplete already on graphs of diameter 2.…”

Section: Rainbow Connectivity (Rc)mentioning

confidence: 99%

“…• In Section 4, we show both problems are solvable in polynomial time when restricted to the class of block graphs. Furthermore, we extend the algorithm of Uchizawa et al [13] for deciding Rainbow Vertex Connectivity on cactus graphs to decide Strong Rainbow Vertex Connectivity for the same graph class.…”

Section: Strong Rainbow Vertex Connectivity (Srvc)mentioning

confidence: 99%

“…An independent set in Table 1: Complexity results for rainbow connectivity problems along with some of our new results marked by . The symbol † stands for [13] and the symbol ‡ for [14].…”

Section: Preliminariesmentioning

confidence: 99%

“…Later on, Huang et al [12] showed the problem remains NP-complete even when the input graph is a line graph. A more systematic study into the complexity of Rainbow Vertex Connectivity was performed by Uchizawa et al [13]. They proved the problem remains NP-complete for both series-parallel graphs, and graphs of bounded diameter.…”

Section: Introductionmentioning

confidence: 99%

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Complexity of rainbow vertex connectivity problems for restricted graph classes

Lauri

2017

Discrete Applied Mathematics

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A path in a vertex-colored graph G is vertex rainbow if all of its internal vertices have a distinct color. The graph G is said to be rainbow vertex connected if there is a vertex rainbow path between every pair of its vertices. Similarly, the graph G is strongly rainbow vertex connected if there is a shortest path which is vertex rainbow between every pair of its vertices. We consider the complexity of deciding if a given vertex-colored graph is rainbow or strongly rainbow vertex connected. We call these problems Rainbow Vertex Connectivity and Strong Rainbow Vertex Connectivity, respectively. We prove both problems remain NP-complete on very restricted graph classes including bipartite planar graphs of maximum degree 3, interval graphs, and k-regular graphs for k ≥ 3. We settle precisely the complexity of both problems from the viewpoint of two width parameters: pathwidth and tree-depth. More precisely, we show both problems remain NP-complete for bounded pathwidth graphs, while being fixed-parameter tractable parameterized by tree-depth. Moreover, we show both problems are solvable in polynomial time for block graphs, while Strong Rainbow Vertex Connectivity is tractable for cactus graphs and split graphs.

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An integer program and new lower bounds for computing the strong rainbow connection numbers of graphs

2021

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We present an integer programming model to compute the strong rainbow connection number, src(G), of any simple graph G. We introduce several enhancements to the proposed model, including a fast heuristic, and a variable elimination scheme. Moreover, we present a novel lower bound for src(G) which may be of independent research interest. We solve the integer program both directly and using an alternative method based on iterative lower bound improvement, the latter of which we show to be highly effective in practice. To our knowledge, these are the first computational methods for the strong rainbow connection problem. We demonstrate the efficacy of our methods by computing the strong rainbow connection numbers of graphs containing up to 379 vertices.

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On the Rainbow Connectivity of Graphs: Complexity and FPT Algorithms

Cited by 26 publications

References 10 publications

Algorithms and Bounds for Very Strong Rainbow Coloring

Algorithms and Bounds for Very Strong Rainbow Coloring

Complexity of rainbow vertex connectivity problems for restricted graph classes

An integer program and new lower bounds for computing the strong rainbow connection numbers of graphs

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