Further Hardness Results on Rainbow and Strong Rainbow Connectivity
Abstract
A path in an edge-colored graph is \textit{rainbow} if no two edges of it are colored the same. The graph is said to be \textit{rainbow connected} if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph is \textit{strong rainbow connected}. We consider the complexity of the problem of deciding if a given edge-colored graph is rainbow or strong rainbow connected. These problems are called \textsc{Rainbow connectivity} and \textsc{Strong rainbow connectivity}, respectively. We prove both problems remain $\NP$\hyp{}complete on interval outerplanar graphs and $k$-regular graphs for $k \geq 3$. Previously, no graph class was known where the complexity of the two problems would differ. We show that for block graphs, which form a subclass of chordal graphs, \textsc{Rainbow connectivity} is $\NP$\hyp{}complete while \textsc{Strong rainbow connectivity} is in $¶$. We conclude by considering some tractable special cases, and show for instance that both problems are in $\XP$ when parameterized by tree-depth.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2014
- DOI:
- 10.48550/arXiv.1404.3082
- arXiv:
- arXiv:1404.3082
- Bibcode:
- 2014arXiv1404.3082L
- Keywords:
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- Computer Science - Computational Complexity;
- Computer Science - Discrete Mathematics;
- 68Q25
- E-Print:
- 13 pages, 4 figures