Abstract
Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, and acyclicity of the monochromatic components.
In this paper we focus on defective colorings in which the monochromatic components are acyclic and have small diameter, namely, they form stars. For outerplanar graphs, we give a linear-time algorithm to decide if such a defective coloring exists with two colors and, in the positive case, to construct one. Also, we prove that an outerpath (i.e., an outerplanar graph whose weak-dual is a path) always admits such a two-coloring. Finally, we present NP-completeness results for non-planar and planar graphs of bounded degree for the cases of two and three colors.
This work has been supported by DFG grant Ka812/17-1a and by the MIUR project AMANDA, prot. 2012C4E3KT_001.
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Notes
- 1.
Recall that the weak-dual of a plane graph is the subgraph of its dual induced by neglecting the face-vertex corresponding to its unbounded face.
- 2.
Note that the spine of G coincides with the spine of the caterpillar obtained from the outerpath G by removing all the edges incident to its outer face, neglecting the additional spine vertex \(v_{m+1}\).
- 3.
Fan \(f_i\) contains all the leaves of the caterpillar incident to \(v_i\), plus the following spine vertex \(v_{i+1}\).
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We thank the participants of the special session GNV of IISA’15 inspiring this work.
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Angelini, P., Bekos, M.A., Kaufmann, M., Roselli, V. (2016). Vertex-Coloring with Star-Defects. In: Kaykobad, M., Petreschi, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2016. Lecture Notes in Computer Science(), vol 9627. Springer, Cham. https://doi.org/10.1007/978-3-319-30139-6_4
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