The computational complexity of the backbone coloring problem for planar graphs with connected backbones
In the paper we study the computational complexity of the backbone coloring problem for planar graphs with connected backbones. For every possible value of integer parameters λ≥2 and k≥1 we show that the following problem: Instance:A simple planar graph G, its connected spanning subgraph (backbone)...
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| Veröffentlicht in: | Discrete Applied Mathematics Jg. 184; S. 237 - 242 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Elsevier B.V
31.03.2015
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| ISSN: | 0166-218X, 1872-6771 |
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| Abstract | In the paper we study the computational complexity of the backbone coloring problem for planar graphs with connected backbones. For every possible value of integer parameters λ≥2 and k≥1 we show that the following problem: Instance:A simple planar graph G, its connected spanning subgraph (backbone) H.Question:Is there a λ-backbone coloring c of G with backbone H such that maxc(V(G))≤k? is either NP-complete or polynomially solvable (by algorithms that run in constant, linear or quadratic time). As a result of these considerations we obtain a complete classification of the computational complexity with respect to the values of λ and k.
We also study the problem of computing the backbone chromatic number for two special classes of planar graphs: cacti and thorny graphs. We construct an algorithm that runs in O(n3) time and solves this problem for cacti and another polynomial algorithm that is 1-absolute approximate for thorny graphs. |
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| AbstractList | In the paper we study the computational complexity of the backbone coloring problem for planar graphs with connected backbones. For every possible value of integer parameters and we show that the following problem: * Instance: A simple planar graph , its connected spanning subgraph (backbone) . * Question: Is there a -backbone coloring of with backbone such that ? is either NP-complete or polynomially solvable (by algorithms that run in constant, linear or quadratic time). As a result of these considerations we obtain a complete classification of the computational complexity with respect to the values of and . We also study the problem of computing the backbone chromatic number for two special classes of planar graphs: cacti and thorny graphs. We construct an algorithm that runs in time and solves this problem for cacti and another polynomial algorithm that is -absolute approximate for thorny graphs. In the paper we study the computational complexity of the backbone coloring problem for planar graphs with connected backbones. For every possible value of integer parameters λ≥2 and k≥1 we show that the following problem: Instance:A simple planar graph G, its connected spanning subgraph (backbone) H.Question:Is there a λ-backbone coloring c of G with backbone H such that maxc(V(G))≤k? is either NP-complete or polynomially solvable (by algorithms that run in constant, linear or quadratic time). As a result of these considerations we obtain a complete classification of the computational complexity with respect to the values of λ and k. We also study the problem of computing the backbone chromatic number for two special classes of planar graphs: cacti and thorny graphs. We construct an algorithm that runs in O(n3) time and solves this problem for cacti and another polynomial algorithm that is 1-absolute approximate for thorny graphs. |
| Author | Janczewski, Robert Turowski, Krzysztof |
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| Cites_doi | 10.1073/pnas.39.4.315 10.1016/j.dam.2014.01.011 10.1016/S0166-218X(02)00575-9 10.1016/0012-365X(80)90236-8 10.1016/j.disc.2008.04.007 |
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| References | Dailey (br000020) 1980; 30 Giaro, Janczewski, Małafiejski (br000030) 2003; 129 Broersma, Fujisawa, Marchal, Paulusma, Salman, Yoshimoto (br000015) 2009; 309 Havet, King, Liedloff, Todinca (br000040) 2014; 169 Broersma (br000005) 2003 Broersma, Fomin, Golovach, Woeginger (br000010) 2003; vol. 2880 Garey, Johnson (br000025) 1979 R. Janczewski, K. Turowski, The backbone coloring problem for bipartite backbones, Graphs Combin. (in press). Harary, Uhlenbeck (br000035) 1953; 39 Giaro (10.1016/j.dam.2014.10.028_br000030) 2003; 129 Garey (10.1016/j.dam.2014.10.028_br000025) 1979 Harary (10.1016/j.dam.2014.10.028_br000035) 1953; 39 10.1016/j.dam.2014.10.028_br000045 Broersma (10.1016/j.dam.2014.10.028_br000015) 2009; 309 Havet (10.1016/j.dam.2014.10.028_br000040) 2014; 169 Broersma (10.1016/j.dam.2014.10.028_br000010) 2003; vol. 2880 Broersma (10.1016/j.dam.2014.10.028_br000005) 2003 Dailey (10.1016/j.dam.2014.10.028_br000020) 1980; 30 |
| References_xml | – volume: 129 start-page: 371 year: 2003 end-page: 382 ident: br000030 article-title: A polynomial algorithm for finding publication-title: Discrete Appl. Math. – volume: vol. 2880 start-page: 131 year: 2003 end-page: 142 ident: br000010 article-title: Backbone colorings for networks publication-title: Proceedings of the 29th International Workshop on Graph-Theoretic Concepts in Computer Science – volume: 309 start-page: 5596 year: 2009 end-page: 5609 ident: br000015 article-title: -backbone colorings along pairwise disjoint stars and matchings publication-title: Discrete Math. – start-page: 65 year: 2003 end-page: 79 ident: br000005 article-title: A general framework for coloring problems: old results, new results, and open problems publication-title: Combinatorial Geometry and Graph Theory – volume: 30 start-page: 289 year: 1980 end-page: 293 ident: br000020 article-title: Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete publication-title: Discrete Math. – volume: 39 start-page: 315 year: 1953 end-page: 322 ident: br000035 article-title: On the number of Husimi trees. I. publication-title: Proc. Natl. Acad. Sci. – volume: 169 start-page: 119 year: 2014 end-page: 134 ident: br000040 article-title: (Circular) backbone colouring: Forest backbones in planar graphs publication-title: Discrete Appl. Math. – reference: R. Janczewski, K. Turowski, The backbone coloring problem for bipartite backbones, Graphs Combin. (in press). – year: 1979 ident: br000025 article-title: Computers and Intractability: A Guide to the Theory of NP-Completeness – year: 1979 ident: 10.1016/j.dam.2014.10.028_br000025 – volume: 39 start-page: 315 issue: 4 year: 1953 ident: 10.1016/j.dam.2014.10.028_br000035 article-title: On the number of Husimi trees. I. publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.39.4.315 – ident: 10.1016/j.dam.2014.10.028_br000045 – volume: 169 start-page: 119 year: 2014 ident: 10.1016/j.dam.2014.10.028_br000040 article-title: (Circular) backbone colouring: Forest backbones in planar graphs publication-title: Discrete Appl. Math. doi: 10.1016/j.dam.2014.01.011 – start-page: 65 year: 2003 ident: 10.1016/j.dam.2014.10.028_br000005 article-title: A general framework for coloring problems: old results, new results, and open problems – volume: 129 start-page: 371 year: 2003 ident: 10.1016/j.dam.2014.10.028_br000030 article-title: A polynomial algorithm for finding T-span of generalized cacti publication-title: Discrete Appl. Math. doi: 10.1016/S0166-218X(02)00575-9 – volume: 30 start-page: 289 issue: 3 year: 1980 ident: 10.1016/j.dam.2014.10.028_br000020 article-title: Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete publication-title: Discrete Math. doi: 10.1016/0012-365X(80)90236-8 – volume: 309 start-page: 5596 issue: 18 year: 2009 ident: 10.1016/j.dam.2014.10.028_br000015 article-title: λ-backbone colorings along pairwise disjoint stars and matchings publication-title: Discrete Math. doi: 10.1016/j.disc.2008.04.007 – volume: vol. 2880 start-page: 131 year: 2003 ident: 10.1016/j.dam.2014.10.028_br000010 article-title: Backbone colorings for networks |
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| SubjectTerms | Algorithms Approximation Backbone Backbone chromatic number Cacti Coloring Complexity Computation Graphs Planar graphs |
| Title | The computational complexity of the backbone coloring problem for planar graphs with connected backbones |
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