The computational complexity of the backbone coloring problem for bounded-degree graphs with connected backbones
Given a graph G, a spanning subgraph H of G and an integer λ≥2, a λ-backbone coloring of G with backbone H is a proper vertex coloring of G using colors 1,2,…, in which the color difference between vertices adjacent in H is greater than or equal to λ. The backbone coloring problem is that of finding...
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| Vydané v: | Information processing letters Ročník 115; číslo 2; s. 232 - 236 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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Amsterdam
Elsevier B.V
01.02.2015
Elsevier Sequoia S.A |
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| ISSN: | 0020-0190, 1872-6119 |
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| Abstract | Given a graph G, a spanning subgraph H of G and an integer λ≥2, a λ-backbone coloring of G with backbone H is a proper vertex coloring of G using colors 1,2,…, in which the color difference between vertices adjacent in H is greater than or equal to λ. The backbone coloring problem is that of finding such a coloring whose maximum color does not exceed a given limit k. In this paper, we study the backbone coloring problem for bounded-degree graphs with connected backbones and we give a complete computational complexity classification of this problem. We present a polynomial algorithm for optimal backbone coloring for subcubic graphs with arbitrary backbones. We also prove that the backbone coloring problem for graphs with arbitrary backbones and with fixed maximum degree (at least 4) is NP-complete. Furthermore, we show that for the special case of graphs with fixed maximum degree at least 5 and λ≥4 the problem remains NP-complete even for spanning tree backbones.
•The backbone coloring problem is solvable in quadratic time for subcubic graphs.•The backbone coloring problem is NP-hard for graphs with degree greater than 4.•The backbone coloring problem is NP-hard for tree backbones with bounded degree. |
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| AbstractList | Given a graph G, a spanning subgraph H of G and an integer λ≥2, a λ-backbone coloring of G with backbone H is a proper vertex coloring of G using colors 1,2,…, in which the color difference between vertices adjacent in H is greater than or equal to λ. The backbone coloring problem is that of finding such a coloring whose maximum color does not exceed a given limit k. In this paper, we study the backbone coloring problem for bounded-degree graphs with connected backbones and we give a complete computational complexity classification of this problem. We present a polynomial algorithm for optimal backbone coloring for subcubic graphs with arbitrary backbones. We also prove that the backbone coloring problem for graphs with arbitrary backbones and with fixed maximum degree (at least 4) is NP-complete. Furthermore, we show that for the special case of graphs with fixed maximum degree at least 5 and λ≥4 the problem remains NP-complete even for spanning tree backbones.
•The backbone coloring problem is solvable in quadratic time for subcubic graphs.•The backbone coloring problem is NP-hard for graphs with degree greater than 4.•The backbone coloring problem is NP-hard for tree backbones with bounded degree. Given a graph G , a spanning subgraph H of G and an integer lambda greater than or equal to 2 lambda greater than or equal to 2, a lambda -backbone coloring of G with backbone H is a proper vertex coloring of G using colors 1,2,...1,2,..., in which the color difference between vertices adjacent in H is greater than or equal to lambda . The backbone coloring problem is that of finding such a coloring whose maximum color does not exceed a given limit k . In this paper, we study the backbone coloring problem for bounded-degree graphs with connected backbones and we give a complete computational complexity classification of this problem. We present a polynomial algorithm for optimal backbone coloring for subcubic graphs with arbitrary backbones. We also prove that the backbone coloring problem for graphs with arbitrary backbones and with fixed maximum degree (at least 4) is NP-complete. Furthermore, we show that for the special case of graphs with fixed maximum degree at least 5 and lambda greater than or equal to 4 lambda greater than or equal to 4 the problem remains NP-complete even for spanning tree backbones. Given a graph G , a spanning subgraph H of G and an integer ...≥2, a ... -backbone coloring of G with backbone H is a proper vertex coloring of G using colors 1,2,..., in which the color difference between vertices adjacent in H is greater than or equal to ... . The backbone coloring problem is that of finding such a coloring whose maximum color does not exceed a given limit k . This paper studies the backbone coloring problem for bounded-degree graphs with connected backbones and it gives a complete computational complexity classification of this problem. This paper presents a polynomial algorithm for optimal backbone coloring for subcubic graphs with arbitrary backbones. It also proves that the backbone coloring problem for graphs with arbitrary backbones and with fixed maximum degree (at least 4) is NP-complete. Furthermore, this paper shows that for the special case of graphs with fixed maximum degree at least 5 and ...≥4 the problem remains NP-complete even for spanning tree backbones. (ProQuest: ... denotes formulae/symbols omitted.) |
| Author | Janczewski, Robert Turowski, Krzysztof |
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| References | Broersma (br0010) 2003 Broersma, Marchal, Paulusma, Salman (br0020) 2007; vol. 4362 Broersma, Fujisawa, Marchal, Paulusma, Salman, Yoshimoto (br0040) 2009; 309 Broersma, Fomin, Golovach, Woeginger (br0030) 2007; 55 Garey, Johnson, Stockmeyer (br0050) 1976; 1 Salman (br0100) 2005 R. Janczewski, K. Turowski, The backbone coloring problem for bipartite backbones, submitted for publication. 10.1016/j.ipl.2014.09.018_br0090 Broersma (10.1016/j.ipl.2014.09.018_br0020) 2007; vol. 4362 Broersma (10.1016/j.ipl.2014.09.018_br0010) 2003 Salman (10.1016/j.ipl.2014.09.018_br0100) 2005 Broersma (10.1016/j.ipl.2014.09.018_br0040) 2009; 309 Broersma (10.1016/j.ipl.2014.09.018_br0030) 2007; 55 Garey (10.1016/j.ipl.2014.09.018_br0050) 1976; 1 |
| References_xml | – year: 2005 ident: br0100 article-title: Contributions to graph theory – volume: vol. 4362 start-page: 188 year: 2007 end-page: 199 ident: br0020 article-title: Improved upper bounds for publication-title: SOFSEM 2007: Theory and Practice of Computer Science, 33rd Conference on Current Trends in Theory and Practice of Computer Science – start-page: 65 year: 2003 end-page: 79 ident: br0010 article-title: A general framework for coloring problems: old results, new results, and open problems publication-title: Combinatorial Geometry and Graph Theory – volume: 309 start-page: 5596 year: 2009 end-page: 5609 ident: br0040 article-title: -Backbone colorings along pairwise disjoint stars and matchings publication-title: Discrete Math. – volume: 1 start-page: 237 year: 1976 end-page: 267 ident: br0050 article-title: Some simplified NP-complete graph problems publication-title: Theor. Comput. Sci. – reference: R. Janczewski, K. Turowski, The backbone coloring problem for bipartite backbones, submitted for publication. – volume: 55 start-page: 137 year: 2007 end-page: 152 ident: br0030 article-title: Backbone colorings for graphs: tree and path backbones publication-title: J. Graph Theory – ident: 10.1016/j.ipl.2014.09.018_br0090 – volume: vol. 4362 start-page: 188 year: 2007 ident: 10.1016/j.ipl.2014.09.018_br0020 article-title: Improved upper bounds for λ-backbone colorings along matchings and stars – volume: 1 start-page: 237 year: 1976 ident: 10.1016/j.ipl.2014.09.018_br0050 article-title: Some simplified NP-complete graph problems publication-title: Theor. Comput. Sci. doi: 10.1016/0304-3975(76)90059-1 – volume: 309 start-page: 5596 issue: 18 year: 2009 ident: 10.1016/j.ipl.2014.09.018_br0040 article-title: λ-Backbone colorings along pairwise disjoint stars and matchings publication-title: Discrete Math. doi: 10.1016/j.disc.2008.04.007 – start-page: 65 year: 2003 ident: 10.1016/j.ipl.2014.09.018_br0010 article-title: A general framework for coloring problems: old results, new results, and open problems – volume: 55 start-page: 137 issue: 2 year: 2007 ident: 10.1016/j.ipl.2014.09.018_br0030 article-title: Backbone colorings for graphs: tree and path backbones publication-title: J. Graph Theory doi: 10.1002/jgt.20228 – year: 2005 ident: 10.1016/j.ipl.2014.09.018_br0100 |
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| SubjectTerms | Backbone Backbone chromatic number Bounded-degree graphs Classification Coloring Complexity Computation Computer science Graph algorithms Graph theory Graphs Integer programming Mathematical problems Optimization Polynomials Studies |
| Title | The computational complexity of the backbone coloring problem for bounded-degree graphs with connected backbones |
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