Complexity of Coloring Graphs without Forbidden Induced Subgraphs
14 Jun 2001-pp 254-262
TL;DR: A complete characterization of parameter graphs H for which the problem of coloring H-free graphs is polynomial and for which it is NP-complete is given.
Abstract: We give a complete characterization of parameter graphs H for which the problem of coloring H-free graphs is polynomial and for which it is NP-complete. We further initiate a study of this problem for two forbidden subgraphs.
Citations
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TL;DR: In this paper, it was shown that for every fixed integer k, there exists a polynomial-time algorithm for determining whether a P5-free graph admits a k-coloring, and finding one, if it does.
Abstract: The problem of computing the chromatic number of a P5-free graph (a graph which contains no path on 5 vertices as an induced subgraph) is known to be NP-hard. However, we show that for every fixed integer k, there exists a polynomial-time algorithm determining whether or not a P5-free graph admits a k-coloring, and finding one, if it does.
160 citations
TL;DR: This work surveys results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions in order to obtain useful results from a graph coloring formulation of his problem.
Abstract: There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions. Thus, one who wishes to obtain useful results from a graph coloring formulation of his problem must do more than just show that the problem is equivalent to the general problem of coloring a graph. If there is to be any hope, one must also obtain information about the structure of the graphs that need to be colored (D.S. Johnson [66]).
150 citations
Cites background or result from "Complexity of Coloring Graphs witho..."
...Very recently Sgall and Woeginger [113] also studied chromatic aspects of graphs without long induced paths....
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...Furthermore, in [79] the authors pose the following meta-problem: Given a nite set A, what is the computational complexity of deciding the chromatic number of A-free graphs?...
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...Another dichotomy result has been obtained by Kr al, Kratochv il, Tuza and Woeginger [79]....
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...Theorem 79 [79] The problem H-free COLOURING is solvable in polynomial time if H is an induced subgraph of P4 or of K1[P3, and NP-complete for any other H....
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TL;DR: In this article, the authors survey known results on the computational complexity of k-coloring and k-COLORING for graph classes that are characterized by one or two forbidden induced subgraphs, and also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex.
Abstract: For a positive integer k, a k-coloring of a graph inline image is a mapping inline image such that inline image whenever inline image. The COLORING problem is to decide, for a given G and k, whether a k-coloring of G exists. If k is fixed (i.e., it is not part of the input), we have the decision problem k-COLORING instead. We survey known results on the computational complexity of COLORING and k-COLORING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex.
128 citations
Posted Content•
TL;DR: In this article, the authors survey known results on the computational complexity of coloring and coloring for graph classes that are characterized by one or two forbidden induced subgraphs, and also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex.
Abstract: For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u)
eq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring of $G$ exists. If $k$ is fixed (that is, it is not part of the input), we have the decision problem $k$-Colouring instead. We survey known results on the computational complexity of Colouring and $k$-Colouring for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex.
109 citations
TL;DR: It is proved that 4-COLORING is NP-complete for P 7 -free graphs, and that 5-COLORing is NP -complete forP 6 -free graph, and the second result is the first NP-hardness result for k -COLORING P 6 - free graphs.
Abstract: A graph is H -free if it does not contain an induced subgraph isomorphic to H . We denote by P t and C t the path and the cycle on t vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for P 7 -free graphs, and that 5-COLORING is NP-complete for P 6 -free graphs. The second result is the first NP-hardness result for k -COLORING P 6 -free graphs. These two results improve all previous NP-complete results on k -coloring P t -free graphs, and almost complete the classification of complexity of k -COLORING P t -free graphs for k ? 4 and t ? 1 , leaving as the only missing case 4-COLORING of P 6 -free graphs. We expect that 4-COLORING is polynomial time solvable for P 6 -free graphs; in support of this, we describe a polynomial time algorithm for 4-COLORING P 6 -free graphs which are also banner-free, where banner is the graph obtained from C 4 by adding a new vertex and making it adjacent to exactly one vertex on the C 4 .
100 citations
Cites background from "Complexity of Coloring Graphs witho..."
...This problem has been given wide attention in recent years and much progress has been made through substantial efforts by different groups of researchers [2,3,4,7,11,13,16,17,18,21,24]....
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References
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TL;DR: It is shown that it is NP-complete to determine the chromatic index of an arbitrary graph, even for cubic graphs.
Abstract: We show that it is NP-complete to determine the chromatic index of an arbitrary graph. The problem remains NP-complete even for cubic graphs.
1,249 citations
"Complexity of Coloring Graphs witho..." refers background in this paper
...Since 3-coloring is NP–complete for line graphs by Holyer [ 4 ] and since line graphs are claw–free, K1,3-Free Coloring is NP–complete, and so is H-Free Coloring in this case....
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TL;DR: Using these results, it is able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness forPlanar generalized geography.
Abstract: We define the set of planar boolean formulae, and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is NP-complete. Using these results, we are able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness for planar generalized geography.The NP-completeness of planar node cover and planar Hamiltonian circuit and line were first proved elsewhere [M. R. Garey and D. S. Johnson, The rectilinear Steiner tree is NP-complete, SIAM J. Appl. Math., 32 (1977), pp. 826–834] and [M. R. Garey, D. S. Johnson and R. E. Tarjan, The planar Hamilton circuit problem is NP-complete, SIAM J. Comp., 5 (1976), pp. 704–714].
796 citations
"Complexity of Coloring Graphs witho..." refers background in this paper
...NP–completeness of planar formulas was first established by Lichtenstein [ 5 ], and NP–completeness of the above described restricted version can be found e.g....
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TL;DR: A well-known theorem of Ramsay (8, 9) states that to every n there exists a smallest integer g(n) such that every graph of g n contains either a set of n independent points or a complete graph of order n.
Abstract: A well-known theorem of Ramsay (8; 9) states that to every n there exists a smallest integer g(n) so that every graph of g(n) vertices contains either a set of n independent points or a complete graph of order n, but there exists a graph of g(n) – 1 vertices which does not contain a complete subgraph of n vertices and also does not contain a set of n independent points. (A graph is called complete if every two of its vertices are connected by an edge; a set of points is called independent if no two of its points are connected by an edge.)
770 citations
TL;DR: In this article, a good characterization of strongly perfect graphs is presented, which includes all comparability graphs, all triangulated graphs and all complements of triangulation graphs.
Abstract: This note presents a good characterization of a class of strongly perfect graphs which includes all comparability graphs, all triangulated graphs and all complements of triangulated graphs.
175 citations
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TL;DR: The decomposition theorem for a class of graphs is given and it is shown how this theorem suggests efficient algorithms to optimize thisclass of graphs.
149 citations