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10
On the Structure of (P_5,Gem)Free Graphs
, 2002
"... We give a complete structure description of (P 5 ,gem)free graphs. By the results of a related paper, this implies bounded clique width for this graph class. Hereby, as usual, the P 5 is the induced path with five vertices a; b; c; d; e and four edges ab; bc; cd; de, and the gem consists of a P ..."
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Cited by 12 (3 self)
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We give a complete structure description of (P 5 ,gem)free graphs. By the results of a related paper, this implies bounded clique width for this graph class. Hereby, as usual, the P 5 is the induced path with five vertices a; b; c; d; e and four edges ab; bc; cd; de, and the gem consists of a P 4 a; b; c; d with edges ab; bc; cd plus a universal vertex e adjacent to a; b; c; d.
On Minimal Prime Extensions of a FourVertex Graph in a Prime Graph
, 2002
"... In a finite undirected graph G = (V; E), a homogeneous set is a set U V of at least two vertices such that every vertex in V n U is either adjacent to all vertices of U or nonadjacent to all of them. A graph is prime if it does not have a homogeneous set. We investigate the minimum... ..."
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Cited by 6 (1 self)
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In a finite undirected graph G = (V; E), a homogeneous set is a set U V of at least two vertices such that every vertex in V n U is either adjacent to all vertices of U or nonadjacent to all of them. A graph is prime if it does not have a homogeneous set. We investigate the minimum...
New Graph Classes of Bounded CliqueWidth
, 2003
"... Cliquewidth of graphs is a major new concept with respect to efficiency of graph algorithms; it is known that every problem expressible in a certain kind of Monadic Second Order Logic called LinEMSOL(τ1,L ) by Courcelle, Makowsky and Rotics, is lineartime solvable on any graph class with bounded c ..."
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Cited by 5 (0 self)
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Cliquewidth of graphs is a major new concept with respect to efficiency of graph algorithms; it is known that every problem expressible in a certain kind of Monadic Second Order Logic called LinEMSOL(τ1,L ) by Courcelle, Makowsky and Rotics, is lineartime solvable on any graph class with bounded cliquewidth for which a kexpression for the input graph can be constructed in linear time. The notion of cliquewidth...
P_6 and trianglefree graph revisited: structure and bounded cliquewidth
 Discrete Mathematics and Theoretical Computer Science
, 2006
"... The Maximum Weight Stable Set (MWS) Problem is one of the fundamental problems on graphs. It is wellknown to be NPcomplete for trianglefree graphs, and Mosca has shown that it is solvable in polynomial time when restricted to P6 and trianglefree graphs. We give a complete structure analysis of ..."
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Cited by 5 (0 self)
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The Maximum Weight Stable Set (MWS) Problem is one of the fundamental problems on graphs. It is wellknown to be NPcomplete for trianglefree graphs, and Mosca has shown that it is solvable in polynomial time when restricted to P6 and trianglefree graphs. We give a complete structure analysis of (nonbipartite) P6 and trianglefree graphs which are prime in the sense of modular decomposition. It turns out that the structure of these graphs is simple implying bounded cliquewidth and thus, efficient algorithms exist for all problems expressible in terms of Monadic Second Order Logic with quantification only over vertex predicates. The problems Vertex Cover, MWS, Maximum Clique, Minimum Dominating Set, Steiner Tree, and Maximum Induced Matching are among them. Our results improve the previous one on the MWS problem by Mosca with respect to structure and time bound but also extends a previous result by Fouquet, Giakoumakis, and Vanherpe which have shown that bipartite P6free graphs have bounded cliquewidth. Moreover, it covers a result by Randerath, Schiermeyer, and Tewes on polynomial time 3colorability of P6 and trianglefree graphs.
A Survey on the Computational Complexity of Colouring Graphs with Forbidden Subgraphs, arXiv:1407.1482 [cs.CC
"... Abstract. For a positive integer k, a kcolouring of a graph G = (V,E) is a mapping c: V → {1, 2,..., k} such that c(u) 6 = c(v) whenever uv ∈ E. The COLOURING problem is to decide, for a given G and k, whether a kcolouring of G exists. If k is fixed (that is, it is not part of the input), we have ..."
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Cited by 3 (2 self)
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Abstract. For a positive integer k, a kcolouring of a graph G = (V,E) is a mapping c: V → {1, 2,..., k} such that c(u) 6 = c(v) whenever uv ∈ E. The COLOURING problem is to decide, for a given G and k, whether a kcolouring of G exists. If k is fixed (that is, it is not part of the input), we have the decision problem kCOLOURING instead. We survey known results on the computational complexity of COLOURING and kCOLOURING 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. 1
Cliquewidth for fourvertex forbidden subgraphs
 Theory Comput. Syst
, 2006
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On Algorithms for (P5,Gem)Free Graphs
, 2003
"... A graph is (P5,gem)free, when it does not contain P5 (an induced path with five vertices) or a gem (a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path P4) as an induced subgraph. We present O(n 2) time recognition algorithms for chordal gemfree a ..."
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Cited by 2 (1 self)
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A graph is (P5,gem)free, when it does not contain P5 (an induced path with five vertices) or a gem (a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path P4) as an induced subgraph. We present O(n 2) time recognition algorithms for chordal gemfree and for (P5,gem)free graphs. Using a characterization of (P5,gem)free graphs by their prime graphs with respect to modular decomposition and their modular decomposition trees [6], we give linear time algorithms for the following NPcomplete problems on (P5,gem)free graphs: Minimum
New Graph Classes of Bounded CliqueWidth II
, 2003
"... Cliquewidth of graphs is a major new concept with respect to eciency of graph algorithms. It is known that every problem expressible in a certain kind of Monadic Second Order Logic, called LinEMSOL( 1;L ) by Courcelle, Makowsky and Rotics, is lineartime solvable on any graph class with bounded ..."
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Cliquewidth of graphs is a major new concept with respect to eciency of graph algorithms. It is known that every problem expressible in a certain kind of Monadic Second Order Logic, called LinEMSOL( 1;L ) by Courcelle, Makowsky and Rotics, is lineartime solvable on any graph class with bounded cliquewidth for which a kexpression for the input graph can be constructed in linear time. The notion of cliquewidth extends the one of treewidth, since bounded treewidth implies bounded cliquewidth. We give