E. Boros, V. Gurvich, and S. Hougardy. Recursive generation of partitionable graphs. J. Graph Theory, 41(4):259--285, 2002.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....smallest partitionable graph C[2; 2; 2; 2] considered in this paper is not minimal circular imperfect. 2 Circular partitionable graphs and the main result In the study of partitionable graphs, some recursive constructions of sub families of partitionable graphs are discussed in the literature [6,4]. The class of circular partitionable graphs was introduced by Chvatal, Graham, Perold and Whitesides [6] For two sets of integers X; Y , let X Y denote the set fx y : x 2 X; y 2 Y g. If X = fxg is a singleton, we write x Y instead of fxg Y . Suppose m i 2 (i = 1; 2; 2r) are ....

E. Boros, V. Gurvich, and S. Hougardy. Recursive generation of partitionable graphs. J. Graph Theory, 41(4):259--285, 2002.

....subgraph of with vertex set V ( n fvg and edge set ffx; yg j fx; yg 2 V ( x 6= v; y 6= vg. An edge e of a graph is said to be an critical edge if and only if ( e) It is known that in a partitionable graph , a maximum clique intersects at least 2 2 other maximum cliques. Following [3], we call critical clique any maximum clique which intersects exactly 2 2 other maximum cliques. Seb [13] proved that a maximum clique Q of a partitionable graph is a critical clique if and only if the critical edges in Q form a spanning tree of Q. Let be any partitionable graph with a critical ....

....Seb [13] proved that a maximum clique Q of a partitionable graph is a critical clique if and only if the critical edges in Q form a spanning tree of Q. Let be any partitionable graph with a critical clique Q and let T be the tree made of the critical edges of Q. Boros, Gurvich and Hougardy [3] noticed that for every edge e of T , there exist two maximum cliques Q 0 and Q 00 such that Q Q 0 is equal to one of the two connected components of T e and that Q Q 00 is equal to the other connected component of T e. Thus if e is any edge containing a leaf of T , one of the maximum ....

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E. Boros, V. Gurvich, and S. Hougardy, Recursive generation of partitionable graphs, Tech. Report RR 1099, Rutcor Research Report, June 1999.

....1998, Bacs, Boros, Gurvich, Ma ray and Preissmann extended this result to any graph of the second one [1] In 1996, Seb proved that no graph of the rst one is a counter example [10] These two constructions remained the only known ones for quite a long time. Recently, Boros, Gurvich and Hougardy [3] managed to de ne a construction extending the rst one of Chvtal, Graham, Perold and Whitesides. It is unknown whether there is a counter example to the SPGC in this wider class. Due to the initials of these three authors, we call BGH graphs, the graphs of this class. A graph is said to be ....

....are ffi; i 1g; i 2 Z 1 g. If Q is any maximum clique of a partitionable graph, then there is a unique maximum stable set S(Q) which does not meet Q. Let be any partitionable graph with a critical clique Q and let T be the tree made of the 2 critical edges of Q. Boros, Gurvich and Hougardy [3] noticed that for every edge e of T , there exists a unique pair of maximum cliques Q 0 and Q 00 such that Q Q 0 is equal to one of the two connected components of T e and that Q Q 00 is equal to the other connected component of T e. Thus if e is any edge containing a leaf of T , one of ....

[Article contains additional citation context not shown here]

E. Boros, V. Gurvich, and S. Hougardy, Recursive generation of partitionable graphs, Tech. Report RR 1099, Rutcor Research Report, June 1999.

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