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The nonexistence of reduction rules giving an embedding into a $$k$$-tree. (English) Zbl 0809.05034
This paper considers the $$k$$-tree embedding problem: given a graph $$G= (V,E)$$, is $$G$$ a subgraph of a $$k$$-tree and if so, find such an embedding. This is equivalent to determining whether $$G$$ has treewidth at most $$k$$, and if so, finding a tree-decomposition with optimal width of $$G$$.
A locally triggered reduction rule, applied to a graph $$G$$, takes a vertex $$v$$ which belongs to a subgraph of $$G$$ with a certain well- described structure, connects all neighbors of $$v$$, and removes $$v$$. For $$k= 2$$, $$k= 3$$, a set of such reduction rules exist, such that the rule can be applied repeatedly until the empty graph results, if and only if $$G$$ is in the class to be recognized. These sets can be used to obtain linear time algorithms for the $$k$$-embedding problem.
This paper shows that for $$k=4$$, such reduction rules do not exist. More general sets of reduction rules for the $$k$$-tree embedding problem (for arbitrary fixed $$k$$) are known to exist, see S. Arnborg, D. G. Corneil and A. Proskurowski [SIAM J. Algebraic Discrete Methods 8, 277-284 (1987; Zbl 0611.05022)]. Also, linear time algorithms for the $$k$$-embedding problem, using a different method, are known ($$k$$ fixed), see H. L. Bodlaender [A linear time algorithm for finding tree- decompositions of small treewidth, Proc. 25th Ann. Symp. Theor. Comp. Sci., 226-234 (1993)].

##### MSC:
 05C05 Trees 05C10 Planar graphs; geometric and topological aspects of graph theory 68R10 Graph theory (including graph drawing) in computer science
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##### References:
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