an:00681639 Zbl 0809.05034 Lagergren, Jens The nonexistence of reduction rules giving an embedding into a $$k$$-tree EN Discrete Appl. Math. 54, No. 2-3, 219-223 (1994). 00022489 1994
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05C05 05C10 68R10 $$k$$-tree embedding problem; embedding; treewidth; tree-decomposition; triggered reduction rule; linear time algorithms 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 \textit{S. Arnborg}, \textit{D. G. Corneil} and \textit{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 \textit{H. L. Bodlaender} [A linear time algorithm for finding tree- decompositions of small treewidth, Proc. 25th Ann. Symp. Theor. Comp. Sci., 226-234 (1993)]. H.Bodlaender (Utrecht) Zbl 0611.05022