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$$k$$-NLC graphs and polynomial algorithms. (English) Zbl 0812.68106
Summary: We introduce the class of $$k$$-node label controlled (NLC) graphs and the class of $$k$$-NLC trees. Each $$k$$-NLC graph is an undirected tree- structured graph, where $$k$$ is a positive integer. The class of $$k$$-NLC trees is a proper subset of the class of $$k$$-NLC graphs. Both classes include many interesting graph families. For instance, each partial $$k$$- tree is a $$(2^{k+ 1}-1)$$-NLC tree and each co-graph is a 1-NLC graph. Furthermore, we introduce a very general method for the design of polynomial algorithms for NP-complete graph problems, where the input graphs are restricted to tree-structured graphs. We exemplify our method with the simple max-cut problem and the Hamiltonian circuit property on $$k$$-NLC graphs.

##### MSC:
 68R10 Graph theory (including graph drawing) in computer science 68Q25 Analysis of algorithms and problem complexity 05C78 Graph labelling (graceful graphs, bandwidth, etc.) 05C05 Trees
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