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Generalized vertex covering in interval graphs. (English) Zbl 0766.05082
The following decision problem is treated in the paper: Given a graph $$G$$ and two integers $$i\geq 2$$, $$k\geq 1$$, is it possible to find $$k$$ vertices of the graph such that any complete subgraph of $$G$$ with $$i$$ vertices contains (at least) one of these distinguished vertices. This problem is NP-complete even for chordal graphs. In this paper a greedy linear-time algorithm for interval graphs is presented — in fact the algorithm solves the corresponding optimization problem. In particular the minimum vertex cover problem $$(i=2)$$ and the minimum feedback vertex set problem $$(i=3)$$ can be solved in linear time for interval graphs.

##### MSC:
 05C85 Graph algorithms (graph-theoretic aspects) 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 68R10 Graph theory (including graph drawing) in computer science 68Q25 Analysis of algorithms and problem complexity
##### Keywords:
greedy linear-time algorithm; interval graphs; vertex cover
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##### References:
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