Ki-covers I: Complexity and polytopes

Abstract A K i in a graph is a complete subgraph of size i . A K i -cover of a graph G ( V , E is a set C of K i − 1 's of G such that every K i in G contains at least one K i − 1 in C. Thus a K 2 -cover is a vertex cover. The problem of determining whether a graph has a K i -cover ( i ⩾ 2) of cardinality ⩽ k is shown to be NP-complete for graphs in general. For chordal graphs with fixed maximum clique size, the problem is polynomial; however, it is NP-complete for arbitrary chordal graphs when i ⩾ 3. The NP-completeness results motivate the examination of some facets of the corresponding polytope. In particular we show that various induced subgraphs of G define facets of the K i -cover polytope. Further results of this type are also produced for the K 3 -cover polytope. We conclude by describing polynomial algorithms for solving the separation problem for some classes of facets of the K i -cover polytope.

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