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The complexity of generalized clique covering. (English) Zbl 0685.68046
A vertex cover of a graph G is a minimal set of vertices of G such that every edge of G contains at least one vertex in the set. The paper studies the following generalized clique cover problem. Let $$c_{ij}(G)$$ denote the minimum number of complete graphs $$K_ j$$ in G such that every $$K_ i$$ in G $$(i>j)$$ contains at least one such $$K_ j$$. Given an input graph G and an integer a, decide whether $$c_{ij}(G)\leq a$$. The problem has been known to be NP-complete if $$i=j+1$$, for $$j\geq 1$$. This result is extended to arbitrary i,j, $$j>j\geq 1$$, by generalizing the construction of proof. Moreover, when the input graph is restricted to be chordal (i.e., each cycle of length greater than 3 has some chord) then the NP-completeness of the problem is shown for $$i>j\geq 2$$ by reduction from the general problem. (The restricted problem has been settled for $$i=j+1.)$$ Finally, a polynomial-time algorithm for chordal graphs, $$j=1$$, and fixed i is given whose complexity, however, is exponential in i. It is shown that even this problem becomes NP-complete if i is regarded as part of the input.
Reviewer: F.Aurenhammer

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 68R10 Graph theory (including graph drawing) in computer science 05C35 Extremal problems in graph theory
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