The complexity of generalized clique covering.

*(English)*Zbl 0685.68046A 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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\textit{D. G. Corneil} and \textit{J. Fonlupt}, Discrete Appl. Math. 22, No. 2, 109--118 (1989; Zbl 0685.68046)

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