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On the complete width and edge clique cover problems

Author

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  • Van Bang Le

    (Universität Rostock)

  • Sheng-Lung Peng

    (National Dong Hwa University)

Abstract

A complete graph is the graph in which every two vertices are adjacent. For a graph $$G=(V,E)$$ G = ( V , E ) , the complete width of G is the minimum k such that there exist k independent sets $$\mathtt {N}_i\subseteq V$$ N i ⊆ V , $$1\le i\le k$$ 1 ≤ i ≤ k , such that the graph $$G'$$ G ′ obtained from G by adding some new edges between certain vertices inside the sets $$\mathtt {N}_i$$ N i , $$1\le i\le k$$ 1 ≤ i ≤ k , is a complete graph. The complete width problem is to decide whether the complete width of a given graph is at most k or not. In this paper we study the complete width problem. We show that the complete width problem is NP-complete on $$3K_2$$ 3 K 2 -free bipartite graphs and polynomially solvable on $$2K_2$$ 2 K 2 -free bipartite graphs and on $$(2K_2,C_4)$$ ( 2 K 2 , C 4 ) -free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on $$\overline{3K_2}$$ 3 K 2 ¯ -free co-bipartite graphs and polynomially solvable on $$C_4$$ C 4 -free co-bipartite graphs and on $$(2K_2, C_4)$$ ( 2 K 2 , C 4 ) -free graphs. We also give a characterization for k-probe complete graphs which implies that the complete width problem admits a kernel of at most $$2^k$$ 2 k vertices. This provides another proof for the known fact that the edge clique cover problem admits a kernel of at most $$2^k$$ 2 k vertices. Finally we determine all graphs of small complete width $$k\le 3$$ k ≤ 3 .

Suggested Citation

  • Van Bang Le & Sheng-Lung Peng, 2018. "On the complete width and edge clique cover problems," Journal of Combinatorial Optimization, Springer, vol. 36(2), pages 532-548, August.
  • Handle: RePEc:spr:jcomop:v:36:y:2018:i:2:d:10.1007_s10878-016-0106-9
    DOI: 10.1007/s10878-016-0106-9
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    References listed on IDEAS

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    1. Renhua Li & Leonie U Hempel & Tingbo Jiang, 2015. "A Non-Parametric Peak Calling Algorithm for DamID-Seq," PLOS ONE, Public Library of Science, vol. 10(3), pages 1-12, March.
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