Good characterizations and linear time recognition for 2-probe block graphs

Lê, Van Bang; Peng, Sheng‐Lung

doi:10.1016/j.dam.2016.11.015

Cited by 1 publication

(3 citation statements)

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“…(iii) G is obtained from G[2] = (K 2 + K 1 ) ⋆ K 1 by substituting the universal vertex by a clique and the other vertices by independent sets. Proof The equivalence of (i) and (iii) has been shown in [21]. The equivalence of (i) and (iii) follows from Lemma 2.…”

Section: Theorem 12mentioning

confidence: 71%

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On the Complete Width and Edge Clique Cover Problems

Lê

Peng

2015

Lecture Notes in Computer Science

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A complete graph is the graph in which every two vertices are adjacent. For a graph G = (V, E), the complete width of G is the minimum k such that there exist k independent sets N i ⊆ V , 1 ≤ i ≤ k, such that the graph G ′ obtained from G by adding some new edges between certain vertices inside the sets N i , 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 -free bipartite graphs and polynomially solvable on 2K 2 -free bipartite graphs and on (2K 2 , C 4 )-free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on 3K 2 -free co-bipartite graphs and polynomially solvable on C 4 -free co-bipartite graphs and on (2K 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 vertices. This provides another proof for the known fact that the edge clique cover problem admits a kernel of at most 2 k vertices. Finally we determine all graphs of small complete width k ≤ 3.

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Section: Theorem 12mentioning

confidence: 71%

“…Proof The equivalence of (i) and (iii) has been shown in [21]. The equivalence of (i) and (iii) follows from Lemma 2.…”

Section: Theorem 12mentioning

confidence: 71%