The Biclique Covering Number of Grids

Guo, Krystal; Huynh, Tony; Macchia, Marco

doi:10.37236/8316

Cited by 3 publications

(2 citation statements)

References 9 publications

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“…Especially interesting should be classes where an exponential separation between the best and an HMU-realisation exists. We have seen the example of complete graphs; a more complex class are the grid graphs ( [16]).…”

Section: Discussionmentioning

confidence: 99%

Transforming Quantified Boolean Formulas Using Biclique Covers

Kullmann

Shukla

2023

Lecture Notes in Computer Science

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We introduce the global conflict graph of DQCNFs (dependency quantified conjunctive normal forms), recording clashes between clauses on such universal variables on which all existential variables depend (called “global variables”). The biclique covers of this graph correspond to the eligible clause-slices of the DQCNF which consider only the global variables. We show that all such slices yield satisfiability-equivalent variations. This opens the possibility to realise this slice using as few global variables as possible. We give basic theoretical results and first supporting experimental data.

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Section: Discussionmentioning

confidence: 99%

Transforming Quantified Boolean Formulas Using Biclique Covers

Kullmann

Shukla

2023

Lecture Notes in Computer Science

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“…Despite of the hardness of solving MBCP on general graphs, the biclique cover number of some well-structured graphs are known, such as complete graphs and 2nvertex crown graphs [5], and grid graphs [13] and there exists polynomial time algorithm to solve the problem exactly on some special classes of graphs such as C4-free graphs [24], and domino-free graphs [2].…”

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confidence: 99%

Maximal Clique and Edge-Ranking Bounds of Biclique Cover Number

Lyu¹,

Hicks²

2023

Preprint

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The biclique cover number (bc) of a graph G is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover all the edges of the graph. In this paper, we show that the biclique cover number (bc) of a graph G is no less than ⌈log 2 (mc(G c ))⌉, where mc(G c ) is the number of maximal cliques of the complementary graph G c , i.e., the number of maximal independent sets of G. We also show that bc(G) ≤ χ ′ r (TKc ) where G is a co-chordal graph such that each vertex is in at most two maximal independent sets and χ ′ r (TKc ) is the optimal edge-ranking number of a clique tree of G c . By identifying the new lower and upper bounds of bc(G), we prove that bc(G) = ⌈log 2 (mc(G c ))⌉ if G c is a path or windmill graph.

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