Biclique Covers and Partitions

Pinto, Trevor

doi:10.37236/3595

Cited by 9 publications

(4 citation statements)

References 18 publications

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“…In the classical work of Fishburn and Hammer [8] vc CB (G) is called the bipartite degree of G. Dong and Liu showed in [6] that vc CB (K n ) = vp CB (K n ) = ⌈log n⌉, 1 and that vp CB (G) ≤ 4 for planar graphs. Pinto [14] calls vc CB (G) and vp CB (G) the local biclique cover and partition number of G, respectively, and shows that there are graphs with vc CB (G) = 2 while vp CB (G) can be arbitrary large. V. Watts investigates fractional partitions and covers in [17].…”

Section: Graph Decompositionmentioning

confidence: 99%

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Erdős–Pyber Theorem for Hypergraphs and Secret Sharing

Csirmaz¹,

Ligeti

Tardos

2014

Graphs and Combinatorics

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A new, constructive proof with a small explicit constant is given to the Erdős-Pyber theorem which says that the edges of a graph on n vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at most O(n/ log n) times. The theorem is generalized to uniform hypergraphs. Similar bounds with smaller constant value is provided for fractional partitioning both for graphs and for uniform hypergraphs. We show that these latter constants cannot be improved by more than a factor of 1.89 even for fractional covering by arbitrary complete multipartite subgraphs or subhypergraphs. In the case every vertex of the graph is connected to at least n − m other vertices, we prove the existence of a fractional covering of the edges by complete bipartite graphs such that every vertex is covered at most O(m/ log m) times, with only a slightly worse explicit constant. This result also generalizes to uniform hypergraphs. Our results give new improved bounds on the complexity of graph and uniform hypergraph based secret sharing schemes, and show the limits of the method at the same time.

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Section: Graph Decompositionmentioning

confidence: 99%

“…While graph decomposition is an interesting topic by itself [1,3,8,9,10,11,14], our interest comes mainly from a particular application in cryptography. Upper bounds on the worst case complexity of secret sharing schemes often use graph decomposition techniques [2,13,16].…”

Section: Introductionmentioning

confidence: 99%

Erdős–Pyber Theorem for Hypergraphs and Secret Sharing

Csirmaz¹,

Ligeti

Tardos

2014

Graphs and Combinatorics

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“…The biclique partition also has a strong connection with biclique cover number (bc), where the edges of a graph are covered by bicliques but not necessarily disjointed. Pinto [19] shows that bp(G) ≤ 1 2 (3 bc(G) − 1). Moreover, finding a biclique partition with the minimum size is NP-complete even on the graphs without 4-cycles [14].…”

Section: Introductionmentioning

confidence: 99%

Finding Bipartite Partitions on Co-Chordal Graphs

Lyu¹,

Hicks²

2022

Preprint

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In this paper, we show that the biclique partition number (bp) of a co-chordal (complementary graph of chordal) graph G = (V, E) is less than the number of maximal cliques (mc) of its complementary graph: a chordal graph G c = (V, E c ). We first provide a general framework of the "divided and conquer" heuristic of finding minimum biclique partition on co-chordal graphs based on clique trees. Then, an O[|V |(|V | + |E c |)]-time heuristic is proposed by applying lexicographic breadth-first search. Either heuristic gives us a biclique partition of G with a size of mc(G c ) − 1. Eventually, we prove that our heuristic can solve the minimum biclique partition problem on G exactly if its complement G c is chordal and clique vertex irreducible. We also show that mc(G c ) − 2 ≤ bp(G) ≤ mc(G c ) − 1 if G is a split graph.

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“…In the local covering number of H with covering class G one also tries to cover the edges of H with graphs from G but now minimizes the largest number of graphs in the covering containing a common vertex of H. We are aware of only two local covering numbers in the literature: The bipartite degree introduced by Fishburn and Hammer [21] is the local covering number where the covering class is the class of complete bipartite graphs. It was rediscovered by Dong and Liu [16] as the local biclique cover number, and recently it has been studied in comparison with its global variant by Pinto [49]. The local clique cover number is another local covering number, where the covering class is the class of complete graphs.…”

Section: Introductionmentioning

confidence: 99%

Three ways to cover a graph

Knauer

Ueckerdt

2016

Discrete Mathematics

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International audienceWe consider the problem of covering an input graph H with graphs from a fixed covering class G. The classical covering number of H with respect to G is the minimum number of graphs from G needed to cover the edges of H without covering non-edges of H. We introduce a unifying notion of three covering parameters with respect to G, two of which are novel concepts only considered in special cases before: the local and the folded covering number. Each parameter measures " how far " H is from G in a different way. Whereas the folded covering number has been investigated thoroughly for some covering classes, e.g., interval graphs and planar graphs, the local covering number has received little attention. We provide new bounds on each covering number with respect to the following covering classes: linear forests, star forests, caterpillar forests, and interval graphs. The classical graph parameters that result this way are interval number, track number, linear arboricity, star arboricity, and caterpillar arboricity. As input graphs we consider graphs of bounded degeneracy, bounded degree, bounded tree-width or bounded simple tree-width, as well as outerplanar, planar bipartite, and planar graphs. For several pairs of an input class and a covering class we determine exactly the maximum ordinary, local, and folded covering number of an input graph with respect to that covering class

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Biclique Covers and Partitions

Cited by 9 publications

References 18 publications

Erdős–Pyber Theorem for Hypergraphs and Secret Sharing

Erdős–Pyber Theorem for Hypergraphs and Secret Sharing

Finding Bipartite Partitions on Co-Chordal Graphs

Three ways to cover a graph

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