Connected matchings in chordal bipartite graphs

Jobson, Adam S.; Kézdy, André E.; White, Susan C.

doi:10.1016/j.disopt.2014.06.003

Cited by 3 publications

References 15 publications

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Hadwiger Number of Graphs with Small Chordality

Golovach¹,

Heggernes²,

Hof³

et al. 2015

SIAM J. Discrete Math.

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The Hadwiger number of a graph G is the largest integer h such that G has the complete graph K h as a minor. We show that the problem of determining the Hadwiger number of a graph is NP-hard on co-bipartite graphs, but can be solved in polynomial time on cographs and on bipartite permutation graphs. We also consider a natural generalization of this problem that asks for the largest integer h such that G has a minor with h vertices and diameter at most s. We show that this problem can be solved in polynomial time on AT-free graphs when s ≥ 2, but is NP-hard on chordal graphs for every fixed s ≥ 2.

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Hadwiger Number of Graphs with Small Chordality

Golovach¹,

Heggernes²,

Hof³

et al. 2015

SIAM J. Discrete Math.

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Multitasking Capacity: Hardness Results and Improved Constructions

Alon¹,

Cohen²,

Griffiths³

et al. 2020

SIAM J. Discrete Math.

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Multitasking Capacity: Hardness Results and Improved Constructions

Alon¹,

Cohen²,

Griffiths³

et al. 2018

Preprint

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We consider the problem of determining the maximal α ∈ (0, 1] such that every matching M of size k (or at most k) in a bipartite graph G contains an induced matching of size at least α|M |. This measure was recently introduced in [ARS + 17] and is motivated by connectionist models of cognition as well as modeling interference in wireless and communication networks.We prove various hardness results for computing α either exactly or approximately. En route to our results, we also consider the maximum connected matching problem: determining the largest matching N in a graph G such that every two edges in N are connected by an edge. We prove a nearly optimal n 1−ε hardness of approximation result (under randomized reductions) for connected matching in bipartite graphs (with both sides of cardinality n). Towards this end we define bipartite half-covers: A new combinatorial object that may be of independent interest. To the best of our knowledge, the best previous hardness result for the connected matching problem was some constant β > 1.Finally, we demonstrate the existence of bipartite graphs with n vertices on each side of average degree d, that achieve α = 1/2 − ε for matchings of size sufficiently smaller than n/poly(d). This nearly matches the trivial upper bound of 1/2 on α which holds for any graph containing a path of length 3.

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Connected matchings in chordal bipartite graphs

Cited by 3 publications

References 15 publications

Hadwiger Number of Graphs with Small Chordality

Hadwiger Number of Graphs with Small Chordality

Multitasking Capacity: Hardness Results and Improved Constructions

Multitasking Capacity: Hardness Results and Improved Constructions

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