Tight bounds on maximal and maximum matchings

Biedl, Thérèse; Demaine, Erik D.; Duncan, Christian A.; Fleischer, Rudolf; Kobourov, Stephen G.

doi:10.1016/j.disc.2004.05.003

Cited by 75 publications

(56 citation statements)

References 12 publications

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“…As ( ′ , ) = ( , ) for ∈ [3] and 4 ( ′ , ) < 4 ( , ) we obtain a contradiction to the lexicographical minimality of ( 1 ( , ), 2…”

Section: Proof Of Theoremmentioning

confidence: 89%

“…Various lower bounds on the matching number for regular graphs have appeared in the literature. For example, Biedl et al [2] proved that if is a cubic graph, then ′ ( ) ≥ (4 − 1)∕9. This result was generalized to regular graphs of higher degree by Henning and Yeo [7] (see also, O and West [14]).…”

Section: Motivationmentioning

confidence: 99%

“…If is odd, then = ( − 1)∕( ( 2 − 3)) and = ( 2 − − 2)∕( ( 2 − 3)), which implies the following (as 0 < < and ≥ 3). 2…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…(1 − ) 2 )| ( )| + ( 1 + (1 − )2 )| ( )| − 1 , 1 − (1 − ) 2 , 2 .holds for all ∈  ′ . Therefore,( 1 + (1 − ) 2 , 1 + (1 − ) 2 ) is -tight letting 1 +(1− ) 2 , 1 +(1− ) 2 = 1 , 1 + (1 − ) 2 , 2 .…”

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Tight lower bounds on the matching number in a graph with given maximum degree

Henning

Yeo

2018

Journal of Graph Theory

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Let k≥3. We prove the following three bounds for the matching number, α′false(Gfalse), of a graph, G, of order n size m and maximum degree at most k. If k is odd, then α′false(Gfalse)≥false(k−1kfalse(k2−3false)false)n+false(k2−k−2kfalse(k2−3false)false)m−k−1k(k2−3). If k is even, then α′false(Gfalse)≥nk(k+1)+mk+1−1k. If k is even, then α′false(Gfalse)≥false(k+2k2+k+2false)m−false(k−2k2+k+2false)n−k+2k2+k+2. In this article, we actually prove a slight strengthening of the above for which the bounds are tight for essentially all densities of graphs. The above three bounds are in fact powerful enough to give a complete description of the set Lk of pairs (γ,β) of real numbers with the following property. There exists a constant K such that α′false(Gfalse)≥γn+βm−K for every connected graph G with maximum degree at most k, where n and m denote the number of vertices and the number of edges, respectively, in G. We show that Lk is a convex set. Further, if k is odd, then Lk is the intersection of two closed half‐spaces, and there is exactly one extreme point of Lk, while if k is even, then Lk is the intersection of three closed half‐spaces, and there are precisely two extreme points of Lk.

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“…As ( ′ , ) = ( , ) for ∈ [3] and 4 ( ′ , ) < 4 ( , ) we obtain a contradiction to the lexicographical minimality of ( 1 ( , ), 2…”

Section: Proof Of Theoremmentioning

confidence: 89%

Section: Motivationmentioning

confidence: 99%

“…If is odd, then = ( − 1)∕( ( 2 − 3)) and = ( 2 − − 2)∕( ( 2 − 3)), which implies the following (as 0 < < and ≥ 3). 2…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Tight lower bounds on the matching number in a graph with given maximum degree

Henning

Yeo

2018

Journal of Graph Theory

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“…We continue this process for N time steps, and at each time step all tasks are required to migrate. We know that a perfect matching will exist at every time step from [8], leaving K 2 total migrations. In contrast, had the edges in all future time steps been known in advance, then the matching remaining in the last time step could have been chosen first, resulting in zero total migrations.…”

Section: Motivating Examplesmentioning

confidence: 99%

Kinetically stable task assignment for networks of microservers

Abrams

Chen

Guibas

et al. 2006

Proceedings of the Fifth International Conference on Information Processing in Sensor Networks - IPSN '06

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This paper studies task assignment in a network of resource constrained computing platforms (called microservers). A task is an abstraction of a computational agent or data that is hosted by the microservers. For example, in an object tracking scenario, a task represents a mobile tracking agent, such as a vehicle location update computation, that runs on microservers, which can receive sensor data pertaining to the object of interest. Due to object motion, the microservers that can observe a particular object change over time and there is overhead involved in migrating tasks among microservers. Furthermore, communication, processing, or memory constraints, allow a microserver to only serve a limited number of objects at the same time. Our overall goal is to assign tasks to microservers so as to minimize the number of migrations, and thus be kinetically stable, while guaranteeing that as many tasks as possible are monitored at all times. When the task trajectories are known in advance, we show that this problem is NP-complete (even over just two time steps), has an integrality gap of at least 2, and can be solved optimally in polynomial time if we allow tasks to be assigned fractionally. When only probabilistic information about future movement of the tasks is known, we propose two algorithms: a multi-commodity flow based algorithm and a maximum matching algorithm. We use simulations to compare the performance of these algorithms against the optimum task allocation strategy.

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Balloons, cut‐edges, matchings, and total domination in regular graphs of odd degree

Suil¹,

West²

2009

Journal of Graph Theory

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A balloon in a graph G is a maximal 2-edge-connected subgraph incident to exactly one cut-edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut-edges, and let α ′ (G) be the maximum size of a matching. Let F n,r be the family of connected (2r + 1)-regular graphs with n vertices, and let b = max{b(G) : G ∈ F n,r }. For G ∈ F n,r , we prove the sharp inequalities c(G) ≤ r(n−2)−2 2r 2 +2r−14r 2 +4r−2 , we obtain a simple proof of the bound α ′ (G) ≥ proved by Henning and Yeo. For each of these bounds and each r, the approach using balloons allows us to determine the infinite family where equality holds. For the total domination number γ t (G) of a cubic graph, we prove γ t (G) ≤ 3 for cubic graphs, this improves the known inequality γ t (G) ≤ α ′ (G).

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Tight bounds on maximal and maximum matchings

Cited by 75 publications

References 12 publications

Tight lower bounds on the matching number in a graph with given maximum degree

Tight lower bounds on the matching number in a graph with given maximum degree

Kinetically stable task assignment for networks of microservers

Balloons, cut‐edges, matchings, and total domination in regular graphs of odd degree

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