On the smallest sets blocking simple perfect matchings in a convex geometric graph

Keller, Chaya; Perles, Micha A.

doi:10.1007/s11856-011-0090-9

Cited by 13 publications

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“…We show that these two families of blockers are actually equal. The proof of Theorem 1.6 is rather short, and in particular, it yields a short proof for Theorem 1.5 -much shorter than the proof presented in [3].…”

Section: Introductionmentioning

confidence: 93%

“…In [3] we provided a complete characterization of the blockers for the family M of SPMs of G. The characterization involves the notion of a caterpillar tree [6]. Definition 1.4.…”

Section: Introductionmentioning

confidence: 99%

“…A longest (simple) path in a caterpillar T is called a spine of T . 3]). Let V be the set of vertices of G (viewed as the vertex set of a convex polygon P in the plane), labelled cyclically from 0 to 2m − 1.…”

Section: Introductionmentioning

confidence: 99%

“…

Let G be a complete convex geometric graph on 2m vertices, and let F be a family of subgraphs of G. A blocker for F is a set of edges, of smallest possible size, that meets every element of F . In [3] we gave an explicit description of all blockers for the family of simple perfect matchings (SPMs) of G. In this paper we show that the family of simple Hamiltonian paths (SHPs) in G has exactly the same blockers as the family of SPMs. Our argument is rather short, and provides a much simpler proof of the result of [3].

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Blockers for Simple Hamiltonian Paths in Convex Geometric Graphs of Even Order

Keller

Perles

2017

Discrete Comput Geom

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Section: Introductionmentioning

confidence: 93%

“…In [3] we provided a complete characterization of the blockers for the family M of SPMs of G. The characterization involves the notion of a caterpillar tree [6]. Definition 1.4.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

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confidence: 99%

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Blockers for Simple Hamiltonian Paths in Convex Geometric Graphs of Even Order

Keller

Perles

2017

Discrete Comput Geom

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“…In [3] we considered the 'even' case G = CK(2m), and provided a complete characterization of the blockers for the family M of simple (i.e., non-crossing) perfect matchings (SPMs) of G. We described the blockers as certain simple caterpillar subtrees of G of size m. (Roughly speaking, a caterpillar is a tree whose derivative is a path. See [7] for the exact definition of caterpillars.)…”

Section: Introductionmentioning

confidence: 99%

Blockers for Simple Hamiltonian Paths in Convex Geometric Graphs of Odd Order

Keller

Perles

2019

Discrete Comput Geom

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Let G be a complete convex geometric graph, and let F be a family of subgraphs of G. A blocker for F is a set of edges, of smallest possible size, that has an edge in common with every element of F . In [4] we gave an explicit description of all blockers for the family of simple (i.e., non-crossing) Hamiltonian paths (SHPs) in G in the 'even' case |V (G)| = 2m. It turned out that all the blockers are simple caterpillar trees of a certain class. In this paper we give an explicit description of all blockers for the family of SHPs in the 'odd' case |V (G)| = 2m − 1. In this case, the structure of the blockers is more complex, and in particular, they are not necessarily simple. Correspondingly, the proof is more complicated.Our main theorem is the following:Theorem. Let G = CK(2m − 1), and let H be the family of simple Hamiltonian paths in E(G). Any blocker for H consists of m edges in m boundary-consecutive directions, and up to cyclical rotation by 0 ≤ k ≤ 2m − 2, it has one of the following two forms. Moreover, all the sets described below (Class A and Class B) are indeed blockers. Class A. Blockers that contain a consecutive boundary path. The edges of the blocker are parallel to the boundary edges [0, 1], [1, 2], . . . , [m − 1, m]. They consist of three parts: 1. The boundary path BP = α, α + 1, . . . , m − δ , for some α, δ ≥ 0 with 0 ≤ α + δ ≤ m − 2. The length of BP is m − α − δ, and ranges between 2 and m. This path is illustrated in the left part of Figure 1. 2. The edges u i = [i − 1 − ǫ i , i + ǫ i ], 1 ≤ i ≤ α (where indices are taken modulo 2m − 1), for ǫ 1 > ǫ 2 . . . > ǫ α > 0, α − i + 1 ≤ ǫ i ≤ m − δ − i − 1. (These are the edges parallel to [0, 1], [1, 2], . . . , [α − 1, α]).

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Ramsey-Type Problems for Geometric Graphs

Károlyi

2012

Thirty Essays on Geometric Graph Theory

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On the smallest sets blocking simple perfect matchings in a convex geometric graph

Cited by 13 publications

References 1 publication

Blockers for Simple Hamiltonian Paths in Convex Geometric Graphs of Even Order

Blockers for Simple Hamiltonian Paths in Convex Geometric Graphs of Even Order

Blockers for Simple Hamiltonian Paths in Convex Geometric Graphs of Odd Order

Ramsey-Type Problems for Geometric Graphs

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