Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures

Schmerl, James H.; Trotter, William T.

doi:10.1016/0012-365x(93)90516-v

Cited by 125 publications

(118 citation statements)

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“…These notions are defined in terms of critical vertices. A vertex x of a prime tournament T is critical [10] if T − x is not prime. The set of noncritical vertices of a prime tournament T was introduced in [9].…”

Section: Partially Critical Tournaments and The Class Tmentioning

confidence: 99%

“…Figure 3 illustrates a tournament obtained from such information. We refer to [10,Discussion] for more details about this purpose. Figure 3);…”

Section: The Tournament T Is Exactly Determined By Giving In Additiomentioning

confidence: 99%

“…Schmerl and W.T. Trotter in [10], we introduce the tournaments T 2n+1 and U 2n+1 defined on N 2n+1 , where n ≥ 2 , as follows:…”

Section: Critical Tournaments and Tournaments Omitting Wmentioning

confidence: 99%

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The prime tournaments $T$ with $\mid\! W_{5}(T) \!\mid = \mid\! T \!\mid -2$

Belkhechine¹,

Boudabbous²,

Hzami³

2015

Turk J Math

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We consider a tournament T = (V, A) . For X ⊆ V , the subtournament of T induced by X is TThe trivial modules of T are ∅ , {x}(x ∈ V ) , and V . A tournament is prime if all its modules are trivial.For n ≥ 2 , W2n+1 denotes the unique prime tournament defined on {0, . . . , 2n} such that W2n+1[{0, . . . , 2n − 1}] is the usual total order. Given a prime tournament T , W5(T ) denotes the set ofIn this article, we characterize the prime tournaments T such that | W5(T ) |=| V | −2 .

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Section: Partially Critical Tournaments and The Class Tmentioning

confidence: 99%

“…Figure 3 illustrates a tournament obtained from such information. We refer to [10,Discussion] for more details about this purpose. Figure 3);…”

Section: The Tournament T Is Exactly Determined By Giving In Additiomentioning

confidence: 99%

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The prime tournaments $T$ with $\mid\! W_{5}(T) \!\mid = \mid\! T \!\mid -2$

Belkhechine¹,

Boudabbous²,

Hzami³

2015

Turk J Math

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“…We say that G is critical if G − v is not a prime for any v ∈ V. It is known that a critical graph G = (V, E) is isomorphic to H(|V|) or to H(|V|) [17]. Hence the number of vertices in a critical graph is always even.…”

Section: Modular Decompositionmentioning

confidence: 99%

Reconstruction Algorithms for Permutation Graphs and Distance-Hereditary Graphs

Kiyomi

Saitoh

Uehara³

2013

IEICE Trans. Inf. & Syst.

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“…The Schmerl-Trotter Theorem for Permutations [6]. Every simple permutation which is not a parallel alternation contains an entry whose removal leaves a simple permutation.…”

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confidence: 99%

A simple proof of a theorem of Schmerl and Trotter for permutations

Brignall

Vatter

2015

Journal of Combinatorics

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When specialized to the context of permutations, Schmerl and Trotter's Theorem states that every simple permutation which is not a parallel alternation contains a simple permutation with one fewer entry. We give an elementary proof of this result.An interval in the permutation π (thought of in one-line notation) is a contiguous set of entries whose values also form a contiguous set. Every permutation of length n has trivial intervals of lengths 0, 1, and n, and permutations with only trivial intervals are called simple. We take a graphical view of permutations, in which we identify a permutation π with its plot, the set of points (i, π(i)) in the plane. Three examples of simple permutations are plotted in Figure 1. Note that 1, 12, and 21 are all simple and that there are no simple permutations of length three. A bit more examination shows that 2413 and 3142 are the only simple permutations of length four.The second and third simple permutations in Figure 1 are called parallel alternations. A parallel alternation is, formally, a permutation whose entries can be divided into two halves of equal length, either both increasing or both decreasing, such that the entries of the halves interleave perfectly. In particular, 12, 21, 2413, and 3142 are parallel alternations. While parallel alternations need not be simple (1324 is a nonsimple parallel alternation), from any parallel alternation we may obtain a simple permutation by removing at most two entries. Specialized to permutations3 , the main result of Schmerl and Trotter is as follows. The Schmerl-Trotter Theorem for Permutations [6]. Every simple permutation which is not a parallel alternation contains an entry whose removal leaves a simple permutation.Schmerl and Trotter's theorem has found wide application both theoretically and practically. For example, it has been used in [2,3] to show that certain classes of permutations are defined by a finite set of restrictions, and in Albert's PermLab package [1] to efficiently generate the simple permutations in a permutation class.1 Both authors were partially supported by the EPSRC Grant EP/J006130/1. 2 Vatter was partially sponsored by the National Security Agency under Grant Number H98230-12-1-0207 and the National Science Foundation under Grant Number DMS-1301692. The United States Government is authorized to reproduce and distribute reprints not-withstanding any copyright notation herein. 3 The notions of intervals and simplicity extend naturally to all relational structures (though with different names, such as modules and primality). Schmerl and Trotter proved their result for simple, irreflexive, binary relational structures.

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Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures

Cited by 125 publications

References 1 publication

The prime tournaments $T$ with $\mid\! W_{5}(T) \!\mid = \mid\! T \!\mid -2$

The prime tournaments $T$ with $\mid\! W_{5}(T) \!\mid = \mid\! T \!\mid -2$

Reconstruction Algorithms for Permutation Graphs and Distance-Hereditary Graphs

A simple proof of a theorem of Schmerl and Trotter for permutations

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