Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0809.1828

Diametral Pairs of Linear Extensions

Graham Brightwell·Mareike Massow
View original source ↗Read source document ↗Export metadata ↓

Abstract

Given a finite poset P, we consider pairs of linear extensions of P with maximal distance, where the distance between two linear extensions L_1, L_2 is the number of pairs of elements of P appearing in different orders in L_1 and L_2. A diametral pair maximizes the distance among all pairs of linear extensions of P. Felsner and Reuter defined the linear extension diameter of P as the distance between a diametral pair of linear extensions. We show that computing the linear extension diameter is NP-complete in general, but can be solved in polynomial time for posets of width 3. Felsner and Reuter conjectured that, in every diametral pair, at least one of the linear extensions reverses a critical pair. We construct a counterexample to this conjecture. On the other hand, we show that a slightly stronger property holds for many classes of posets: We call a poset "diametrally reversing" if, in every diametral pair, both linear extensions reverse a critical pair. Among other results we show that interval orders and 3-layer posets are diametrally reversing. From the latter it follows that almost all posets are diametrally reversing.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Graham Brightwell, Mareike Massow. 2008-09-10. Diametral Pairs of Linear Extensions. https://arxiv.org/abs/0809.1828

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗