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Generating Linear Extensions Fast
"... One of the most important sets associated with a poset P is its set of linear extensions, E(P) . "ExtensionFast.html" 87 lines, 2635 characters One of the most important sets associated with a poset P is its set of linear extensions, E(P) . In this paper, we present an algorithm to generate all of t ..."
Abstract

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One of the most important sets associated with a poset P is its set of linear extensions, E(P) . "ExtensionFast.html" 87 lines, 2635 characters One of the most important sets associated with a poset P is its set of linear extensions, E(P) . In this paper, we present an algorithm to generate all of the linear extensions of a poset in constant amortized time; that is, in time O(e(P)) , where e ( P ) =  E(P) . The fastest previously known algorithm for generating the linear extensions of a poset runs in time O(n e(P)) , where n is the number of elements of the poset. Our algorithm is the first constant amortized time algorithm for generating a ``naturally defined'' class of combinatorial objects for which the corresponding counting problem is #Pcomplete. Furthermore, we show that linear extensions can be generated in constant amortized time where each extension differs from its predecessor by one or two adjacent transpositions. The algorithm is practical and can be modified to efficiently count linear extensions, and to compute P(x < y) , for all pairs x,y , in time O( n^2 + e ( P )).
A counterexample to a conjecture of
"... A counterexample is presented to the following conjecture of Jackson and Wormald: If j ~ 1, k ~ 2 and a graph is connected, locally j connected and K l, (j+l)(kl)+2free then it has a ktree. Preliminaries All graphs considered here are finite and without loops or multiple edges. As usual, we let V ..."
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A counterexample is presented to the following conjecture of Jackson and Wormald: If j ~ 1, k ~ 2 and a graph is connected, locally j connected and K l, (j+l)(kl)+2free then it has a ktree. Preliminaries All graphs considered here are finite and without loops or multiple edges. As usual, we let V(G) and E(G) denote respectively the vertex set and the edge set of the graph G. The cardinality of the set 8 is denoted by 181. A K l,kfree graph is a graph containing no copy of K l,k as an induced subgraph. Also, a graph is locally jconnected if every subgraph induced by the set of neighbours of a vertex v is jconnected. A ktree of a graph is a spanning tree with maximum degree at most k. The join of two disjoint graphs Gl and G2, denoted by Gl + G2, is obtained by joining each vertex of G l to each vertex of G2 • The union of m disjoint copies of the same graph G is denoted by mG. In [1], Bill Jackson and Nicholas C. Wormald made the following conjecture: If j ~ 1, k ~ 2 and a graph is connected, locally jconnected and K l, (j+l)(kl)+2free then it has a ktree. A counterexample For any integers <5 ~ 2 and k ~ 2, first we construct the graph Gl + G 21 where G l = Ko and G2 = {<5(k 1) + I}Ko. Then join a K o(kl) to each copy of Ko in G 2; the graph is depicted in Figure 1.