Combinatorial generation via permutation languages. IV. Elimination trees

Abstract

An elimination tree for a connected graph G is a rooted tree on the vertices of G obtained by choosing a root x and recursing on the connected components of G − x to produce the subtrees of x. Elimination trees appear in many guises in computer science and discrete mathematics, and they encode many interesting combinatorial objects, such as bitstrings, permutations and binary trees. We apply the recent Hartung-Hoang-Mütze-Williams combinatorial generation framework to elimination trees, and prove that all elimination trees for a chordal graph G can be generated by tree rotations using a simple greedy algorithm. This yields a short proof for the existence of Hamilton paths on graph associahedra of chordal graphs. Graph associahedra are a general class of high-dimensional polytopes introduced by Carr, Devadoss, and Postnikov, whose vertices correspond to elimination trees and whose edges correspond to tree rotations. As special cases of our results, we recover several classical Gray codes for bitstrings, permutations and binary trees, and we obtain a new Gray code for partial permutations. Our algorithm for generating all elimination trees for a chordal graph G can be implemented in time O(m + n) per generated elimination tree, where m and n are the number of edges and vertices of G, respectively. If G is a tree, we improve this to a loopless algorithm running in time O(1) per generated elimination tree. We also prove that our algorithm produces a Hamilton cycle on the graph associahedron of G, rather than just Hamilton path, if the graph G is chordal and 2-connected. Moreover, our algorithm characterizes chordality, i.e., it computes a Hamilton path on the graph associahedron of G if and only if G is chordal.

Combinatorial Gray codes-an updated survey

“…• Graph associahedra, introduced by Carr and Devadoss [CD06,Dev09] and Postnikov [Pos09], are another large class of polytopes that generalize hypercube, permutahedron and associahedron. They yield a family of 2 n 2 /2(1+o(1)) many flip graphs, most of which have a Hamilton cycle [MP16,CMM21]. • The Johnson graph and the flip graph of spanning trees of a graph under edge exchanges (recall Figure 1 (b)+(g)) arise as skeleta of matroid base polytopes [NP81], and a Hamilton path on all of those can be computed by a simple greedy algorithm (see Section 9.1).…”

“…This generation approach therefore generalizes several known Gray codes, such as the BRGC (for X = {231, 132}), the SJT algorithm (for X = ∅ minimal jumps are adjacent transpositions), the Gray code for binary trees by rotations due to Lucas, Roelants van Baronaigien and Ruskey [LRvBR93] (for X = {231}), and the Gray code for set partitions by element exchanges described by Kaye [Kay76] (for X = {231}). It also yields Gray codes for many different classes of rectangulations [MM21], and it produces Hamilton paths and cycles on large classes of polytopes [HM21,CMM21]. The reason why minimal jumps are natural becomes clear when considering the inversion vector of the permutations.…”

“…Cardinal, Merino and Mütze [CMM21] consider partial permutations, which are all linearly ordered subsets of [n] of arbitrary size [OEIS A000522]. Their Gray code uses either an adjacent transposition, deletion or insertion of a trailing element in each step.…”

“…Subsequently, Lucas, Roelants van Baronaigien and Ruskey [LRvBR93] provided a simpler proof for the existence of a Hamilton path in G n , which yields a loopless algorithm. Possibly the simplest description of their path was given by Williams [Wil13] via a greedy algorithm, which is to repeatedly rotate the edge with the largest inorder label so that a new tree is generated; see also [CMM21] and Figure 11. Another proof for the existence of a Hamilton cycle in G n was obtained by Hurtado and Noy [HN99] via a generation tree.…”

“…The corresponding flip graphs arise as skeleta of graph associahedra, which are a class of polytopes introduced by Carr and Devadoss [CD06,Dev09] and Postnikov [Pos09]. Cardinal, Merino and Mütze [CMM21] present a simple and efficient algorithm to compute a Gray code for the elimination forests of H by tree rotations for the case when H is a chordal graph, and this Gray code is cyclic if H is also 2-connected. This Gray code generalizes the Gray code for binary trees due to Lucas, Roelants van Baronaigien and Ruskey [LRvBR93], the SJT algorithm for permutations, and the BRGC for bitstrings.…”