Pinar Heggernes

Minimal triangulations of graphs: A survey

Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms.

Maximum Cardinality Search for Computing Minimal Triangulations of Graphs

Abstract We present a new algorithm, called MCS-M, for computing minimal triangulations of graphs. Lex-BFS, a seminal algorithm for recognizing chordal graphs, was the genesis for two other classical algorithms: LEX M and MCS. LEX M extends the fundamental concept used in Lex-BFS, resulting in an algorithm that not only recognizes chordality, but also computes a minimal triangulation of an arbitrary graph. MCS simplifies the fundamental concept used in Lex-BFS, resulting in a simpler algorithm for recognizing chordal graphs. The new algorithm MCS-M combines the extension of LEX M with the simplification of MCS, achieving all the results of LEX M in the same time complexity.

A practical algorithm for making filled graphs minimal

For an arbitrary filled graph G+ of a given original graph G, we consider the problem of removing fill edges from G+ in order to obtain a graph M that is both a minimal filled graph of G and a subgraph of G+. For G+ with f fill edges and e original edges, we give a simple O(f(e+f)) algorithm which solves the problem and computes a corresponding minimal elimination ordering of G. We report on experiments with an implementation of our algorithm, where we test graphs G corresponding to some real sparse matrix applications and apply well-known and widely used ordering heuristics to find G+. Our findings show the amount of fill that is commonly removed by a minimalization for each of these heuristics, and also indicate that the runtime of our algorithm on these practical graphs is better than the presented worst-case bound.

Interval Completion Is Fixed Parameter Tractable

We present an algorithm with runtime

The Computational Complexity of the Minimum Degree Algorithm

O(k^{2k}n^3m)

Interval completion with few edges

for the following NP-complete problem [M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco, 1979, problem GT35]: Given an arbitrary graph

A vertex incremental approach for maintaining chordality

G

The Minimum Degree Heuristic and the Minimal Triangulation Process

on

Methods for Large Scale Total Least Squares Problems

n

Obtaining a bipartite graph by contracting few edges

vertices and