Geneviève Simonet

An Introduction to Clique Minimal Separator Decomposition

This paper is a review which presents and explains the decomposition of graphs by clique minimal separators. The pace is leisurely, we give many examples and figures. Easy algorithms are provided to implement this decomposition. The historical and theoretical background is given, as well as sketches of proofs of the structural results involved.

Maximal Label Search Algorithms to Compute Perfect and Minimal Elimination Orderings

Many graph search algorithms use a vertex labeling to compute an ordering of the vertices. We examine such algorithms which compute a peo (perfect elimination ordering) of a chordal graph and corresponding algorithms which compute an meo (minimal elimination ordering) of a non-chordal graph, an ordering used to compute a minimal triangulation of the input graph. We express all known peo-computing search algorithms as instances of a generic algorithm called MLS (maximal label search) and generalize Algorithm MLS into CompMLS, which can compute any peo. We then extend these algorithms to versions which compute an meo and likewise generalize all known meo-computing search algorithms. We show that not all minimal triangulations can be computed by such a graph search, and, more surprisingly, that all these search algorithms compute the same set of minimal triangulations, even though the computed meos are different. Finally, we present a complexity analysis of these algorithms. An extended abstract of part of this paper was published in WG 2005 .

Ultimate generalizations of LexBFS and LEX m

Many graph search algorithms use a labelling of the vertices to compute an ordering of the vertices. We examine such algorithms which compute a peo (perfect elimination ordering) of a chordal graph, and corresponding algorithms which compute an meo (minimal elimination ordering) of a non-chordal graph. We express all known peo-computing search algorithms as instances of a generic algorithm called MLS (Maximal Label Search) and generalize Algorithm MLS into CompMLS, which can compute any peo. We then extend these algorithms to versions which compute an meo, and likewise generalize all known meo-computing search algorithms. We show the surprising result that all these search algorithms compute the same set of minimal triangulations, even though the computed meos are different.

A General Label Search to investigate classical graph search algorithms

Many graph search algorithms use a labeling of the vertices to compute an ordering of the vertices. We generalize this idea by devising a general vertex labeling algorithmic process called General Label Search (GLS), which uses a labeling structure which, when specified, defines specific algorithms. We characterize the vertex orderings computable by the basic types of searches in terms of properties of their associated labeling structures. We then consider performing graph searches in the complement without computing it, and provide characterizations for some searches, but show that for some searches such as the basic Depth-First Search, no algorithm of the GLS family can exactly find all the orderings of the complement. Finally, we present some implementations and complexity results of GLS on a graph and on its complement.

Two FOL Semantics for Simple and Nested Conceptual Graphs

J.F. Sowa has defined a FOL semantics for Simple Conceptual Graphs and proved the soundness of the graph operation called projection with respect to this semantics. M. Chein and M.L. Mugnier have proved the completeness result, with a restriction on the form of the target graph of the projection. I propose here another FOL semantics for Simple Conceptual Graphs corresponding to a slightly different interpretation of a Conceptual Graph. Soundness and completeness of the projection with respect to this semantics are true without any restriction. I extend the definitions and results on both semantics to Conceptual Graphs containing co-reference links and to Nested Conceptual Graphs.

Sequential and parallel triangulating algorithms for Elimination Game and new insights on Minimum Degree

Elimination Game is a well-known algorithm that simulates Gaussian elimination of matrices on graphs, and it computes a triangulation of the input graph. The number of fill edges in the computed triangulation is highly dependent on the order in which Elimination Game processes the vertices, and in general the produced triangulations are neither minimum nor minimal. In order to obtain a triangulation which is close to minimum, the Minimum Degree heuristic is widely used in practice, but until now little was known on the theoretical mechanisms involved. In this paper we show some interesting properties of Elimination Game; in particular that it is able to compute a partial minimal triangulation of the input graph regardless of the order in which the vertices are processed. This results in a new algorithm to compute minimal triangulations that are sandwiched between the input graph and the triangulation resulting from Elimination Game. One of the strengths of the new approach is that it is easily parallelizable, and thus we are able to present the first parallel algorithm to compute such sandwiched minimal triangulations. In addition, the insight that we gain through Elimination Game is used to partly explain the good behavior of the Minimum Degree algorithm. We also give a new algorithm for producing minimal triangulations that is able to use the minimum degree idea to a wider extent.

Computing a Clique Tree with the Algorithm Maximal Label Search

The algorithm MLS (Maximal Label Search) is a graph search algorithm that generalizes the algorithms Maximum Cardinality Search (MCS), Lexicographic Breadth-First Search (LexBFS), Lexicographic Depth-First Search (LexDFS) and Maximal Neighborhood Search (MNS). On a chordal graph, MLS computes a PEO (perfect elimination ordering) of the graph. We show how the algorithm MLS can be modified to compute a PMO (perfect moplex ordering), as well as a clique tree and the minimal separators of a chordal graph. We give a necessary and sufficient condition on the labeling structure of MLS for the beginning of a new clique in the clique tree to be detected by a condition on labels. MLS is also used to compute a clique tree of the complement graph, and new cliques in the complement graph can be detected by a condition on labels for any labeling structure. We provide a linear time algorithm computing a PMO and the corresponding generators of the maximal cliques and minimal separators of the complement graph. On a non-chordal graph, the algorithm MLSM, a graph search algorithm computing an MEO and a minimal triangulation of the graph, is used to compute an atom tree of the clique minimal separator decomposition of any graph.

Extremities and orderings defined by generalized graph search algorithms

Abstract We define Algorithm MLS which uses a labelling structure to generalize all known graph searches which compute a peo on a chordal graph, as well as its extension MLSM, which computes an meo on an arbitrary graph. We also examine the vertices numbered as 1 by these algorithms, and show how they can be seen as extremities of the graph.