Lilian Markenzon

Subclasses of k-trees: Characterization and recognition

A k-tree is either a complete graph on k vertices or a graph G = (V, E) that contains a vertex whose neighbourhood in G induces a complete graph on k vertices and whose removal results in a k-tree. We present two new subclasses of k-trees and their properties. First, we present the definition and characterization of k-path graphs, based on the concept of k-paths, that generalizes the classic concept of paths. We also introduce the simple-clique k-trees, of which the maximal outerplanar graphs and the planar 3-trees are particular cases. Based on Characterization Theorems, we show recognition algorithms for both families. Finally, we establish the inclusion relations among these new classes and k-trees.

One‐phase algorithm for the determination of minimal vertex separators of chordal graphs

The set of minimal vertex separators of chordal graphs is usually obtained by two-phase algorithms. Based on properties of the lexicographic breadth-first search, we propose a new one-phase algorithm. We present also a characterization for planar chordal graphs; using our proposed algorithm as an initial step, the implementation of the recognition algorithm becomes trivial.

The Reduced Prüfer Code for Rooted Labelled k-Trees

Abstract Based on the redundant Prufer code for k -trees, developed by Renyi and Renyi, a more compact new encoding scheme is proposed and the one-to-one correspondence between both codes is proved.

DIRECTED HYPERGRAPH PLANARITY

Directed hypergraphs are generalizations of digraphs and can be used to model binary relations among subsets of a given set. Planarity of hypergraphs was studied by Johnson and Pollak; in this paper we extend the planarity concept to directed hypergraphs. It is known that the planarity of a digraph relies on the planarity of its underlying graph. However, for directed hypergraphs, this property do not apply and we propose a new approach which generalizes the usual concept. We also show that the complexity of the recognition of a directed hypergraph as planar is linear on the size of the hypergraph.

Solving problems for maximal reducible flowgraphs

In this paper, the family of Maximal Reducible Flowgraphs (MRFs) is recursively defined, based on a decomposition theorem. A one-to-one association between MRFs and extended binary trees allows to deduce some numerical properties of the family. Hamiltonian problems, testing isomorphism and finding a minimum cardinality feedback arc set are efficiently solved for MRFs. The results concerning hamiltonian paths and cycles also hold for reducible flowgraphs.

Two methods for the generation of chordal graphs

Abstract In this paper two methods for automatic generation of connected chordal graphs are proposed: the first one is based on new results concerning the dynamic maintenance of chordality under edge insertions; the second is based on expansion/merging of maximal cliques. Theoretical and experimental results are presented. In both methods, chordality is preserved along the whole generation process.

New results on ptolemaic graphs

In this paper, we analyze ptolemaic graphs for their properties as chordal graphs. First, two characterizations of ptolemaic graphs are proved. The first one is based on the reduced clique graph, a structure that was defined by Habib and Stacho (Habib and Stacho, 2012). In the second one, we simplify the characterization presented by Uehara and Uno (Uehara and Uno, 2009) with a new proof. Then, known subclasses of ptolemaic graphs are reviewed in terms of minimal vertex separators. We also define another subclass, the laminar chordal graphs, and we show that a hierarchy of ptolemaic graphs can be built based on characteristics of the minimal vertex separators in each subclass.

Generating and counting unlabeled k-path graphs

A subfamily of k-trees, the k-path graphs generalize path graphs in the same way k-trees generalize trees. This paper presents a code for unlabeled k-path graphs. The effect of structural properties of the family on the code is investigated, leading to the solution of two problems: determining the exact number of unlabeled k-path graphs with n vertices and generating all elements of the family.

Some results about the connectivity of trees

The second smallest Laplacian eigenvalue of a graph G is called algebraic connectivity, denoted a(G). The ordering of trees via this graph invariant is frequently studied in the literature. In this paper, we present a new invariant, the Internal Degree Sequence (IDS), that also supports an accurate evaluation of the connectivity of trees. We compare the IDS with a(G) for all elements in six classes of trees known to have the largest algebraic connectivity and we show that the IDS provides a strict total ordering of the elements of these classes. This result is also proved for a subclass of trees of diameter 4.

A clique-difference encoding scheme for labelled k-path graphs

We present in this paper a codeword for labelled k-path graphs. Structural properties of this codeword are investigated, leading to the solution of two important problems: determining the exact number of labelled k-path graphs with n vertices and locating a hamiltonian path in a given k-path graph in time O(n). The corresponding encoding scheme is also presented, providing linear-time algorithms for encoding and decoding.