Clique graphs and Helly graphs

AbstractAmong the graphs for which the system of cliques has the Helly property those are characterized which are clique-convergent to the one-vertex graph. These graphs, also known as the so-called absolute retracts of reflexive graphs, are the line graphs of conformal Helly hypergraphs possessing a certain elimination scheme. From particular classes of such hypergraphs one can readily construct various classes G of graphs such that each member of G has its clique graph in G and is itself the clique graph of some other member of G. Examples include the classes of strongly chordal graphs and Ptolemaic graphs, respectively

The parallel solution of domination problems on chordal and strongly chordal graphs

AbstractWe present efficient parallel algorithms for the domination problem on strongly chordal graphs and related problems, such as the set cover problem for α-acyclic hypergraphs and the dominating clique problem for strongly chordal graphs. Moreover, we present an efficient parallel algorithm which checks, for any chordal graph, whether it has a dominating clique

Equivalence between Hypergraph Convexities

Let G be a connected graph on V. A subset X of V is all-paths convex (or ap -convex) if X contains each vertex on every path joining two vertices in X and is monophonically convex (or m-convex) if X contains each vertex on every chordless path joining two vertices in X. First of all, we prove that ap -convexity and m-convexity coincide in G if and only if G is a tree. Next, in order to generalize this result to a connected hypergraph H, in addition to the hypergraph versions of ap -convexity and m-convexity, we consider canonical convexity (or c-convexity) and simple-path convexity (or sp -convexity) for which it is well known that m-convexity is finer than both c-convexity and sp -convexity and sp -convexity is finer than ap -convexity. After proving sp -convexity is coarser than c-convexity, we characterize the hypergraphs in which each pair of the four convexities above is equivalent. As a result, we obtain a convexity-theoretic characterization of Berge-acyclic hypergraphs and of γ-acyclic hypergraphs

Fast Parallel Algorithms on a Class of Graph Structures With Applications in Relational Databases and Computer Networks.

The quest for efficient parallel algorithms for graph related problems necessitates not only fast computational schemes but also requires insights into their inherent structures that lend themselves to elegant problem solving methods. Towards this objective efficient parallel algorithms on a class of hypergraphs called acyclic hypergraphs and directed hypergraphs are developed in this thesis. Acyclic hypergraphs are precisely chordal graphs and their subclasses, and they have applications in relational databases and computer networks. In this thesis, first, we present efficient parallel algorithms for the following problems on graphs. (1) determining whether a graph is strongly chordal, ptolemaic, or a block graph. If the graph is strongly chordal, determine the strongly perfect vertex elimination ordering. (2) determining the minimal set of edges needed to make an arbitrary graph strongly chordal, ptolemaic, or a block graph. (3) determining the minimum cardinality dominating set, connected dominating set, total dominating set, and the domatic number of a strongly chordal graph. Secondly, we show that the query implication problem (Q\sb1\ \to\ Q\sb2) on two queries, which is to determine whether the data retrieved by query Q\sb1 is always a subset of the data retrieved by query Q\sb2, is not even in NP and in fact complete in \Pi\sb2\sp{p}. We present several \u27fine-grain\u27 analyses of the query implication problem and show that the query implication can be solved in polynomial time given chordal queries. Thirdly, we develop efficient parallel algorithms for manipulating directed hypergraphs H such as finding a directed path in H, closure of H, and minimum equivalent hypergraph of H. We show that finding a directed path in a directed hypergraph is inherently sequential. For directed hypergraphs with fixed degree and diameter we present NC algorithms for manipulations. Directed hypergraphs are representation schemes for functional dependencies in relational databases. Finally, we also present an efficient parallel algorithm for multi-dimensional range search. We show that a set of points in a rectangular parallelepiped can be obtained in O(logn) time with only 2.log\sp2 n $-$ 10.logn + 14 processors on a EREW-PRAM. A nontrivial implementation technique on the hypercube parallel architecture is also presented. Our method can be easily generalized to the case of d-dimensional range search

On hypergraph acyclicity and graph chordality

Partially supported by MP1 National projects on Formal Aspects of Databases and on Theory of Algorithms.SIGLEITItal