259 research outputs found
On some Graphs with a Unique Perfect Matching
We show that deciding whether a given graph of size has a unique
perfect matching as well as finding that matching, if it exists, can be done in
time if is either a cograph, or a split graph, or an interval graph,
or claw-free. Furthermore, we provide a constructive characterization of the
claw-free graphs with a unique perfect matching
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Partitioning Perfect Graphs into Stars
The partition of graphs into "nice" subgraphs is a central algorithmic
problem with strong ties to matching theory. We study the partitioning of
undirected graphs into same-size stars, a problem known to be NP-complete even
for the case of stars on three vertices. We perform a thorough computational
complexity study of the problem on subclasses of perfect graphs and identify
several polynomial-time solvable cases, for example, on interval graphs and
bipartite permutation graphs, and also NP-complete cases, for example, on grid
graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor
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
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