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Intersection Graph Algorithms

Author: Paul Frederick Dietz
Publisher: Ithaca, N.Y. : Cornell University, 1984.
Dissertation: Ph. D. Cornell University 1984
Series: Cornell University, Department of Computer Science, TR 84-628.
Edition/Format:   Thesis/dissertation : Thesis/dissertation : EnglishView all editions and formats
Database:WorldCat
Summary:
An intersection graph for a set of sets $C$ is a graph $G$ together with a bijection from the vertices of $G$ to $C$ such that distinct vertices in $G$ are adjacent if and only if their images under this bijection intersect. Of particular interest have been the classes of chordal graphs, the intersection graphs of sets of subtrees of a tree, and interval graphs, the intersection graphs of sets of intervals of the
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Additional Physical Format: Online version:
Dietz, Paul Frederick, 1959-
Intersection Graph Algorithms.
Ithaca, N.Y. : Cornell University, 1984
(OCoLC)704420872
Material Type: Thesis/dissertation
Document Type: Book
All Authors / Contributors: Paul Frederick Dietz
OCLC Number: 13399697
Description: viii, 134 pages : illustrations ; 28 cm.
Series Title: Cornell University, Department of Computer Science, TR 84-628.
Responsibility: Paul F. Dietz.

Abstract:

An intersection graph for a set of sets $C$ is a graph $G$ together with a bijection from the vertices of $G$ to $C$ such that distinct vertices in $G$ are adjacent if and only if their images under this bijection intersect. Of particular interest have been the classes of chordal graphs, the intersection graphs of sets of subtrees of a tree, and interval graphs, the intersection graphs of sets of intervals of the real line.

I examine another class of intersection graphs, the class of directed path graphs: intersection graphs of sets of paths in a directed tree. This class properly contains the class of interval graphs, and is properly contained by the class of chordal graphs. I give a linear time algorithm for recognizing directed path graphs and for constructing intersection representations, and a polynomial time algorithm for deciding directed path graph isomorphism.

Both algorithms use a data structure called a partial path tree, which is a kind of skeletal representation of the clique hypergraph of a directed path graph. I present linear time algorithms for finding partial path trees with specific roots and for finding partial path trees with arbitrary roots. I prove that partial path trees with identical roots are identical. Using this fact I develop a polynomial time algorithm for directed path graph isomorphism.

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