http://swrc.ontoware.org/ontology#Thesis
Studies on Subgraph and Supergraph Enumeration Algorithms
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清見 礼
キヨミ マサシ
KIYOMI Masashi
総研大甲第998号
Enumeration is listing all objects that satisfy given properties. We call<br />enumeration of subgraphs of a given graph, such that those subgraphs have<br />specified properties, as subgraph enumeration. Similarly We call enumeration<br />of supergraphs of a given graph as subgraph enumeration. In this thesis, we will<br />consider about subgraph/supergraph enumeration algorithms. In areas such as<br />data mining or statistics, subgraph enumeration and supergraph enumeration<br />play important roles to find frequent patterns or to draw on some rules satisfied<br />by the inputs, etc.<br /> We developed two types of algorithms of subgraph/supergraph enumeration<br />for chordal and related graphs; one searches graphs to be enumerated by an<br />edge addition or an edge removal; the other defines a neighbor of searching by<br />a simplicial vertex elimination, which is specific for chordal graphs. The first<br />type uses the fact that there are only O(n<sup>2</sup>) edges in a complete graph K<sub>n</sub>,<br />and achieves polynomial time delay algorithms. We can use this method to<br />develop both subgraph enumeration algorithms and super graph enumeration<br />algorithms. The second type uses nice properties of simplicial vertices and the<br />fact that we can enumerate cliques in a chordal graph quickly. Using this type<br />of algorithm for chordal subgraph enumeration is faster than doing so using<br />the first type (it needs only constant time to enumerate each chordal graph).<br />However, this method is only for the subgraph enumeration.<br /> The organization of this thesis is as follows. We first introduce enumeration,<br />focusing particularly on graph enumeration. Chapter 2 provides the preliminaries,<br />notes about terms that we use in this thesis, and explanations about graph<br />classes. In Chapter 3, we discuss the difficulties of our enumeration problems,<br />and explain the framework of the reverse search method. In Chapter 4, we<br />develop algorithms for our enumeration problems. These algorithms are based<br />on the reverse search method. They are of two types: one defines parents such<br />that the difference between a graph and its parent is exactly one, and the other<br />defines parents such that the parent of a graph is obtained by eliminating a<br />simplicial vertex. And, we conclude the thesis in Chapter 5.
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