CSE 222Systems Programming

Graph Theory Basics

Dr. Jim Holten

CSE 222201/21/09

Introduction

• Basic Definitions

• Concepts and Algorithms

• Applications

CSE 222301/21/09

Basic Definitions

• Nodes aka Vertices

• Links aka Edges

aka Arcs

CSE 222401/21/09

Basic Definitions

• Subgraphs

• Undirected vs Directed Graphs

• Paths, Distances, and Graph Diameter

• Connected Graphs

• Components of a Graph

• Fully connected graph or subgraph

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Undirected Graph

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Directed Graph

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Basic Definitions

• Node Attributes aka Labels, Indices,

Weights, Categories, etc.

• Link Attributes aka Labels, Indices,

Weights, Categories, etc.

• A Semantic Graph is a graph partitioned

based on node or link attribute values

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Semantic Graph

AuthorsDocuments

Concepts

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More Definitions

• A Hypernode replaces a collection of

nodes and their included links.

• A Hypergraph is any graph that includes

one or more hypernodes.

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Example of Hypergraph

AuthorsDocuments

Concepts

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Hypergraph of Social Network

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Some Concepts

• A cluster is a subgraph made up of set of

highly interconnected nodes.

• A critical node is a node on a unique path

between two nodes or subgraphs of

interest.

• A critical link is a link on a unique path

between two nodes or subgraphs of

interest.

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Related Algorithms

• Finding clusters is defined many ways

– Fully connected subgraph or just better than

some threshold on connectedness?

– Is a node only connected to a node in the

cluster to be included in the cluster?

• Finding critical paths, nodes, or links

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Related Concepts

• Projections of nodes or subgraphs are

mappings via directed links.

• The domain subgraph includes the projection's

"from" nodes.

• The range subgraph includes the projection's

"to" nodes.

• Basis Node Set is a minimal set of nodes that

have a path to every other node

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Projection Mapping

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Subset Projections

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Related Concepts

• There are multiple ways of looking at a

single collection of nodes and links.

• Are multiple graphs just a single multi-

component graph?

• Interdependencies among multiple graphs

can be useful.

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Interdependencies

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Applications

• Software Organization

• Activity Sequence Organization

• Interactions between processes

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Software Organization

• Flow charts

• Data interdependencies

• Interactions between processes

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Data Organization Chart

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Processes Organization Chart

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Activity Sequence Organization

• Gantt and Pert charts

• Project resource planning charts

• Storyboarding

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Storyboard of the

Software Development Process

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Network Modeling Applications• Communications (Internet, etc.)

• Energy distribution (electric power, natural gas,

etc.)

• Transportation (highways, railways, waterways,

etc)

• Social associations

• Financial and product flows

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Infrastructure Networks

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Critical Infrastructure

Interdependencies

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Physics Models

• Mesh Models

– Space segmented into volumes

– Volumes surrounded by faces and edges

– Edges meet at vertices

• Model element type dependencies form a

graph of interdependent element sets

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Mesh Models

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Element Type Dependencies

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Summary

Graphs are a significant factor in the present

and the future of computer science!