Threshold Graphs

Definition

Threshold graphs are a special class of graphs characterized by the property that their vertex set can be partitioned into two subsets: one with a small number of vertices and the other with a large number, where each vertex in the larger subset is connected to all vertices in the smaller subset. This structure connects closely with bipartite graphs, complete graphs, and regular graphs, as threshold graphs can be seen as a simplified form that preserves certain relationships between these different types of graphs.

5 Must Know Facts For Your Next Test

1. Every threshold graph can be constructed from a single vertex by repeatedly adding isolated vertices or connecting a new vertex to all existing vertices.
2. Threshold graphs can be represented using a simple integer sequence known as a threshold sequence, where the integers indicate how many connections each vertex makes.
3. They include important subclasses like complete graphs and star graphs, showcasing their versatility in representing various structures.
4. The degree of each vertex in a threshold graph can be at most one more than the degree of any adjacent vertex.
5. Threshold graphs are closed under taking induced subgraphs, meaning any induced subgraph of a threshold graph is also a threshold graph.

Review Questions

• How do threshold graphs relate to bipartite graphs, and what characteristics distinguish them from one another?
  • Threshold graphs share a connection with bipartite graphs because both can be partitioned into disjoint sets of vertices. However, while bipartite graphs allow for any kind of connection between the two sets, threshold graphs specifically require that every vertex in one set connects to all vertices in the other. This unique structure gives threshold graphs certain combinatorial properties not necessarily present in all bipartite graphs.
• Discuss the significance of threshold sequences in constructing threshold graphs and how they relate to complete and regular graphs.
  • Threshold sequences are critical for constructing threshold graphs as they indicate the connectivity pattern among vertices. Each number in the sequence corresponds to how many connections a vertex makes, helping define the graph's overall structure. While complete graphs connect every pair of vertices, and regular graphs maintain uniformity in degree across all vertices, threshold graphs serve as a bridge by incorporating elements from both types yet retaining their unique partitioning characteristic.
• Evaluate the implications of threshold graphs being closed under induced subgraphs on their application in network theory and algorithm design.
  • The closure property of threshold graphs under induced subgraphs implies that any analysis or algorithm developed for threshold graphs can be extended to their substructures. This means that properties and patterns observed in larger threshold networks remain valid at smaller scales. This characteristic is beneficial in network theory where understanding localized behavior within larger networks is crucial, and it enhances algorithm design by simplifying complex problems through modularity, allowing for recursive application of solutions.

Related terms

Bipartite Graph: A graph whose vertices can be divided into two disjoint sets such that no two graph vertices within the same set are adjacent.

Complete Graph: A type of graph in which every pair of distinct vertices is connected by a unique edge.

Regular Graph: A graph where each vertex has the same number of edges, meaning all vertices have the same degree.