CS614. Advanced Algorithms. L02 Quiz.

CS614. Advanced Algorithms.

L02 Quiz

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Problem 1. Identify the Circuits

Let G be a simple, undirected, and connected graph. Consider the graphic matroid discussed in class, i.e, where:

• the universe U is the set of edges of G, i.e, E(G);
• the family \mathcal{F} of independent sets is the collection of all subsets of edges that are acyclic.

A maximal independent set in a matroid is called a basis, and for this example, the maximal independent sets correspond to spanning trees.

A minimal dependent set in a matroid is called a circuit. In this example, what are the circuits?

Problem 2. Matchings

Let G be a simple, undirected, and connected graph. Consider the following set system:

• the universe U is the set of edges of G, i.e, E(G);
• the family \mathcal{F} of independent sets is the collection of all subsets of edges that are matchings.

Is this a matroid? Why or why not? Justify your answer.

Problem 3. Independent Sets

Let G be a simple, undirected, and connected graph. Consider the following set system:

• the universe U is the set of vertices of G, i.e, V(G);
• the family \mathcal{F} of independent sets is the collection of all subsets S of that are independent in G, i.e, the subgraph G[S] has no edges.

Is this a matroid? Why or why not? Justify your answer.