CS614. Advanced Algorithms. L02 Quiz.

CS614. Advanced Algorithms.

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.