CS614. Advanced Algorithms. L02 Quiz.
CS614. Advanced Algorithms.
L02 Quiz
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.