Matching and Covering Online Test

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Matching and Covering

Description: This quiz covers the concepts of matching and covering in graph theory. Questions will assess your understanding of different types of matchings, covering theorems, and their applications.
Number of Questions: 15
Created by: Aliensbrain Bot
Tags: graph theory matching covering menger's theorem perfect matching maximum matching

In a graph, a matching is a set of:

Disjoint edges
Disjoint vertices
Disjoint paths
Disjoint cycles

Which of the following is a necessary condition for a graph to have a perfect matching?

The graph must be bipartite
The graph must be connected
The graph must have an even number of vertices
The graph must have a cycle

Menger's Theorem states that the maximum number of edge-disjoint paths between two vertices in a graph is equal to:

The minimum number of edges in a cut separating the two vertices
The maximum number of edges in a cut separating the two vertices
The minimum number of vertices in a cut separating the two vertices
The maximum number of vertices in a cut separating the two vertices

In a graph, a vertex cover is a set of vertices that:

Covers all the edges of the graph
Covers all the paths of the graph
Covers all the cycles of the graph
Covers all the connected components of the graph

Which of the following is a necessary condition for a graph to have a vertex cover of size k?

The graph must have at least k vertices
The graph must have at most k edges
The graph must have a cycle of length k
The graph must have a path of length k

Which of the following is a sufficient condition for a graph to have a perfect matching?

The graph must be bipartite and have an even number of vertices
The graph must be connected and have an even number of vertices
The graph must be regular and have an even number of vertices
The graph must be complete and have an even number of vertices

Which of the following is a sufficient condition for a graph to have a vertex cover of size k?

The graph must have a cycle of length k
The graph must have a path of length k
The graph must have a clique of size k
The graph must have an independent set of size k

In a graph, an edge cover is a set of edges that:

Covers all the vertices of the graph
Covers all the paths of the graph
Covers all the cycles of the graph
Covers all the connected components of the graph

Which of the following is a necessary condition for a graph to have an edge cover of size k?

The graph must have at least k edges
The graph must have at most k vertices
The graph must have a cycle of length k
The graph must have a path of length k

Which of the following is a sufficient condition for a graph to have an edge cover of size k?

The graph must have a cycle of length k
The graph must have a path of length k
The graph must have a clique of size k
The graph must have an independent set of size k

In a graph, a maximum matching is a matching that:

Has the maximum number of edges
Has the maximum number of vertices
Has the maximum weight
Has the minimum weight

Which of the following is a necessary condition for a graph to have a maximum matching of size k?

The graph must have at least k vertices
The graph must have at most k edges
The graph must have a cycle of length k
The graph must have a path of length k

Which of the following is a sufficient condition for a graph to have a maximum matching of size k?

The graph must have a cycle of length k
The graph must have a path of length k
The graph must have a clique of size k
The graph must have an independent set of size k

In a graph, a minimum vertex cover is a vertex cover that:

Has the minimum number of vertices
Has the minimum number of edges
Has the minimum weight
Has the maximum weight

Which of the following is a necessary condition for a graph to have a minimum vertex cover of size k?

The graph must have at least k vertices
The graph must have at most k edges
The graph must have a cycle of length k
The graph must have a path of length k

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