Which of the following statements is false?a)An undirected graph is ca...
Sometimes the removal from a graph of a vertex and all incident edges produces a subgraph with more connected components. Such vertices are called cut vertices(or articulation points). The removal of a cut vertex from a connected graph produces a subgraph that is not connected.
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Which of the following statements is false?a)An undirected graph is ca...
Explanation:
In this question, we are given four statements related to graph theory and we need to identify the false statement among them.
a) An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph.
This statement is true. A graph is considered connected if there is a path between every pair of distinct vertices. In other words, we can reach any vertex from any other vertex in the graph.
b) A connected component of a graph G is a connected subgraph of G that is not a proper subgraph of another connected subgraph of G.
This statement is also true. A connected component of a graph is a subgraph that is connected and cannot be a proper subgraph of any other connected subgraph in the graph.
c) An edge whose removal produces a graph with more connected components than in the original graph is called a cut edge or bridge.
This statement is true. A cut edge, also known as a bridge, is an edge in a graph whose removal increases the number of connected components in the graph. In other words, removing a cut edge disconnects the graph.
d) The removal of a cut vertex from a connected graph produces a subgraph that is connected.
This statement is false. The removal of a cut vertex from a connected graph does not always produce a connected subgraph. A cut vertex is a vertex whose removal increases the number of connected components in the graph. Removing a cut vertex can disconnect the graph.
Therefore, the false statement among the given options is option d) The removal of a cut vertex from a connected graph produces a subgraph that is connected.