Questions: Which of the following graphs are trees?
Transcript text: Which of the following graphs are trees?
Solution
{
"options": [
{
"isCorrect": "yes",
"label": "1",
"text": "A graph with 5 vertices and 4 edges, where all vertices are connected."
},
{
"isCorrect": "no",
"label": "2",
"text": "A graph with 5 vertices and 5 edges, where all vertices are connected."
},
{
"isCorrect": "yes",
"label": "3",
"text": "A graph with 5 vertices and 4 edges that is disconnected."
},
{
"isCorrect": "no",
"label": "4",
"text": "A graph with 5 vertices, 5 edges, and no cycles."
}
],
"explanation": "A tree is a connected graph with no cycles. Equivalently, a tree with n vertices has n-1 edges. \n\nOption 1: A connected graph with 5 vertices and 4 edges is a tree.\nOption 2: A connected graph with 5 vertices and 5 edges must contain a cycle, so it is not a tree.\nOption 3: A disconnected graph is not a tree, regardless of the number of vertices and edges.\nOption 4: A graph with 5 vertices, 5 edges, and no cycles cannot exist. If it has no cycles, it must be disconnected to have 5 edges."
}