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Problem 5: Let G be an undirected graph. Let P(x, y) mean that there is a path from vertex x to vertex y. Express each of the following sentences in terms of P, quantifiers, logical connectives, and equality, using variables that range over the vertices of G. (Reminder: there is a zero-length path from any vertex to itself.) a) Vertices x and y are in the same connected component. b) G has a vertex of degree zero. (Reminder: undirected graphs only have edges between distinct vertices, that is, no self-loops.) c) G has at least three connected components. d) There is a positive length simple path from x to y.

          Problem 5: Let G be an undirected graph. Let P(x, y) mean that there is a path from vertex x to vertex y. Express each of the following sentences in terms of P, quantifiers, logical connectives, and equality, using variables that range over the vertices of G. (Reminder: there is a zero-length path from any vertex to itself.)

a) Vertices x and y are in the same connected component.
b) G has a vertex of degree zero. (Reminder: undirected graphs only have edges between distinct vertices, that is, no self-loops.)
c) G has at least three connected components.
d) There is a positive length simple path from x to y.

Added by Andrew H.

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