We now focus on the use of Datalog for defining properties and queries m graphs.

(a) Suppose that P is some property of graphs definable in Datalog. Show drat P is preserved under extensions and homomorphisms. That is, if G is a graph satisfying P, then every supergraph of G (i.e., graph extending G) satisfies P, and if h is a graph homomorphism, then h (G) satisfies P.

Which of the following properties and queries on graphs are definable in Datalog?

b) The number of vertices is even.

(c) There is a simple path (i.e., a path without repeated vertices) of even length between two specified vertices.

(d) The binary relation T containing all pairs of vertices (a, D) for which there is a path of even length from o to b. Provide either a Datalog program defining the property or query or an argument why the property or query is not definable in Datalog.