1. Let p **be the property of Example 2.1, and let** φ ∈ MXY *be a partial morphism.*
Since i φ = r for all pairs (i, r) ∈ Γ(φ), we trivially have i ∼p r *for all pairs*
(i, r) ∈ Γ(φ)**. Hence,** Γ(φ) ⊆ RpXY and therefore φ is a p*-morphism. Similarly,*
mono-, homo, and isomorphisms are p*-morphisms.*
2. Let p be the property of Example 2.2, and let φ **be a partial connectivity preserving**
morphism on V (X). As in (1), we find that φ is a p**-morphism. Note that for**
non-edge items (i, j) ∈ E(X) *the restriction* φ 0 of φ to {i, j} **does not preserve**
connectivity. But what is important is that φ 0*can be extended to a partial morphism* φ *that preserves connectivity.*
Let MpXY ⊆ MXY denote the set of all p-morphisms. We say, a subset M of MXY is p*-closed* if M = MpXY . The next result provides basic p-closed sets.

Proposition 1. The following subsets of MXY are p*-closed:*
1. M = **set of all partial morphisms** 2. M = **set of all partial monomorphisms**

3. M = *set of all partial homomorphisms* 4. M = *set of all partial isomorphisms*
5. M = *set of all partial subgraph-morphisms* Proof: We only show the assertion for the set M of all partial isomorphisms . The proofs for the other sets are similar. Let X and Y be graphs with adjacency matrices X = (xij ) and Y = (yrs). We define a property p on IXY such that

## R P Xy =(I, R) : Xi = Yr,Type(I) = Type(R)	⊆ I(X) × I(Y ),

where type(i) maps item i to its type (vertex, edge, or non-edge).

Next, we show that M = MpXY . Let φ be a partial isomorphism from X to Y , let i be an item of X, and let i φ = r be the image of i in Y . Since φ is a partial isomorphism, we have xi = yj . In addition, φ preserves the type of items i and r. Hence, the set Γ(φ)
is a subset of R
p XY . This proves *M ⊆ M*pXY .

Now assume that φ is a p-morphism from MpXY . Since Γ(φ) is a subset of R
p XY , it is sufficient to show that φ is bijective. Assume that φ is not bijective. Then φ is not injective, because φ is a partial morphism. Hence, there are distinct vertices *i, j* of X with i φ = j φ = r. Then φ maps a non-vertex item (*i, j*) to a vertex item (*r, r*), which contradicts the condition type(i) = type(r). This proves MpXY ⊆ M. Combining both results yields the assertion.

Though the proof of Proposition 1 is fairly simple, it illustrates the basic approach how to show that a certain subset M of MXY is p-closed. The task is to construct an appropriate property p such that we can show M = MpXY .

We conclude this section with a formal proof that the set M2 from Example 4 is not p-closed.