Is it possible to achieve desired permutation of values for vertices with adjacent swaps?

Question

Consider an arbitrary connected (acyclic/cyclic) undirected graph with N Vertices, with vertex numbered from 1 to N. Each vertex has some value assigned to it. Let the values be denoted by A1, A2, A3, ... AN, where A[i] denotes value of ith vertex. Let P be a permutation of A. Each operation, we can swap values of two adjacent vertices. Is it possible to achieve A = P, i.e. after all swapping operation A[i] = P[i] for all 1 <= i <= N. In other words, each vertex i should have value P[i] after the operations. P.S - I was confused about where to ask this - stack overflow or math.stack exchange. Apologies in advance.

Edit 1: I think the answer should be Yes. But I am only saying this on basis of case analysis of different types of graphs of 5 vertices. I tried to modify the permutation to Q where Q1 < Q2 < .. This changes a problem a bit that now final state should be A1 < A2 < A3... AN. So it can be said can the graph be sorted? Please correct me if my assumption is wrong.