How to implement graph search to solve Sudoku puzzle

My teacher pointed out to us during lectures that we could use Graph Search to help us solve Sudoku puzzles which has left me puzzled .

I dont see how this is possible as Graph Search is mostly about getting from Node A to Node B. He mentioned about how its a directed graph where the nodes correspond to partially completed puzzle

What is the general idea behind using Graph Search to solve Sudoku Puzzle

A graph is defined abstractly as a set of vertices together with a set of edges connecting those vertices. You have to give the vertices/edges meaning to make sense of the problem. For Sudoku, a vertex is a board configuration and an edge from configuration $a$ to configuration $b$ exists if $b$ is obtained from $a$ by inserting a number in an empty cell of $a$. There are $(9^2)^9$ possible configurations; searching this graph for a solution will take a lot of time but you will find one eventually. Many of the configurations are invalid Sudoku grids so you can eliminate lots of them.

A typical Sudoku solver would search an implicit graph: the solver is not given a graph as input but it is given a procedure to generate candidate configurations given some configuration. One such procedure may be described as:

generate_candidates(config) {
  candidates = [] is a list of configs
  for every empty cell (call it C) in config
    for every number in [1..9]
      C = number
        if config is a valid grid
          add copy of config to candidates
}

In the implicit graph, the edges are the ones that look like 'config $\to$ candidate' where config is the input to generate_candidates and candidate is a configuration generated by generate_candidates. Of course you have to generate new candidates from the ones just generated; you keep growing that list of candidates until no more can be generated or you've found a solution.

StackExchange Computer Science Q#22695, answer score: 4