BFS traversal of all paths in graph using adjacency list

Question

I am currently trying to traverse all paths from source to destination in a graph which uses adjacency matrix. I have been trying to do it in BFS way.Thanks for the help. I

Answer 1

Apparently it is impossible to retrieve all paths from a given source to a given terminal via Breadth-First search. Consider the following class of graphs.

For any nonnegative integer n, let

V := {v_1,...,v2_n}                             // inner vertices
     union
     {s, t},                                    // source and terminal
E := { {v_i,v+2,} : i < 2n-2 }                  // horizontal edges
     union
     { {v_i,v_i+3} : i < 2n-3, i is odd }       // cross edges from top to bottom
     union
     { {v_i,v_i+3} : i < 2n-3, i is even }      // cross edges from bottom to top
     union
     { {s,v_1}, {s,v_2}, {t,v_2n-1}, {t,v_2n} } // source and terminal

Informally, the graph consists out of two rows of vertices with n columns each, to the left there is a source node and to the right there is a terminal node. For each path from s to t, you can choose for each column to stay in the current row or to switch to the other row.

In total, there are 2^n different paths from s to t, as for each column there are two possibilities to chose the row.

On the other hand, Breadth-First search yields a runtime bound which is polynomial in the encoding length of the graph; this means that Breadth-first search, in general, cannot generate all possible paths from a given source to a given terminal. Furthermore, if the graph contains a cycle, the number of paths might be inifinite via repetition of the cycle.