A simple algorithm I devised to enumerate the disjoint paths between a pair of vertices in a given graph. This algorithm was devised as a part of my course-work for Masters [Tech.] at IIT-Banaras Hindu University.

1. Algorithm To Count Number of
Disjoint Paths Between A Source
and Target, In A Given Graph
Sujith Nair

2. Introduction
• This algorithm finds all the edge-disjoint paths
between a source S ∈ V and a target T ∈ V, in a
given graph G(E,V).
• The algorithm uses Dijkstra’s Single-Source-
Shortest-Path algorithm as a sub-routine.

3. Pseudo-Code
disjoint-path(G[n][n],X,Y){
path=path_find(X,Y);
while(path!=NULL){
count++;
delete_path(G,path);
path=path_find(X,Y);
}
return count;
}

4. delete_path(G,path){
for each edge in path
G:=G-edge;
}

5. path_find(G,X,Y){
for each vertex V in G:
dist[v]:=infinity;
previous[v]:= undefined;
dist[x]:=0;
Q:= set of all nodes in G;
while Q is not empty
u:=vertex in Q with smallest distance in
dist[];

6. If dist[u]=infinity:
break;
remove u from Q;
for each neighbour v of u:
alt:= dist[u]+1;
if alt<dist[v]:
dist[v]:=alt;
previous[v]=u;

7. path:=empty sequence;
k=Y;
while (previous[k]!=NULL)
insert k at the end of path;
k:=previous[k];
return path;
}

8. Example
Source Vertex: a
Target Vertex: f

9. Example Continued.
List Of Edge-
Disjoint Paths:
1. a-b-f

10. Example Continued
List Of Edge-Disjoint
Paths:
1.a-b-f
2.a-c-f

11. Example Continued
List Of Edge-
Disjoint Paths:
1.a-b-f
2.a-c-f
3.a-d-c-f

12. Endnotes
• This algorithm will also work in the case of
weighted graphs. In the case of weighted
graphs, the algorithm returns a list of paths
with the lowest weights.
• This algorithm, with minor modifications, will
also work for the case of vertex-disjoint paths.

13. Thank You