Below is the general code for DFS with logic for marking back edges and tree edges. My doubt is that back edges from a vertex go back and point to an ancestor and those which point to the parent are not back edges (lets assume undirected graph). In an undirected graph we have an edge back and forth between 2 vertices $x$ and $y$. So after visiting $x$ when I process $y$, $y$ has $x$ as an adjacent vertex, but as its already visited, the code will mark it as a back edge. Am I right in saying that? Should we add any extra logic to avoid this, in case my assumption is valid?
DFS(G) for v in vertices[G] do color[v] = white parent[v]= nil time = 0 for v in vertices[G] do if color[v] = white then DFS-Visit(v)
Induce a depth-first tree on a graph starting at $v$.
DFS-Visit(v) color[v]=gray time=time + 1 discovery[v]=time for a in Adj[v] do if color[a] = white then parent[a] = v DFS-Visit(a)<br> v->a is a tree edge elseif color[a] = grey then v->a is a back edge color[v] = black time = time + 1
unexplored, gray means
frontier, black means `processed'