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The following algorithm (written in pseudo-pascal) work on an undirected graph $G$

program Explore (G)
 procedure Visit (u)
   begin
        if Adj (u) is not empty
           {comment:Adj (u) is the list of edges incident to u}
        then
             begin
                  Select an edge from Adj (u);
                  Let edge be e=(u, v)
                  remove e from Adj (u) and Adj (v);
                  Visit (v);
            end
        else
                  mark u as a finished vertex and remove u from LIST
                  {Comment: LIST is the set of vertices in the graph}
   end;
 begin
     While LIST is not empty
       do
            begin
                 Let v ∊ LIST;
                 Visit (v);
    end
end.

Note: Initially $Adj(u)$ is the list of all edges incident to $u$ and LIST is the set of all vertices in the graph. They are globally accessible.

  1. What kind of subgraphs are obtained when this algorithm traverses the graphs $G_{1}$ and $G_{2}$ shown in Fig$.(6)$ and Fig$.(7)$ respectively?

 

 

  1. What is the commonly Known traversal of graphs that can be obtained from the subgraphs generated by Program Explore?
  2. Show that the time complexity of the procedure is $O (v + e)$ for a graph with $v$ vertices and $e$ edges, given that each vertex can be accessed and removed from LIST in constant time. Also, show that all edges of the graph are traversed.
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