document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1044147>Question 1044147</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can anyone tell me if I'm on the right track with this:\r<BR>\n" );
document.write( "\n" );
document.write( "Question:<BR>\n" );
document.write( "Let G = (V, E) be a directed graph. Let u, v &#8712; V be distinct vertices. Define C(u) to be a<BR>\n" );
document.write( "strongly connected component of G that contains u, and C(v) a strongly connected component<BR>\n" );
document.write( "of G that contains v. Prove that either C(u) and C(v) do not share any vertex or C(u) and<BR>\n" );
document.write( "C(v) are equal. [5 marks]<BR>\n" );
document.write( "[Hint: To write such a proof you can begin with: “If C(u) and C(v) do not share any vertex<BR>\n" );
document.write( "then the result is true, so suppose C(u) and C(v) share a vertex.”]\r<BR>\n" );
document.write( "\n" );
document.write( "My answer:<BR>\n" );
document.write( "A directed graph is strongly connected if there is a path from a to b, and a path from b to a, whenever a and b are vertices in the graph. <BR>\n" );
document.write( "therefore if C(u) and c(V) share a vertex say w,<BR>\n" );
document.write( "for the graph to be directed there must be strongly connected components between C(u) and C(w), and strongly connected components between C(v) and C(w) <BR>\n" );
document.write( "Therefore as C(u) is strongly connected to C(w) and C(v) is strongly connected to C(w) (we have previously proved this theorem u,v,w E G(V,E) so c(u) is strongly connected to C(v) so therefore as they are both strongly connected to w in order for the definition to hold they must be equal and part of the same graph </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #659453 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=659453\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Your proof is fairly difficult to read. Not clear what \"strongly connected component *between* C(u) and C(w)\" refers to. It might be easier to draw pictures and think about the problem in terms of paths between vertices.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Suppose C(u) and C(v) share a vertex w. Our goal is to prove that the subgraphs C(u) and C(v) are equal.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To do this, consider some arbitrary vertex in C(u) (could be u, w, or a different vertex). Can you show that this vertex is also in C(v)? hint: use the hypothesis that there are paths from this vertex to w, and to u and v. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );