I am wanting to know why the triangle inequality is always satisfied in a weighted graph. In particular, I want to know why the attached graph (please click the link weighted K_5 below) is not a counter-example. Is it true that the triangle inequality says that for any vertices $u, v, w \in V(G)$, $wt(u,v) < wt(u,w) + wt(w,v)$? Because in the graph, if we consider $u=v_3, v=v_4$ and $w=v_2$, we have that $wt(v_3, v_4)=8, wt(v_3, v_2)=4$ and $wt(v_2, v_4)=2$. In this case $wt(v_3, v_4) \nless wt(v_3, v_2) + wt(v_2, v_4)$, which is $8 \nless 4+2$. I know I am wrong, no doubt, I'm just struggling to see why.
I want to clarify this because I want to see which graphs cannot be used to solve the Travelling Salesman problem (TSP).
Any help would be greatly appreciated.
Weighted K_5:
