Locally connected graphs: metric properties

In this work we show that any connected locally connected graph defines a metric space having at least as many lines as vertices with only three exception: the complete multipartite graphs $K_{1,2,2}$, $K_{2,2,2}$ and $K_{2,2,2,2}$. This proves that this class fulfills a conjecture, proposed by Chen and Chv\'atal, saying that any metric space on n points has at least n lines or a line containing all the points.