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The status of a vertex v in a connected graph G is the sum of the distances between v and all the other vertices of G. The subgraph induced by the vertices of minimum (maximum) status in G is called median (anti-median) of G. Let H = (G 1 , G 2 , r ) denote a graph with G 1 as the median and G 2 as the anti-median of H , d(G 1 , G 2 ) = r and both G 1 and G 2 are convex subgraphs of H . It is known that (G 1 , G 2 , r ) exists for every G 1 , G 2 with r ≥ diam(G 1 )/2 + diam(G 2 )/2 + 2. In this paper we show the existence of (G 1 , G 2 , r ) for every G 1 , G 2 and r ≥ 1. We also obtain a sharp upper bound for the maximum status difference in a graph G.

The Gallai and the anti-Gallai graphs of a graph G are complementary pairs of spanning subgraphs of the line graph of G. In this paper we find some structural relations between these graph classes by finding a partition of the edge set of the line graph of a graph G into the edge sets of the Gallai and anti-Gallai graphs of G. Based on this, an optimal algorithm to find the root graph of a line graph is obtained. Moreover, root graphs of diameter-maximal, distance-hereditary, Ptolemaic and chordal graphs are also discussed.

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