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Let G = ( V , E , w ) be a connected weighted graph in which each edge e in E has weight w

Let G=(V,E,w)
be a connected weighted graph in which each edge e in E
has weight w(e)>0
. Suppose T1
is a Shortest Path Tree (SPT) of G
rooted at some source vertex s
, and T2
is a Minimum Spanning Tree (MST) of G
. Now consider keeping the same set of vertices and edges, but modifying the edge weights to get a modified graph G^=(V,E,w^)
, where the modified edge weights are defined as w^(e)=(4+w(e))(7+w())
, for each edge e in E
.
Then, as a set of edges in G^
,

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