Question

Let G be a graph of n nodes {1, 2, 3, ... , n}. For each permutation f of the nodes of G, we can have an induced weighting of G: the induced weight of each edge (x,y) is |f(x)-f(y)|. The total induced weight of G is the sum of the induced weights of its edges. Note that you can think of f as a relabeling of the nodes of G, where f(x) is the new label of node x, and the new labels induce edge weights. We are interested to find a permutation f that results in a minimum total induced weight of G. Such a permutation can be called a minimum relabeling of G. Consider the graph G=(V,E): V={1, 2, 3,4}, E={(1,2) , (1,3) , (1,4) , (2,3)}. Give two different permutations (relabelings) of the nodes of G that yield 2 different total induced weights of G. Show the induced edge weights for each permutation, along with the corresponding total induced weight.

Answer 1

SOLUTION : G be guaph con baining n nodes ,343 4! 4-4 匕

Given Goa,ph ing od :- 2 2. ろ 4 2 2 es 3

Glabel GiI es 3 2- 4