(Homework Solution): Let T be a minimum spanning tree (MST) of a connected graph G. Given a

Let T be a minimum spanning tree (MST) of a connected graph G. Given a
connected subgraph H of G, show that T ∩H is contained in some MST of H

Expert Answer


HD49 98%| 3:18 PM 0o Document -Saved Cut of graph: A cut of a graph G (V, E) is a partition of the set of nodes V into two sets of nodes, S SV and V-S. we denote this cut as the pair (S, V-S). An edge e of G crosses the cut (S, V-S) if one of the endpoints of e is in S and the other is in V-s Cut Property: Let G-(V, E) be a graph, (S, V - S) be a cut of G, F be a set of edges of G, and e be an edge of G such that (a) F is contained in a MST of G (b) no edge in F crosses the cut (S, V- S), and (c) e is a minimum weight edge that crosses the cut (S, V - S) Then Fu (e is also contained in a MST of G So now applying the same cut property we will prove the solution By using the same cut property Let e be a edge in G which is the least weighed edge. So now it must be present in T, which is a MST ofG Now let us take the e as cut edge which divides the G (V, E) into (S, V-S) This should be same for H as it is Sub-graph of G. So the cut is (SnH, H-S So now if e is in TnH, then the same cut must be present in MST of H

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