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## Therefore, we have a centroidal tree with centroid d.

Nov 25, In Kruskal's algorithm, you keep adding the minimum weight edge in the graph to the set S. You can cut the graph in any manner, but if that cut passes through the minimum weight edge, then that edge will be the lightest edge. And, it has to be added to the minimum spanning tree (provided that it connects two different trees).

I hope that helps. Fundamental Cut - Set Since { b } is a cut-set in spanning tree T, {b} partitions all vertices of T into two disjoint sets: one at each end of {b}. Cut-set contains only one branch {b} of T, and rest of the edges in S are chords w.r.t.

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spanning tree T. Jul 21, Let T be a particular spanning-tree of an undirected graph G. By c we denote a chord and by e an edge of T. Justify the following statements: If S is a fundamental circuit defined by T and c, then c appears in each fundamental cut-set defined by an.

Def. A spanning tree of a graph G is a subgraph T that is connected and acyclic. Property. MST of G is always a spanning tree. 15 Greedy Algorithms Simplifying assumption. All edge costs ce are distinct. Cycle property. Let C be any cycle, and let f be the max cost edge belonging to C.

Then the MST does not contain f. Cut property. Apr 06, Each of the spanning trees has the same weight equal to 2. Cut property: For any cut C of the graph, if the weight of an edge E in the cut-set of C is strictly smaller than the weights of all other edges of the cut-set of C, then this edge belongs to all the MSTs of the graph. Below is the image to illustrate the same: Cycle property. Algorithm1.

### Note that we can think of the centroid or bicentroid as the ' center of gravity ' of the tree.

Remove all the vertices of degree1, together with their incident edges. Repeat the process until we obtain either a single vertex (the center)or two vertices joined by an edge (the bicenter). A tree with a center is called a central tree, and a tree with a bicenteris called a bicentral tree.