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Graph Theory: Trees

Graph Theory: Trees. Pallab Dasgupta , Professor, Dept. of Computer Sc. and Engineering, IIT Kharagpur pallab@cse.iitkgp.ernet.in. Trees and Spanning Trees. A graph having no cycles is acyclic. A forest is an acyclic graph. A leaf is a vertex of degree 1.

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Graph Theory: Trees

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  1. Graph Theory: Trees PallabDasgupta, Professor,Dept. of Computer Sc. and Engineering, IIT Kharagpur pallab@cse.iitkgp.ernet.in

  2. Trees and Spanning Trees • A graph having no cycles is acyclic. • A forest is an acyclic graph. • A leaf is a vertex of degree 1. • A spanning sub-graph of G is a sub-graph with vertex set V(G). • A spanning tree is a spanning sub-graph that is a tree.

  3. Distances • If G has a u,v-path, then the distance from u to v, written dG(u,v) or simply d(u,v), is the least length of a u,v-path. • If G has no such path, then d(u,v) = 

  4. Tree: Characterization • An n-vertex graph G (with n  1) is a tree iff: • G is connected and has no cycles • G is connected and has n1 edges • G has n1 edges and no cycles • For u,v  V(G), G has exactly one u,v-path

  5. Some results … • Every tree with at least two vertices has at least two leaves. • Deleting a leaf from a tree with n vertices produces a tree with n-1 vertices. • If T is a tree with k edges and G is a simple graph with (G)  k, then T is a sub-graph of G.

  6. Some results … • If T and T are two spanning trees of a connected graph G and e  E(T)  E(T), then there is an edge e  E(T)  E(T) such that T  e + e is a spanning tree of G.

  7. Diameter and Radius • The eccentricityof a vertex u, written (u), is the maximum of its distances to other vertices. • In a graph G, the diameter, diamG, and the radius, radG, are the maximum and minimum of the vertex eccentricities respectively. • The centerof G is the subgraph induced by the vertices of minimum eccentricity.

  8. Counting Trees • There are nn2trees with vertex set [n].

  9. Prüfer Code / Sequence Algorithm:Production of f(T) = {a1, …, an-2} Input:A tree T with vertex set S  . Iteration:At the ithstep, delete the least remaining leaf, and let ai be the neighbor of this leaf.

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