1 / 9

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.

kaida
Télécharger la présentation

Graph Theory: Trees

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related