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The Isoperimetric Problem On Trees And Bounded Tree Width Graphs

Electronic Theses of Indian Institute of Science

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Title The Isoperimetric Problem On Trees And Bounded Tree Width Graphs
 
Creator Bharadwaj, Subramanya B V
 
Subject Computer Graphics - Algorithms
Computer Graphics - Mathematical Models
Isoperimetric Inequalities
Meta-Fibonacci Sequences
Graph Theory
Trees (Graph Theory)
Treewidth Graphs
Weighted Graphs
Infinite Binary Tree
Isoperimetric Problem
Computer Science
 
Description In this thesis we study the isoperimetric problem on trees and graphs with bounded treewidth. Let G = (V,E) be a finite, simple and undirected graph. For let δ(S,G)= {(u,v) ε E : u ε S and v ε V – S }be the edge boundary of S. Given an integer i, 1 ≤ i ≤ | V| , let the edge isoperimetric value of G at I be defined as be(i,G)= mins v;|s|= i | δ(S,G)|. For S V, let φ(S,G) = {u ε V – S : ,such that be the vertex boundary of S. Given an integer i, 1 ≤ i ≤ | V| , let the vertex isoperimetric value of G at I be defined as bv(i,G)=
The edge isoperimetric peak of G is defined as be(G) =. Similarly
the vertex isoperimetric peak of G is defined as bv(G)= .The problem
of determining a lower bound for the vertex isoperimetric peak in complete k-ary trees of depth d,Tdkwas recently considered in[32]. In the first part of this thesis we provide lower bounds for the edge and vertex isoperimetric peaks in complete k-ary trees which improve those in[32]. Our results are then generalized to arbitrary (rooted)trees.
Let i be an integer where . For each i define the connected edge
isoperimetric value and the connected vertex isoperimetric value of
G at i as follows: is connected and is connected A meta-Fibonacci sequence is given by the reccurence a(n)= a(x1(n)+ a1′(n-1))+ a(x2(n)+ a2′(n -2)), where xi: Z+ → Z+ , i =1,2, is a linear function of n and ai′(j)= a(j) or ai′(j)= -a(j), for i=1,2. Sequences belonging to this class have been well studied but in general their properties remain intriguing. In the second part of the thesis we show an interesting connection between the problem of determining and certain meta-Fibonacci sequences.
In the third part of the thesis we study the problem of determining be and bv algorithmically for certain special classes of graphs.
Definition 0.1. A tree decomposition of a graph G = (V,E) is a pair where I is an index set, is a collection of subsets of V and T is a tree whose node set is I such that the following conditions are satisfied:
(For mathematical equations pl see the pdf file)
 
Contributor Chandran, Sunil
 
Date 2010-08-26T08:08:31Z
2010-08-26T08:08:31Z
2010-08-26
2008-09
 
Type Thesis
 
Identifier http://etd.iisc.ernet.in/handle/2005/844
 
Language en_US
 
Relation G22614