Isoperimetric Problem and Meta-fibonacci Sequences

• Published in: Computing and Combinatorics Vol. 5092; pp. 22 - 30
• Main Authors: Bharadwaj, B. V. S., Chandran, L. S., Das, Anita
• Format: Book Chapter
• Language: English
• Published: Germany Springer Berlin / Heidelberg 2008; Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• ISBN: 3540697322; 9783540697329
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-540-69733-6_3

Let G = (V,E) be a simple, finite, undirected graph. For S ⊆ V, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\delta(S,G) = \{ (u,v) \in E : u \in S \mbox { and } v \in V-S \}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi(S,G) = \{ v \in V -S: \exists u \in S$\end{document}, such that (u,v) ∈ E} be the edge and vertex boundary of S, respectively. Given an integer i, 1 ≤ i ≤ ∣ V ∣, the edge and vertex isoperimetric value at i is defined as be(i,G) =  min S ⊆ V; |S| = i |δ(S,G)| and bv(i,G) =  min S ⊆ V; |S| = i |φ(S,G)|, respectively. The edge (vertex) isoperimetric problem is to determine the value of be(i, G) (bv(i, G)) for each i, 1 ≤ i ≤ |V|. If we have the further restriction that the set S should induce a connected subgraph of G, then the corresponding variation of the isoperimetric problem is known as the connected isoperimetric problem. The connected edge (vertex) isoperimetric values are defined in a corresponding way. It turns out that the connected edge isoperimetric and the connected vertex isoperimetric values are equal at each i, 1 ≤ i ≤ |V|, if G is a tree. Therefore we use the notation bc(i, T) to denote the connected edge (vertex) isoperimetric value of T at i. Hofstadter had introduced the interesting concept of meta-fibonacci sequences in his famous book “Gödel, Escher, Bach. An Eternal Golden Braid”. The sequence he introduced is known as the Hofstadter sequences and most of the problems he raised regarding this sequence is still open. Since then mathematicians studied many other closely related meta-fibonacci sequences such as Tanny sequences, Conway sequences, Conolly sequences etc. Let T2 be an infinite complete binary tree. In this paper we related the connected isoperimetric problem on T2 with the Tanny sequences which is defined by the recurrence relation a(i) = a(i − 1 − a(i − 1)) + a(i − 2 − a(i − 2)), a(0) = a(1) = a(2) = 1. In particular, we show that bc(i, T2) = i + 2 − 2a(i), for each i ≥ 1. We also propose efficient polynomial time algorithms to find vertex isoperimetric values at i of bounded pathwidth and bounded treewidth graphs.