Isoperimetric Problem and Meta-fibonacci Sequences

• Authors: B. V. Bharadwaj;L. S. Chandran;Anita Das
• Venue: COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
• Year: 2008

Let G= (V,E) be a simple, finite, undirected graph. For S茂戮驴 V, let $\delta(S,G) = \{ (u,v) \in E : u \in S \mbox { and } v \in V-S \}$ and $\phi(S,G) = \{ v \in V -S: \exists u \in S$, such that (u,v) 茂戮驴 E} be the edge and vertex boundary of S, respectively. Given an integer i, 1 ≤ i≤ 茂戮驴 V茂戮驴, the edge and vertex isoperimetric value at iis defined as be(i,G) = min S茂戮驴 V; |S| = i|茂戮驴(S,G)| and bv(i,G) = min S茂戮驴 V; |S| = i|茂戮驴(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 Sshould 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 Gis a tree. Therefore we use the notation bc(i, T) to denote the connected edge (vertex) isoperimetric value of Tat 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 T2be an infinite complete binary tree. In this paper we related the connected isoperimetric problem on T2with the Tanny sequences which is defined by the recurrence relation a(i) = a(i茂戮驴 1 茂戮驴 a(i茂戮驴 1)) + a(i茂戮驴 2 茂戮驴 a(i茂戮驴 2)), a(0) = a(1) = a(2) = 1. In particular, we show that bc(i, T2) = i+ 2 茂戮驴 2a(i), for each i茂戮驴 1.We also propose efficient polynomial time algorithms to find vertex isoperimetric values at iof bounded pathwidth and bounded treewidth graphs.