Isoperimetric Problem and Meta-fibonacci Sequences
Let G=(V,E) be a simple, finite, undirected graph. For S(C)V, let 6(S,G)=(u,v) ∈E:u € S and v ∈ V-S and ψ(S, G)=v ∈V—S: 3w ∈S, 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)=mingcv; |s|=I |δ (S,G)| and bv(I, G)=minscv; |s|=I |ψ(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≤z≤ |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 Godel, 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-l-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.
B.V.S. Bharadwaj L.S. Chandran Anita Das
Department of Computer Science and Automation, Indian Institute of Science, Bangalore- 560012, India Department of Computer Science and Automation, Indian Institute of Science, Bangalore-560012, India
国际会议
The 4th Annual International Computing and Combinatorics Conference,COCOON 2008(第14届国际计算和组合会议)
大连
英文
22-30
2008-06-01（万方平台首次上网日期，不代表论文的发表时间）