A local–global principle for vertex-isoperimetric problems

We consider the vertex-isoperimetric problem (VIP) for cartesian powers of a graph G. A total order ≼ on the vertex set of G is called isoperimetric if the boundary of sets of a given size k is minimum for any initial segment of ≼, and the ball around any initial segment is again an initial segment...

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Bibliographic Details
Published inDiscrete mathematics Vol. 257; no. 2; pp. 285 - 309
Main Authors Bezrukov, Sergei L., Serra, Oriol
Format Journal Article
LanguageEnglish
Published Elsevier B.V 28.11.2002
Subjects
Cartesian product
Macaulay posets
Nested solutions
Shadows
Vertex-isoperimetric problem
Vertex-isoperimetric problem
Shadows
Nested solutions
Macaulay posets
Cartesian product
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Summary:We consider the vertex-isoperimetric problem (VIP) for cartesian powers of a graph G. A total order ≼ on the vertex set of G is called isoperimetric if the boundary of sets of a given size k is minimum for any initial segment of ≼, and the ball around any initial segment is again an initial segment of ≼. We prove a local–global principle with respect to the so-called simplicial order on G n (see Section 2 for the definition). Namely, we show that the simplicial order ≼ n is isoperimetric for each n⩾1 iff it is so for n=1,2. Some structural properties of graphs that admit simplicial isoperimetric orderings are presented. We also discuss new relations between the VIP and Macaulay posets.
ISSN:0012-365X
1872-681X
DOI:10.1016/S0012-365X(02)00431-4