Efficient Reassembling of Graphs, Part 1: The Linear Case
Abstract
The reassembling of a simple connected graph G = (V,E) is an abstraction of a problem arising in earlier studies of network analysis. Its simplest formulation is in two steps: (1) We cut every edge of G into two halves, thus obtaining a collection of n=V onevertex components. (2) We splice the two halves of every edge together, not of all the edges at once, but in some ordering \Theta of the edges that minimizes two measures that depend on the edgeboundary degrees of assembled components. The edgeboundary degree of a component A (subset of V) is the number of edges in G with one endpoint in A and one endpoint in VA. We call the maximum edgeboundary degree encountered during the reassembling process the alphameasure of the reassembling, and the sum of all edgeboundary degrees is its betameasure. The alphaoptimization (resp. betaoptimization) of the reassembling of G is to determine an order \Theta for splicing the edges that minimizes its alphameasure (resp. betameasure). There are different forms of reassembling. We consider only cases satisfying the condition that if the an edge between disjoint components A and B is spliced, then all the edges between A and B are spliced at the same time. In this report, we examine the particular case of linear reassembling, which requires that the next edge to be spliced must be adjacent to an already spliced edge. We delay other forms of reassembling to followup reports. We prove that alphaoptimization of linear reassembling and minimumcutwidth linear arrangment (CutWidth) are polynomially reducible to each other, and that betaoptimization of linear reassembling and minimumcost linear arrangement (MinArr) are polynomially reducible to each other.
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 arXiv:
 arXiv:1509.06434
 Bibcode:
 2015arXiv150906434K
 Keywords:

 Computer Science  Discrete Mathematics
 EPrint:
 doi:10.1007/s108780160024x