Efficient Reassembling of ThreeRegular Planar Graphs
Abstract
A reassembling of a simple graph G = (V,E) is an abstraction of a problem arising in earlier studies of network analysis. There are several equivalent definitions of graph reassembling; in this report we use a definition which makes it closest to the notion of graph carving. A reassembling is a rooted binary tree whose nodes are subsets of V and whose leaf nodes are singleton sets, with each of the latter containing a distinct vertex of G. The parent of two nodes in the reassembling is the union of the two children's vertex sets. The root node of the reassembling is the full set V. The edgeboundary degree of a node in the reassembling is the number of edges in G that connect vertices in the node's set to vertices not in the node's set. A reassembling's alphameasure is the largest edgeboundary degree of any node in the reassembling. A reassembling of G is alphaoptimal if its alphameasure is the minimum among all alphameasures of G's reassemblings. The problem of finding an alphaoptimal reassembling of a simple graph in general was already shown to be NPhard. In this report we present an algorithm which, given a 3regular plane graph G = (V,E) as input, returns a reassembling of G with an alphameasure independent of n (number of vertices in G) and upperbounded by 2k, where k is the edgeouterplanarity of G. (Edgeouterplanarity is distinct but closely related to the usual notion of outerplanarity; as with outerplanarity, for a fixed edgeouterplanarity k, the number n of vertices can be arbitrarily large.) Our algorithm runs in time linear in n. Moreover, we construct a class of $3$regular plane graphs for which this alphameasure is optimal, by proving that 2k is the lower bound on the alphameasure of any reassembling of a graph in that class.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 arXiv:
 arXiv:1807.03479
 Bibcode:
 2018arXiv180703479K
 Keywords:

 Computer Science  Data Structures and Algorithms;
 68Q99;
 68W40;
 05C10;
 C.m;
 F.2;
 F.m
 EPrint:
 49 pages, 25 figures, 15 references