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Faster Isometric Embedding in Products of Complete Graphs

Summary: Faster Isometric Embedding in
Products of Complete Graphs 
Franz Aurenhammer x Michael Formann x Ramana M. Idury {
Alejandro A. Scha er { Frank Wagner x
An isometric embedding of a connected graph G into a cartesian product of
complete graphs is equivalent to a labeling of each vertex of G by a string of xed
length such that the distance in G between two vertices is equal to the Hamming
distance between their labels. We give a simple O(D(m;n) + n 2 )-time algorithm
for deciding if G admits such an embedding, and for labeling G if one exists, where
D(m;n) is the time needed to compute the all-pairs distance matrix of a graph with
m edges and n vertices. If the distance matrix is part of the input, our algorithm
runs in O(n 2 ) time. We also show that an n-vertex subgraph of (K a ) d , the cartesian
product of d equal-sized complete graphs, cannot have more than a 1
2 n log a n edges.
With this result our algorithm can be used to decide whether a graph G is an a-ary
Hamming graph in O(n 2 log n) time (for xed a).
1 Introduction
Let G be a connected graph with n vertices and let H be any graph. An injective mapping
f : V (G) ! V (H) is an isometric embedding if it preserves distances; that is, if for any


Source: Aurenhammer, Franz - Institute for Theoretical Computer Science, Technische Universitšt Graz


Collections: Computer Technologies and Information Sciences