Constructing Labeling Schemes Through Universal Matrices

Abstract

Let f be a function on pairs of vertices. An f-labeling scheme for a family of graphs ${\mathcal F}$ labels the vertices of all graphs in ${\mathcal F}$ such that for every graph $G\in{\mathcal F}$ and every two vertices u,v∈G, f(u,v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in ${\mathcal F}$. This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes. Let ${\mathcal F}(n)$ be a family of connected graphs of size at most n and let ${\mathcal C}({\mathcal F},n)$ denote the collection of graphs of size at most n, such that each graph in ${\mathcal C}({\mathcal F},n)$ is composed of a disjoint union of some graphs in ${\mathcal F}(n)$. We first investigate methods for translating f-labeling schemes for ${\mathcal F}(n)$ to f-labeling schemes for ${\mathcal C}({\mathcal F},n)$. In particular, we show that in many cases, given an f-labeling scheme of size g(n) for a graph family ${\mathcal F}(n)$, one can construct an f-labeling scheme of size g(n)+loglogn+O(1) for ${\mathcal C}({\mathcal F},n)$. We also show that in several cases, the above mentioned extra additive term of loglogn+O(1) is necessary. In addition, we show that the family of n-node graphs which are unions of disjoint circles enjoys an adjacency labeling scheme of size logn+O(1). This illustrates a non-trivial example showing that the above mentioned extra additive term is sometimes not necessary. We then turn to investigate distance labeling schemes on the class of circles of at most n vertices and show an upper bound of 1.5logn+O(1) and a lower bound of 4/3logn–O(1) for the size of any such labeling scheme.