On the Relation Between the Common Labelling and the Median Graph

TL;DR

This paper investigates the theoretical relationship between the common labelling problem and the median graph problem in structural pattern recognition, establishing bounds and equivalences.

Contribution

It provides the first formal proof of the relation between the common labelling and median graph problems, including bounds and conditions for equivalence.

Findings

The cost of common labelling upper-bounds the median graph cost.

The two problems are equivalent under certain conditions.

The analysis advances understanding of median graph computation.

Abstract

In structural pattern recognition, given a set of graphs, the computation of a Generalized Median Graph is a well known problem. Some methods approach the problem by assuming a relation between the Generalized Median Graph and the Common Labelling problem. However, this relation has still not been formally proved. In this paper, we analyse such relation between both problems. The main result proves that the cost of the common labelling upper-bounds the cost of the median with respect to the given set. In addition, we show that the two problems are equivalent in some cases.