Learning graphs and 1-uniform dcsl graphs

Abstract

A distance compatible set labeling (dcsl) of a connected graph [Formula: see text] is an injective set assignment [Formula: see text] [Formula: see text] being a non-empty ground set, such that the corresponding induced function [Formula: see text] given by [Formula: see text] satisfies [Formula: see text] for every pair of distinct vertices [Formula: see text] where [Formula: see text] denotes the path distance between [Formula: see text] and [Formula: see text] and [Formula: see text] is a constant, not necessarily an integer, depending on the pair of vertices [Formula: see text] chosen. A dcsl [Formula: see text] of [Formula: see text] is [Formula: see text]-uniform if all the constants of proportionality with respect to [Formula: see text] are equal to [Formula: see text] and if [Formula: see text] admits such a dcsl then [Formula: see text] is called a [Formula: see text]-uniform dcsl graph. The family [Formula: see text] is well-graded family, if there is a tight path between any two of its distinct sets. A learning graph is an [Formula: see text]-induced graph of a learning space. In this paper, we initiate a study on subgraphs of 1-uniform graphs which lead to the study of Knowledge Structures, Learning Spaces and Union-closed conjecture using graph theory techniques.