On Equi-Cardinal, Restrictions of a Graph

Abstract

A graph G is an ordered pair (V, E) where V is a set of objects called vertices, and E is a set of unordered pairs of vertices (v, v') in which each such pair can occur at most once in E, and if (v, v') ϵ E then v ≠ v'. The order of G is the cardinality of the set V, and the degree δ(v) of an element v ϵ V is the number of elements of E which contain v. G is said to be regular of degree d if δ(v) = d for each v ϵ V. G is a complete graph if E contains every pair of elements of V. A graph H = (V', E') is a partial graph of G=(V, E) if V' ⊆ v and E' ⊆ E. H is a restriction of G if H is a partial graph of G in which V' = V. Let S = { e₁, …, e_l} be a subset of E such that e_j={v_j-1, v_j} for 1 ≤j ≤l.