A graph G = (L, S) given as such, a sub-graph of G is the graph G1 = (L1, S1) formed by a subset S1 of vertices of G and a sub-set L1 of edges of L.
Example
Consider the graph G below, defined by G = (L, S), where L = {a, b, c, d, e, g, f, g, h, m, n} and S = {A, B, C, D, E, F}.
The graph G1 = (L1, S1), such as L1 = {a, b, g, n} and S1 = {A, B, C, E}, is a sub-graph of G.
