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Let e = uv ∈ E(G), N (u) − N (v) = {u 1 , u 2 , . . . , u p } and N (v) − N (u) = {v 1 , v 2 , . . . , v l }. Also let e i = uu i for i = 1, 2, . . . , p and f j = vv j for j = 1, 2, . . . , l. N Γ(G) (e) = {e 1 , e 2 , . . . , e p , f 1 , f 2 , . . . , f l }. hN (e)i c is clique irreducible if and only if every maximal independent set I in hN (e)i has a pair of vertices of its own. e i is not adjacent to e j if and only if u i is adjacent to u j . Similarly, f i is not adjacent to f j if and only if v i is adjacent to v j . So, I = {e i1
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