Let \(R=K[x_{1},\dots,x_{n}]\) be a polynomial ring over the field \(K\) with the usual grading. Let \(f_{i}=a_{i}x^{d_{i}}+b_{i}m_{i} \ (1\le i\le n)\) be homogeneous polynomials in \(R\), where \(a_{i}\ne0\) and \(b_{i}\) are scalars in \(K\), and \(m_{i}\) is a monomial of degree \(d_{i}\) for each \(i\). Let \(B=\{f_{1},\dots, f_{n}\}\), and let \(I\) be the ideal of \(R\) generated by \(B\). The authors associate with \(B\) a directed graph \(G_{B,d}\) (the reduction graph). Recall that the ideal \(I\) is a complete intersection if the sequence \(f_{1},\dots, f_{n}\) is regular (equivalently the only zero of the ideal \(I\) in the algebraic closure of \(K\) is the origin). Assuming that \(I\) is a complete intersection, and using the graph \(G_{B,d}\), the authors prove that the monomials not divisible by \(x_{i}^{d_{i}}\) for \(1\le i\le n\) form a vector space basis for \(R/I\), that the Macaulay dual generator is algorithmically determined by the graph \(G_{B,d}\), and so are the factors of the resultant of \(B\). The authors discuss presentations of complete intersections, Lefschetz properties of some Gorenstein algebras with squarefree monomial basis, etc.