The paper introduces the one-way quantum computer (QC$_C$), a scheme for quantum computation that relies solely on one-qubit measurements on a specific class of entangled states, known as cluster states. The authors prove the universality of QC$_C$, showing that any unitary quantum logic network can be efficiently simulated on it. They also relate quantum algorithms to mathematical graphs and provide examples of practical circuits, such as those for quantum Fourier transformation and the quantum adder. The paper discusses the scaling of required resources and explores computation with finite-size clusters. The authors explain how to remove redundant cluster qubits and demonstrate the concatenation of gate simulations to form complete circuits. They show that the measurement-based scheme of quantum computation can be decomposed into basic building blocks, which are essential for understanding the functioning of the QC$_C$.The paper introduces the one-way quantum computer (QC$_C$), a scheme for quantum computation that relies solely on one-qubit measurements on a specific class of entangled states, known as cluster states. The authors prove the universality of QC$_C$, showing that any unitary quantum logic network can be efficiently simulated on it. They also relate quantum algorithms to mathematical graphs and provide examples of practical circuits, such as those for quantum Fourier transformation and the quantum adder. The paper discusses the scaling of required resources and explores computation with finite-size clusters. The authors explain how to remove redundant cluster qubits and demonstrate the concatenation of gate simulations to form complete circuits. They show that the measurement-based scheme of quantum computation can be decomposed into basic building blocks, which are essential for understanding the functioning of the QC$_C$.