Algorithms for solving systems of equations over various classes of finite graphs

The aim of the paper is to study and to solve finite systems of equations over finite undirected graphs. Equations over graphs are atomic formulas of the language ${\rm L}$ consisting of the set of constants (graph vertices), the binary vertex adjacency predicate and the equality predicate. It is proved that the problem of checking compatibility of a system of equations $S$ with $k$ variables over an arbitrary simple $n$-vertex graph $\Gamma$ is $\mathcal{NP}$-complete. The computational complexity of the procedure for checking compatibility of a system of equations $S$ over a simple graph $\Gamma$ and the procedure for finding a general solution of this system is calculated. The computational complexity of the algorithm for solving a system of equations $S$ with $k$ variables over an arbitrary simple $n$-vertex graph $\Gamma$ involving these procedures is $O(k^2n^{k/2+1}(k+n)^2)$ for ${n \geq 3}$. Polynomially solvable cases are distinguished: systems of equations over trees, forests, bipartite and complete bipartite graphs. Polynomial time algorithms for solving these systems with running time $O(k^2n(k+n)^2)$ are proposed.