The authors deal with a boundary value problem for the second-order differential equation \(y''= f(x,y,y')\) defined on finite geometric connected graph \(\Gamma\subset\mathbb{R}^n\), whose edges are finite open intervals, under the boundary conditions \(y(a)= P(a)\), \(a\in\partial\Gamma\). Here, the function \(f: \Gamma\times\mathbb{R}\times \mathbb{R}\to\mathbb{R}\) is jointly continuous, \(\partial\Gamma\) is the set of boundary (deadlock) vertices, and \(\{P(a)\}_{a\in\partial\Gamma}\) is a given set of numbers. On the basis of a priori estimates on solutions to the above equation, the authors prove an existence theorem for this boundary value problem. The uniqueness of its solution is guaranteed if \(f(x,y,y')\) is strictly increasing with respect to \(y\). Moreover, sufficient conditions are derived for the desired solution to be continuous with respect to values \(P(a)\) on the boundary \(\partial\Gamma\).