 Boundary-reaching stencil connectivity

A finite elliptic stencil has boundary-reaching connectivity when every unknown node has a directed path of strictly positive neighbor coefficients to a prescribed boundary node. For $L{\psi }_p={\sum }_{r\ne p}{c}_{p,r}{\psi }_r-c_{p,p}{\psi }_p$, with $c_{p,r}\ge 0$ and $c_{p,p}\ge {\sum }_{r\ne p}{c}_{p,r}$, a nonnegative interior maximum of a subsolution $L\psi \ge 0$ propagates along every positive-weight edge. The path condition carries it to the boundary, proving the discrete maximum principle. Connectedness of the underlying geometric domain alone does not imply this graph property.