Elimination-tree column patterns and reach

Let $A\in {\mathbb{C}}^{n\times n}$ satisfy $A=A^{\ast }$, use the Hermitian structural graph of Undirected structural graph of a Hermitian matrix, and fix an elimination ordering $\pi =(v_0,\dots ,v_{n-1})$ of its vertices. For this chosen ordering, construct the filled graphs, parent map, and elimination forest by Structural graphs and ordered symmetric elimination. All column sets, counts, ancestors, descendants, and reaches below are relative to $\pi$. For each $k$, let $N_k(v_k)$ be the later-neighbor set at elimination of $v_k$, and define its structural column set ${\mathcal{C}}_k=\{v_k\}\cup N_k(v_k)\mathrm{.}$ Its off-diagonal size is the column count ${\mathrm{\ell }}_k=\mathrm{\mid }{N}_k(v_k)\mathrm{\mid }$. An ancestor of $v_k$ is obtained by following one or more supplied parent edges; a descendant is the inverse relation. The reach of $v_k$ is the set of vertices visited by following its parent path, including $v_k$.