arXiv:2609.01474v3 Announce Type: replace Abstract: Codes over graphs store symbols on edges of a complete undirected graph on $n$ vertices with a self-loop at each vertex; erasing a node removes all its incident edges. Extending Yohananov, Efron, and Yaakobi's construction, we obtain binary codes correcting any three node erasures at infinitely many prime lengths $n$, unconditionally, with redundancy $3n-2$, one bit above the graph Singleton bound. The earlier construction has the same redundancy but requires $2$ to be primitive modulo $n$, which is not known for infinitely many primes. We give a deterministic decoder for these triple-node codes using $O(n^2)$ binary operations and construct Singleton-optimal binary triple-node codes at $n=6,8,10,12$. When $2$ is primitive modulo an odd prime $n$, we extend their double- and triple-node codes to binary codes correcting $\rho$ node erasures with redundancy $\rho n-(\rho-1)$ for $2\leq\rho\leq(n+1)/2$, saving $\rho-1$ bits over the binary symmetric rank-metric construction when $\rho<(n+1)/2$. Over every finite field of size $q$, the node weight of a word is the least number of vertices whose erasure removes its support. We give exact formulas for ball volumes in this node metric and prove that, for fixed $q,t$, the radius-$t$ volume is asymptotic to $\binom nt q^{tn-\binom t2}$ as $n\to\infty$. We obtain existence and covering bounds; for fixed $q,\rho$, we show that linear $\rho$-node-erasure-correcting codes exist with redundancy at most $\rho\log_q n+O_{q,\rho}(1)$ above the graph Singleton bound. Finally, reformulating Yohananov and Yaakobi's criterion, we show that a linear code over any finite field is Singleton-optimal for $\rho$ node erasures if and only if its dual is Singleton-optimal for erasures of the coordinates inside any $n-\rho$ vertices.