Sharp Edge-Edit Bounds at Every Level for Leaky Positive Semidefinite Forcing

Abstract

This work disproves the 1-leaky positive semidefinite edge-deletion conjecture and replaces it with a sharp theorem. For every leak level $\ell$ and edge $e$, one has $|Z^+_{(\ell)}(G)-Z^+_{(\ell)}(G-e)|\le 2$. More generally, if two graphs differ only on edges with both endpoints in $S$, their parameters differ by at most $|S|$. An endpoint-sensitive refinement recovers an increase of at most one whenever some minimum set for $G-e$ contains an endpoint of $e$. Both signs are sharp for every positive leak level. Joining two copies of $K_{\ell+1}$ by a bridge gives $Z^+_{(\ell)}(G)=2\ell$ and $Z^+_{(\ell)}(G-e)=2\ell+2$. For every $\ell\ge 2$, a connected clique-leaf pair of order $2\ell+3$ gives the opposite difference. The remaining positive-difference one-leak case is attained by connected graphs on nine vertices with $Z^+_{(1)}(H)=4$ and $Z^+_{(1)}(G)=6$. Explicit forcing sequences, fort certificates, and an exact verifier check the finite extremal example and stress-test the general results.