Qid restricted blockade with empty blocks

For a finite simple graph $G$, a QID block sequence $(B_1,\dots ,B_k)$ has integer length $k\ge 1$, pairwise disjoint subsets $B_i\subseteq V(G)$, and width ${\min }_i\mathrm{\mid }{B}_i\mathrm{\mid }$. Empty blocks are permitted, including repeated empty sets.

For $x\ge 0$, it is $x$-restricted if for each $i$ one may choose $K_i\in \{G,\bar{G}\}$ such that every $v\in {\bigcup }_{j>i}{B}_j$ satisfies $\mathrm{\mid }{N}_{K_i}(v)\cap B_i\mathrm{\mid }\le x\mathrm{\mid }{B}_i\mathrm{\mid }$. The choice of $K_i$ is fixed for that index, for all later vertices. It is uniformly $x$-sparse in $K$ if every index uses the same $K$.

This extends Sparsity of one vertex set to another, and weak sparsity of a pair by the degree inequality itself. If $B_i$ is empty both sides are zero; if the later union is empty the condition has no instances. At positive width these are the blockades of Blockades, their length, their width, and their support. For positive width and $0\le x\le 1$, uniform $x$-sparsity in a fixed $K\in \{G,\bar{G}\}$ is precisely the $x$-sparse blockade notion of Complete, anticomplete, pure, weakly sparse, and $x$-sparse blockades applied to the ambient graph $K$: the degree condition on a union holds exactly when it holds on every constituent later block. For $x>1$ the QID degree inequality remains meaningful, but it lies outside that published definition's parameter range.

Proof/convention locator: Bucic, Nguyen, Scott and Seymour, Induced subgraph density I, Section 2, blockade conventions.