Pattern Avoidance parameterized by vertex integrity and pattern edge count

Determine the complexity of Pattern Avoidance parameterized by the sum of the number of pattern edges m_P and the vertex integrity of the input graph G, beginning with the special case of Bandwidth parameterized by vertex integrity.

Background

The paper provides a fixed-parameter algorithm for Pattern Avoidance parameterized by neighborhood diversity plus m_P, and separately by vertex integrity plus n_P. It does not establish an analogous result for vertex integrity plus the number of pattern edges.

This gap is already present in the special case of Bandwidth, which is represented by patterns with one forced edge. The paper explicitly connects the unresolved general parameterization to the open parameterized complexity of Bandwidth with respect to vertex integrity.

References

What is the complexity of Pattern Avoidance\ w.r.t.\ $m_P$ plus the vertex integrity of $G$? This once again is tied to settling a special case first: the parameterized complexity of bandwidth w.r.t.\ the vertex integrity is explicitly noted as open in the recent work targeting that problem.

— The (Parameterized) Complexity of Ordering a Graph While Avoiding a Forbidden Pattern  (2608.30667 - Depian et al., 31 Aug 2026) in Section 7 (Concluding Remarks)