Dominating Vertex Extensions and Higher Homological Shift Ideals of Edge Ideals

Let G be a finite simple graph, let I(G) denote its edge ideal, and let m be the homogeneous maximal ideal of the corresponding polynomial ring. We introduce the notion of shadow-compatible regular powers: for every p,q≥0 with p+q>0, the mixed ideal mpI(G)q admits a linear-quotient order with a regular decomposition function, and these orders are compatible under the adjacent inclusions mpI(G)q⊆mp+1I(G)q−1,q≥1. For graphs with this property, every such mixed ideal has a (p+2q)-linear resolution. Moreover, for every k≥1 and i≥0, the homological shift ideal HSi(I(G)k), whenever nonzero, is generated in degree 2k+i. For i≥2, this generating-degree statement does not imply a linear resolution in general; by contrast, HS1(I(G)k) has linear quotients and a (2k+1)-linear resolution. We prove that shadow-compatible regular powers are preserved under adjoining a dominating vertex and hence under joins with complete graphs. Finally, we show that every Ferrers edge ideal has this property. Consequently, for every Ferrers graph Fλ and every r≥1, the edge ideal of Fλ*Kr provides a family with linear mixed powers and homological shift ideals generated in the expected degrees.