Implications among linkage properties in graphs

Given a graph H with vertices w1, . . . , wm, a graph G with at least m vertices is H-linked if for every choice of vertices v1, . . . , vm in G, there is a subdivision of H in G such that vi is the branch vertex representing wi (for all i). This concept generalizes the notions of k-linked, k-connected, and k-ordered graphs. For graphs H1 and H2 with the same order that are not contained in stars, the property of being H1-linked implies that of being H2-linked if and only if H2 ⊆ H1. The implication also holds when H1 is obtained from H2 by replacing an edge xy with an edge from y to a new vertex x′. Other instances of non-implication are obtained, using a lemma that the number of vertices appearing in minimum vertex covers of a graph G is at most the vertex cover number plus the size of a maximum matching.