Dominating Hadwiger's Conjecture holds for all $2K_2$-free graphs

A dominating $K_t$ minor in a graph $G$ is a sequence $(T_1,\dots ,T_t)$ of pairwise disjoint non-empty connected subgraphs of $G$, such that for $1\le i<j\le t$, every vertex in $T_j$ has a neighbor in $T_i$. Replacing every vertex in $T_j$'' by some vertex in $T_j$’’ retrieves the standard definition of a $K_t$ minor. The strengthened notion was introduced by Illingworth and Wood [arXiv:2405.14299], who asked whether every graph with chromatic number $t$ contains a dominating $K_t$ minor. This is a substantial strengthening of the celebrated Hadwiger’s Conjecture, which asserts that every graph with chromatic number $t$ contains a $K_t$ minor. At the New Perspectives in Colouring and Structure'' workshop held at the Banff International Research Station from September 29 - October 4, 2024, Norin referred to this question as the Dominating Hadwiger’s Conjecture’’ and believes it is likely false. In this paper we prove that the Dominating Hadwiger’s Conjecture holds for all $2K_2$-free graphs. A key component of our proof is the clever use of the existence of an induced banner, obtained by adding a vertex adjacent to exactly one vertex on a cycle of length four.