A hierarchy of dismantlings in Graphs

Given a finite undirected graph $X$, a vertex is $0$-dismantlable if its open neighbourhood is a cone and $X$ is $0$-dismantlable if it is reducible to a single vertex by successive deletions of $0$-dismantlable vertices. By an iterative process, a vertex is $(k+1)$-dismantlable if its open neighbourhood is $k$-dismantlable and a graph is $k$-dismantlable if it is reducible to a single vertex by successive deletions of $k$-dismantlable vertices. We introduce a graph family, the cubion graphs, in order to prove that $k$-dismantlabilities give a strict hierarchy in the class of graphs whose clique complex is non-evasive. We point out how these higher dismantlabilities are related to the derivability of graphs defined by Mazurkievicz and we get a new characterization of the class of closed graphs he defined. By generalising the notion of vertex transitivity, we consider the issue of higher dismantlabilities in link with the evasiveness conjecture.