Neighborhood fixed pendant vertices

Abstract:

If $x$ is pendant in $G$, then ${x^ \ast }$ denotes the unique vertex of $G$ adjacent to $x$. Such an $x$ is said to be neighborhood-fixed whenever ${x^ \ast }$ is fixed by $A(G - x)$. It is shown that if $G$ is not a tree and has a pendant vertex, but no *-fixed pendant vertex, then there is a subgraph ${G^\# }$ of $G$ such that for some $y \in V({G^\# })$, $O(A{({G^\# })_y}) \geqslant t!$ where $t$ is the maximum number of edges in a tree rooted in ${G^\# }$.