Minimal and Maximal $e=1$ Functions

Abstract

An $e=1$ function is a function $f:V(G)\to [0,1]$ such that every non-isolated vertex $u$ is adjacent to some vertex $v$ such that $f(u)+f(v)=1$, and every isolated vertex $w$ has $f(w)=1$. A theory of $e=1$ functions is developed focussing on minimal and maximal $e=1$ functions. Relationships are traced between $e=1$ parameters and some well-known domination parameters, which lead to results about classical and fractional domination parameters.