On average sizes and enumeration of minimal edge covers

Abstract

An edge cover $M$ of a graph $G$ is a subset of edges such that every vertex of $G$ is an end-vertex of some edge in $M$. An edge cover is called a minimal edge cover if it does not properly contain another edge cover. Let $mec\left(G\right)$ be the number of minimal edge covers of $G$, and let ${mec}_{\mathrm{av}}\left(G\right)$ be the average size of a minimal edge cover of $G$. We consider the extremal values of $mec\left(G\right)$ and ${mec}_{\mathrm{av}}\left(G\right)$ when $G$ is restricted to various families of graphs. In particular, we determine the graphs $G$ which minimize $mec\left(G\right)$ among all graphs with fixed order, and among the family of 2-regular graphs with fixed order. We also determine the graphs which maximize $mec\left(G\right)$ within the families of trees, of unicyclic graphs with fixed order, and of 2-regular graphs with a fixed order. Finally, we provide a characterization of all extremal graphs for the maximum and minimum values of ${mec}_{\mathrm{av}}\left(G\right)$ among all graphs $G$ with fixed order.